# Quantum Coherence Spaces Revisited: A von Neumann (Co)Algebraic Approach

Thea Li, Vladimir Zamdzhiev

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Thea Li, Vladimir Zamdzhiev. Quantum Coherence Spaces Revisited: A von Neumann (Co)Algebraic Approach. FOSSACS 2026 - International Conference on Foundations of Software Science and Computational Structures, Apr 2026, Turin, Italy. hal-05456935

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# Quantum Coherence Spaces Revisited: A von Neumann (Co)Algebraic Approach

Thea Li and Vladimir Zamdzhiev

Inria, LMF

**Abstract.** We describe a categorical model of MALL (Multiplicative Additive Linear Logic) inspired by the Heisenberg-Schrödinger duality of finite-dimensional quantum theory. Proofs of formulas with positive logical polarity correspond to CPTP (completely positive trace-preserving) maps in our model, i.e. the quantum operations in the Schrödinger picture, whereas proofs of formulas with negative logical polarity correspond to CPU (completely positive unital) maps, i.e. the quantum operations in the Heisenberg picture. The mathematical development is based on noncommutative geometry and finite-dimensional von Neumann (co)algebras, which can be defined as special kinds of (co)monoid objects internal to the category of finite-dimensional operator spaces.

**Keywords:** Quantum Theory, Linear Logic, Categorical Semantics

## 1 Introduction and Related Work

Linear Logic [19] was discovered by Girard while studying the mathematical model of *coherence spaces* [18,19]. Coherence spaces “played an essential role in the discovery of linear logic” [13, p. 967] and clearly they have had a major impact on the study of (semantics of) Linear Logic (LL). Later, *Probabilistic Coherence Spaces* (PCSs) [13] were described and they give a model of LL which is suitable for modeling discrete probability. The study of PCSs has resulted in impressive results, such as full abstraction for probabilistic PCF [16] and also full abstraction for a more complicated probabilistic programming language with recursive types [17].

A natural next step in the development of these models is to consider *quantum coherence spaces* (QCSs) with the intention of taking the semantic study of LL to quantum theory. Girard already proposed models of QCSs [20,21,22] and they were later studied in [3], but this approach has already been criticised by Selinger [32] as being inappropriate for quantum theory. One of the main problems with this proposal is that the notion of morphism between QCSs does not appear to correspond to a *completely positive map* (see also [3, p. 2]). This is problematic from the point of view of quantum theory, because *quantum operations* (also known as *quantum channels*) are modelled mathematically as certain kinds of completely positive maps.

Inspired by the success of (probabilistic) coherence spaces, in this work we set out to paint a new picture of what a quantum coherence space ought to be. We
consider finite-dimensional quantum theory and we describe a model of MALL based on the following natural ideas: (1) proofs $P \vdash R$ of formulas with positive logical polarities should admit an interpretation as CPTP (completely positive trace-preserving) maps, i.e. as the quantum operations in the *Schrödinger picture* of quantum theory; (2) proofs $M \vdash N$ of formulas with negative logical polarities should admit an interpretation as CPU (completely positive unital) maps, i.e. as the quantum operations in the *Heisenberg picture* of quantum theory; (3) the LL duality should coincide with the Heisenberg-Schrödinger duality of quantum theory on polarised formulas. These desiderata naturally lead to the theory of operator spaces [15,7,31] which has excellent categorical properties [29] and which can be used to construct a model of (full) LL whose duality is compatible with the Heisenberg-Schrödinger duality on the level of objects/formulas [30]. However, in [30], the morphisms/proofs do not correspond to the quantum operations in either picture. We address this problem (which is suggested for future work in [30]) for *finite-dimensional* (f.d.) quantum theory and MALL.

In §2, we provide relevant background on operator spaces. We begin our contributions in §3, where we show that **FdOS**, the category of f.d. operator spaces, is a model of MALL, and we prove other relevant categorical properties. The category has a very important monoidal structure (**FdOS**, $C, \otimes^h$) that is outside the scope of LL – it is given by the Haagerup tensor [15, §9], [7, pp. 30–34], [31, §5], which is one of the highlights of operator space theory. This monoidal structure allows us to give a new, but equivalent, definition of a f.d. *von Neumann algebra* (vN-algebra) in §4 as a certain kind of internal monoid object in (**FdOS**, $C, \otimes^h$). It is well-known that CPU maps may be defined on vN-algebras and this allows us to identify the subcategory of **FdOS** corresponding to negative logical polarity and the Heisenberg picture. The novel categorical definition of f.d. vN-algebras, and the self-duality of the Haagerup tensor, allow us to easily dualise this definition and we introduce f.d. *von Neumann coalgebras* (vN-coalgebras) in §5 as certain kinds of internal comonoid objects in (**FdOS**, $C, \otimes^h$). We show that CPTP maps are a natural notion of morphism between vN-coalgebras and so we identify the subcategory of **FdOS** corresponding to positive logical polarity and the Schrödinger picture. In §6 we prove that there is a categorical duality between the two subcategories, which gives a categorical formulation of the Heisenberg-Schrödinger duality. Finally, in §7, we apply the semantic technique of *gluing and orthogonality* of Hyland and Schalk [24] to **FdOS** which results in a model of MALL that satisfies the desiderata (1) – (3), see Figure 1.

Other relevant works related to quantum theory include [33] which describes a model of MALL and [35] which describes a model of LL. These papers use techniques based on gluing and orthogonality or are inspired by this. Our model exhibits two main differences: first, all of our polarised formulas have physical interpretations and a mathematical formulation based on mathematical physics; second, complete positivity is not assumed a priori, but it is *derived* only where necessary, i.e. for *mixed state* quantum computation. We show in §7 that our model has the advantage that it can also be used to reason about *pure state* quantum computation, where complete positivity obviously makes no sense.
One direction for future work is related to general recursion – our mor-
phisms should be seen as “total” rather than “partial” from the point of view
of domain theory [1]. To get more interesting domain-theoretic properties, one
should replace trace-preserving maps with *trace-non-increasing* (TNI) maps in
the Schrödinger picture, replace unital maps with *subunitary* (SU) ones in the
Heisenberg picture, and modify the polarity/orthogonality in §7 accordingly. In-
deed, in finite dimensions, the Loewner order gives both the CPTNI maps and
the CPSU maps the structure of a continuous domain [32,26], which is a very
nice kind of dcpo from a domain-theoretic point of view. Since our model can
talk about both pure state and mixed state quantum computation, another di-
rection for future work is to use our model to extract suitable design for type
systems and/or logics that can combine classical and quantum control.

## 2 Background on Operator Spaces and Banach Spaces

Operator spaces are widely seen as the “non-commutative” or “quantised” gen-
eralisation of Banach spaces. In this section we recall some basic background
material on these topics from several books on the matter [15], [7], [31].

2.1 Normed Spaces and Banach Spaces

A *normed space* is a pair (X, ||·||<sub>X</sub>) where X is a complex vector space and ||·||<sub>X</sub> a norm on it. A *Banach space* is a normed space that is complete with respect to the induced norm topology. In this paper we work with finite-dimensional normed spaces and every such space is necessarily a Banach space. We usually simply write X to refer to a Banach space and we just write ||·|| for its norm, which should be clear from context.

Recall that, if $X$ and $Y$ are two Banach spaces and $\varphi: X \to Y$ is a linear map between them, then its *operator norm* is defined by the assignment $||\varphi|| \triangleq \sup\{||\varphi(x)|| : x \in X, ||x|| \le 1\}$. Then $\varphi$ is continuous iff $\varphi$ is bounded iff $||\varphi|| < \infty$. Moreover, the space $B(X,Y) \triangleq \{\varphi: X \to Y \mid \varphi \text{ is bounded and linear}\}$ is a Banach space when equipped with the operator norm. If $X$ and $Y$ are finite-dimensional, then any linear map between them is bounded and the underlying vector space of $B(X,Y)$ is simply the vector space $L(X,Y)$ consisting of the linear maps between $X$ and $Y$. We often use the abbreviation $B(X) \triangleq B(X,X)$.

A linear map $f: X \to Y$ between two Banach spaces is called an *isometry* if $||f(x)|| = ||x||$ for each $x \in X$ and it is called a *contraction* whenever $||f|| \le 1$, equivalently when $||f(x)|| \le ||x||$ for every $x \in X$. We write **FdBan** for the category whose objects are finite-dimensional Banach spaces and whose morphisms are the linear contractions between them. In order to gain some intuition for the constructions that follow, it is useful to recall some of the categorical properties of **FdBan**. It has a zero object given by the zero-dimensional Banach space 0. The binary categorical product of X and Y is given by the Banach space $X \underset{0}{\oplus} Y \triangleq (X \oplus Y, ||\cdot||_{\ell^{\infty}})$, where $||(x,y)||_{\ell^{\infty}} \triangleq \max(||x||, ||y||)$ is simply the $\ell^{\infty}$-norm and $X \oplus Y$ is the usual direct sum of vector spaces. The binary categorical
coproduct of $X$ and $Y$ is given by the Banach space $X \oplus_1 Y \triangleq (X \oplus Y, \|\cdot\|_{\ell^1})$, where
$\|(x, y)\|_{\ell^1} \triangleq \|x\| + \|y\|$ is simply the $\ell^1$-norm. The category **FdBan** is symmetric
monoidal closed with respect to the *projective tensor product* $\otimes^\gamma$ (details omit-
ted) with internal hom given by $B(X, Y)$. In fact, **FdBan** is also $*$-autonomous
and therefore a model of MALL [6, §2.6]; the additive conjunction (disjunction)
is modelled by the categorical (co)products; the multiplicative conjunction (dis-
junction) corresponds to $\otimes^\gamma (\otimes^\delta)$, where $\otimes^\delta$ stands for the *injective tensor product*;
negation corresponds to the dual Banach space $X^* \triangleq B(X, \mathbb{C})$.

## 2.2 Operator Spaces and their Morphisms

An (abstract) operator space may be seen as a vector space $X$ equipped with additional norms on matrices with entries in $X$ that satisfy some conditions.

**Definition 1 (Matrix Space).** Given a complex vector space $X$, we write $M_{n,m}(X)$ for the vector space consisting of the $n \times m$ matrices with entries in $X$. The vector space structure is defined in the obvious way, i.e. componentwise. We also use the shorthand notation $M_n(X) \triangleq M_{n,m}(X)$ and $M_n \triangleq M_n(\mathbb{C})$.

**Definition 2 (Matrix Norm [15, p. 20]).** A matrix norm on a vector space $X$ is an assignment of a norm $\| \cdot \|_n$ on $M_n(X)$ for each $n \in \mathbb{N}$. Given such a norm, we write $M_n(X)$ for the normed space $(M_n(X), \| \cdot \|_n)$.

*Example 3.* We write $M_{n,m}$ for the normed space $(M_{n,m}, \|\cdot\|_{\text{op}})$, where $\|\cdot\|_{\text{op}}$ is the usual operator norm on $M_{n,m}$. The normed spaces $M_n \triangleq M_{n,n}$ for $n \in \mathbb{N}$ determine a matrix norm on $\mathbb{C}$ that we use throughout the paper.

**Definition 4 (Operator Space [31, pp. 34–35]).** An (abstract) operator space is a complex vector space $X$ equipped with a matrix norm that satisfies:

(B) The pair $(M_1(X), \|\cdot\|_1)$ is a Banach space;

$$
(M1) \quad \|x \oplus y\|_{m+n} = \max\{\|x\|_m, \|y\|_n\}
$$

$$
(M2) \quad \|\alpha x \beta\|_m \le \|\alpha\| \|x\|_m \|\beta\|
$$

for $n, m \in \mathbb{N}$, $x \in M_m(X)$, $y \in M_n(X)$, $\alpha, \beta \in M_m$. We write $x \oplus y \triangleq \begin{bmatrix} x & 0 \\ 0 & y \end{bmatrix} \in M_{m+n}(X)$ for the indicated matrix and $\alpha x \beta \in M_m$ for the matrix that results from the obvious generalisation of matrix multiplication.

Note that $\alpha, \beta$ in the above definition are scalar (i.e. complex) matrices, but $x$ need not be one. A matrix norm that satisfies the above criteria is called an *operator space structure* (o.s.s.) on $X$. The obvious isomorphism $M_1(X) \cong X$ determines a norm on $X$ for which we write $\|\cdot\|_X$ or simply $\|\cdot\|$ in the sequel. If $x \in M_n(X)$ we often simply write $\|x\|$ for $\|x\|_n$. Standard textbook results show that for each operator space $X$, both $M_n(X)$ and also $X$ are Banach spaces. In this paper we work with *finite-dimensional* operator spaces, i.e. operator spaces whose underlying vector space is finite-dimensional. The condition (B) is then automatically satisfied.
**Example 5.** The complex numbers $\mathbb{C}$ is an operator space when equipped with the matrix norm from Example 3. In fact, this is the unique o.s.s. on $\mathbb{C}$. More generally, each vector space $M_{n,m}$ can be equipped with an o.s.s by defining a norm on $M_k(M_{n,m})$ via the linear isomorphism $M_k(M_{n,m}) \cong M_{nk,mk}$ and the norm on the latter space (see Example 3). We write $M_{n,m}$ for this operator space. Note that this notation is compatible with Example 3, i.e. the norm $\|\cdot\|_{M_{n,m}}$ on $M_{n,m}$ is exactly the operator norm. The operator spaces $M_n \triangleq M_{n,n}$ are fundamental for the theory.

**Example 6.** For an operator space $X$, each of the spaces $M_n(X)$ has a canonical o.s.s that is determined by the linear isomorphism $M_m(M_n(X)) \cong M_{mn}(X)$ and the already given norm on the latter space. More generally, each matrix space $M_{m,n}(X)$ also has a canonical o.s.s that can be defined via the linear embedding $M_{m,n}(X) \hookrightarrow M_p(X)$, where $p = \max(m, n)$, and the o.s.s of the latter space [15, §2.1] and we write $M_{m,n}(X)$ for this operator space.

Next, we introduce important classes of morphisms between operator spaces.

**Definition 7 ([15, §2.2]).** Let $X$ and $Y$ be two vector spaces and $\varphi: X \to Y$ a linear map between them. The $n$-th amplification of $\varphi$ is the linear function $\varphi_n: M_n(X) \to M_n(Y) :: [x_{ij}] \mapsto [\varphi(x_{ij})]$, i.e. the componentwise application of $\varphi$ to the matrix $[x_{ij}]$.

**Definition 8 ([15, §2.2]).** Let $X$ and $Y$ be operator spaces and $\varphi: X \to Y$ a linear map. Consider the assignment $\|\varphi\|_{cb} \triangleq \sup \{\|\varphi_n\| : n \in \mathbb{N}\}$, where $\|\varphi_n\|$ is the operator norm of $\varphi_n: M_n(X) \to M_n(Y)$. We say that $\varphi$ is

- a completely bounded map (c.b) if $\|\varphi\|_{cb} < \infty$;

- a complete contraction (c.c) if $\|\varphi\|_{cb} \le 1$, i.e. each $\varphi_n$ is a contraction;

- a complete isometry (c.i) if each $\varphi_n$ is an isometry;

- a complete quotient map (c.q) if each $\varphi_n$ is a quotient map, i.e. $\varphi_n$ maps the open unit ball of $M_n(X)$ onto the open unit ball of $M_n(Y)$.

- a completely isometric isomorphism (c.i.i) if $\varphi$ is a surjective c.i.

Every linear map $\varphi: X \to Y$ between two finite-dimensional operator spaces is (completely) bounded [15, Corollary 2.2.4] and therefore $\varphi$ and all of its amplifications $\varphi_n$ are continuous (and bounded). The morphisms that are central to our development are the complete contractions. We write (**Fd**)OS for the category whose objects are the (finite-dimensional) operator spaces with morphisms the complete contractions between them. We write **FdVect** for the category of finite-dimensional complex vector spaces with linear maps as morphisms and we write $U: \text{FdOS} \to \text{FdBan}$ and $V: \text{FdOS} \to \text{FdVect}$ for the obvious forgetful functors.

The next proposition is useful because it simplifies some norm computations.

**Proposition 9 ([15, §2.2, §3.3]).** Let $\varphi: X \to Y$ be a c.b. map. If $X = \mathbb{C}$ or $Y = \mathbb{C}$, then $\|\varphi\|_{cb} = \|\varphi\|$, i.e. the cb-norm and operator norm coincide.
The vector space $CB(X, Y)$, consisting of the completely bounded maps between $X$ and $Y$, becomes a Banach space when equipped with the cb-norm $||\cdot||_{cb}$ and it becomes an operator space when equipped with the matrix norm given by the linear isomorphism $M_n(CB(X, Y)) \cong CB(X, M_n(Y))$ and the Banach space structure of the latter space [15, §3.2]. Note that in **FdOS**, the underlying vector space of $CB(X, Y)$ is exactly $L(X, Y)$, i.e. $V(CB(X, Y)) = L(X, Y)$.

If $X$ is an operator space, then its (operator space) *dual* is the operator space $X^* \triangleq CB(X, \mathbb{C})$. This is consistent with the notation for the Banach space dual, i.e. we have a Banach space equality $U(X^*) = (UX)^*$ [15, §3.2]. It is also consistent with the vector space dual for *finite-dimensional* spaces. That is, for $Y \in \text{FdVect}$, we define $Y^* \triangleq L(Y, \mathbb{C})$ and then for $X \in \text{FdOS}$ we also have $V(X^*) = ( VX )^*$. In the sequel, we show that $CB(X, Y)$ is the internal hom of **FdOS** (just like it is for **OS** [30]) and $X^*$ is the dual in the sense of *-autonomy.

**Example 10.** The vector space $M_n$ can be equipped with another important o.s.s. via the linear isomorphism $M_n \cong M_n^* :: a \mapsto (b \mapsto \text{tr}(ab))$ and the o.s.s. of the latter space. We write $T_n$ for the resulting operator space. Here, tr stands for the usual trace functional and $ab$ for matrix multiplication. The induced norm on $T_n$ coincides with the *trace norm* and we have a completely isometric isomorphism $T_n \cong M_n^*$. This is known as the *trace class* o.s.s. [7, 1.4.5]. It follows that $\text{tr} : T_n \to \mathbb{C}$ is a complete contraction, but $\text{tr} : M_n \to \mathbb{C}$ is not one for $n > 1$.

**Remark 11.** Complete isometries and complete quotient maps are dual notions. A c.b. map $\varphi: X \to Y$ is a complete isometry (complete quotient map) iff $\varphi^*: Y^* \to X^*$ is a complete quotient map (complete isometry) [7, 1.4.3]. The complete isometries coincide with the strong monomorphisms and the complete quotient maps coincide with the strong epimorphisms in **FdOS** [29, §4.3].

Since $V(M_n) = M_n = V(T_n)$, the usual definition of a completely positive map makes sense for linear maps $M_n \to M_m$ and $T_n \to T_m$. The next example shows that the quantum operations in the Heisenberg and Schrödinger pictures provide two important classes of completely contractive maps.

**Example 12.** Every CPTP map $\varphi: T_n \to T_m$ is a complete contraction and every CPU map $\psi: M_m \to M_n$ is also a complete contraction with $||\varphi||_{cb} = 1 = ||\psi||_{cb}$.

## 2.3 Direct Sums and (Co)products

The category **FdOS** has finite (co)products which follows immediately from [29, §4.2]. We recall these constructions, which are actually completely analogous and compatible with the (co)products in **FdBan**. The category **FdOS** has a zero object given by the zero-dimensional operator space $0$. If $X, Y \in \text{FdOS}$, then the vector space $X \oplus Y$ can be equipped with a matrix norm via the linear isomorphism $M_n(X \oplus Y) \cong M_n(X) \oplus_M M_n(Y)$ and the Banach space structure of the latter space (see §2.1). This gives an o.s.s. on $X \oplus Y$ and we write $X \underset{\oplus}{\otimes} Y$ for the resulting operator space, which is the categorical product of $X$ and $Y$.
The notation is consistent with the one in **FdBan**, i.e. $U(X \underset{\oplus}{\stackrel{\infty}{Y}}) = UX \underset{\oplus}{\stackrel{\infty}{U}} Y$ as Banach spaces. The categorical coproduct of $X$ and $Y$ in **FdOS**, written $X \underset{\oplus}{^{1}} Y$, is the vector space $X \oplus Y$ together with the o.s.s inherited via the linear isomorphism $X \oplus Y \cong (X^* \underset{\oplus}{\stackrel{\infty}{Y}})^*$ and the o.s.s. of the latter space. This is also consistent with the notation in **FdBan**, i.e. $U(X \underset{\oplus}{^{1}} Y) = UX \underset{\oplus}{\stackrel{\infty}{U}} Y$ as Banach spaces. Note that $V(X \underset{\oplus}{^{1}} Y) = VX \oplus VY = V(X \underset{\oplus}{\stackrel{\infty}{Y}})$, i.e. $V$ sends (co)products of **FdOS** to biproducts in **FdVect**. The functorial action of $\underset{\oplus}{^{1}}$ and $\underset{\oplus}{\stackrel{\infty}{}}$ coincides with that of $\oplus$ on morphisms, i.e. $f \underset{\oplus}{^{1}} g = f \oplus g = f \underset{\oplus}{\stackrel{\infty}{g}}$.

## 2.4 Tensor Products of Operator Spaces

We recall the construction of three operator space tensor products that we use in the sequel. The first two are the completely projective tensor $\hat{\otimes}$ and the completely injective one $\tilde{\otimes}$. They are analogous to their non-complete Banach space counterparts $\hat{\otimes}$ and $\tilde{\otimes}$ in that $\hat{\otimes}$ ($\tilde{\otimes}$) is the largest (smallest) tensor o.s.s that can be assigned on the vector space tensor product $X \otimes Y$, but we elide the details of what this means. The third tensor, called the Haagerup tensor, is rather unique to operator space theory and it allows us to define von Neumann (co)algebras as certain kinds of internal (co)monoid objects in the sequel. We define all three tensors for finite-dimensional operator spaces for simplicity.

**Remark 13.** The completely projective tensor is usually referred to as the “projective tensor” of operator spaces and the completely injective one as the “minimal” or “injective” tensor of operator spaces. We have chosen the alternative names in order to avoid confusion with their Banach space counterparts.

**Definition 14 ([7, (1.5.1)])**. Let $X, Y \in \text{FdOS}$. The completely injective tensor product, written $X \hat{\otimes} Y$, is the vector space $X \otimes Y$ with the o.s.s inherited via the linear isomorphism $X \otimes Y \cong \text{CB}(Y^*, X)$ and the o.s.s of the latter space.

We show in the sequel that the above tensor product serves as the multiplicative disjunction and that the next one serves as the multiplicative conjunction.

**Definition 15.** Let $X, Y \in \text{FdOS}$. The completely projective tensor product, written $X \hat{\otimes} Y$, is the vector space $X \otimes Y$ with the o.s.s inherited via the linear isomorphism $X \otimes Y \cong (X^* \hat{\otimes} Y^*)^*$ and the o.s.s of the latter space.

The above definition is not the standard one in the literature, but we prove in Appendix A that for finite-dimensional operator spaces the two are equivalent. Before we may recall the third tensor, we need an auxiliary definition.

**Definition 16 ([15, §9.1])**. Given $x \in \mathbb{M}_{n,r}(X)$ and $y \in \mathbb{M}_{r,m}(Y)$ the matrix inner product of $x$ and $y$ is the matrix $x \odot y \in \mathbb{M}_{n,m}(X \otimes Y)$ defined as

$$ (x \odot y)_{i,j} \triangleq \sum_{k=1}^{r} x_{i,k} \otimes y_{k,j}. $$
**Definition 17 ([15, §9.2]).** Let $X, Y \in \text{FdOS}$. The Haagerup tensor product, written $X \hat{\otimes} Y$, is the vector space $X \otimes Y$ together with the o.s.s defined for $v \in \mathbb{M}_n(X \otimes Y)$ by:

$$ \|v\|_h \triangleq \inf\{\|x\|\|y\|: v = x \odot y, x \in M_{n,r}(X), y \in M_{r,n}(Y), r \in \mathbb{N}\}. $$

An interesting property of the Haagerup tensor is that it is self-dual in **FdOS**.

**Proposition 18 ([15, Cor. 9.4.8]).** Given $X, Y \in \text{FdOS}$, the linear isomorphism $\theta: X^* \hat{\otimes} Y^* \cong (X \hat{\otimes} Y)^*$ :: $f \otimes g \mapsto (x \otimes y \mapsto f(x)g(y))$ is a c.i.i.

Another fact that distinguishes it (again) from the other two tensors is that the Haagerup tensor is not symmetric in **FdOS**. What they do have in common is that $V(X \hat{\otimes} Y) = V(X \hat{\otimes} Y) = V(X \tilde{\otimes} Y) = VX \otimes VY$, i.e. the underlying vector space of all three tensors coincides with the vector space tensor product. Furthermore, the functorial action of all three tensors on morphisms coincides with $\otimes$, i.e. $(f \hat{\otimes} g) = (f \tilde{\otimes} g) = f \otimes g$. The identity map (on $X \otimes Y$) gives complete contractions $X \hat{\otimes} Y \to X \tilde{\otimes} Y \to X \tilde{\otimes} Y$, because the respective norms on the three spaces satisfy $\|\cdot\|_{\Lambda} \ge \|\cdot\|_{h} \ge \|\cdot\|_{\vee}$. The next proposition (special case of [14, Theorem 6.1]) shows even more.

**Proposition 19.** Given operator spaces $A, B, C, D \in \text{FdOS}$, the permutation $u_{ABCD}: (A \otimes B) \otimes (C \otimes D) \to (A \otimes C) \otimes (B \otimes D) :: (a \otimes b) \otimes (c \otimes d) \mapsto (a \otimes c) \otimes (b \otimes d)$ gives the two complete contractions $v_{ABCD} = u_{ABCD} = w_{ABCD}$ with types

$$ w_{ABCD}: (A \hat{\otimes} B) \hat{\otimes} (C \hat{\otimes} D) \to (A \hat{\otimes} C) \hat{\otimes} (B \hat{\otimes} D) $$

$$ v_{ABCD}: (A \tilde{\otimes} B) \hat{\otimes} (C \tilde{\otimes} D) \to (A \hat{\otimes} C) \tilde{\otimes} (B \hat{\otimes} D) $$

In operator space theory, maps of the above form are known as *shuffle maps*, whereas in category theory they are known as *interchange laws*.

## 2.5 Conjugate Operator Space and Opposite Operator Space

In the sequel, we need the operation $(\cdot)^{\dagger}$ of taking the conjugate transpose of a matrix. This is not a linear map, nor a complete contraction $M_n \to M_n$ (when $n > 1$). We deal with this by recalling two involutive constructions.

An anti-linear map $f: V \to W$ between two complex vector spaces may be equivalently described as a linear map $f: V_c \to W$, where $V_c$ is the *conjugate* vector space of $V$. Recall that $V_c$ is a complex vector space with the same underlying set as $V$ and same additive structure as $V$, but whose scalar multiplication is defined by $\lambda * v \triangleq \bar{\lambda}v$. This makes sense for operator spaces as well.

**Definition 20 ([31, Sec. 2.9]).** Let $X$ be an operator space. The conjugate vector space $X_c$ can be equipped with an o.s.s. defined by $\|x\|_{M_n(X_c)} \triangleq \|x\|_{M_n(X)}$ for $x \in \mathbb{M}_n(X_c)$. We say that $X_c$ is the conjugate operator space of $X$.
<table><caption>**Table 1.** MALL and BV categorical structure.</caption><thead><tr><th>MALL / BV</th><th>FdOS</th><th>FdBan</th><th>FdVect</th></tr></thead><tbody><tr><td>X &rarr; Y</td><td>CB(X, Y)</td><td>B(X, Y)</td><td>L(X, Y)</td></tr><tr><td>X &amp; Y</td><td>X &oplus; Y</td><td>X &oplus; Y</td><td>X &oplus; Y</td></tr><tr><td>X &oplus; Y</td><td>X &oplus; Y</td><td>X &oplus; Y</td><td>X &oplus; Y</td></tr><tr><td>X &otimes; Y</td><td>X &otimes; Y</td><td>X &otimes; Y</td><td>X &otimes; Y</td></tr><tr><td>X &not; Y</td><td>X &not; Y</td><td>X &not; Y</td><td>X &otimes; Y</td></tr><tr><td>X &larr; Y</td><td>X &otimes; Y</td><td></td><td>X &otimes; Y</td></tr></tbody></table>

**Definition 21 ([31, Sec. 2.10]).** Let $X$ be an operator space. The opposite operator space of $X$, written $X_o$, has the same underlying vector space but its o.s.s. is defined by $\|[x_{ji}]\|_{M_n(X_o)} \triangleq \|[x_{ij}]\|_{M_n(X)}$, for $x \in M_n(X_o)$.

For convenience, we also sometimes write $X^o$ or $X^c$ for the opposite/conjugate space, e.g. we write $M_n^o$ instead of $(M_n)_o$.

*Example 22.* The conjugate transpose is a completely isometric isomorphism $(\cdot)^\dagger: M_n^c \cong M_n^o$. This is just a special case of a result for $C^*$-algebras [31, p. 65].

## 3 Categorical Properties of FdOS

We begin our contributions by proving relevant categorical properties of **FdOS**. Our first proposition follows easily from existing results (see Appendix A). Our first theorem, given after it, shows the relevance of **FdOS** to Linear Logic.

**Proposition 23.** The category **FdOS** is a monoidal category when equipped with any of the tensors $\hat{\otimes} / \tilde{\otimes} / \otimes^h$ as monoidal product and $\mathbb{C}$ as monoidal unit. The first two monoidal structures are symmetric and in all cases the (symmetric) monoidal natural isomorphisms coincide with those in **FdVect**, i.e. the forgetful functor $V: \mathbf{FdOS} \to \mathbf{FdVect}$ is a strict (symmetric) monoidal functor.

**Theorem 24.** The category **FdOS** has finite (co)products, it is $*-\text{autonomous}$ with relevant data given in Table 1, and is therefore a model of MALL.

The Haagerup tensor is outside the scope of Linear Logic, but **FdOS** together with this tensor and the maps from Proposition 19 constitute a BV-category (with negation) in the sense of [8] (see Appendix A). The binary connective $\triangleleft$ from Table 1 stands for the “seq” tensor of BV-categories, which corresponds to $\otimes^h$ in **FdOS**. In the sequel, we show that $\otimes^h$ has even more categorical properties.

Next, we consider some of the categorical properties of the assignments $(\cdot)_c$ and $(\cdot)_o$ of taking conjugate/opposite operator spaces. The second equality from our next proposition follows from [11, Lemma 2.1] where it is shown that the identity map is a c.i.i (and therefore an equality). The first equality is easy.
**Proposition 25.** Given operator spaces X and Y, we have operator space equalities $CB(X, Y)_c = CB(X_c, Y_c)$ and $CB(X, Y)_o = CB(X_o, Y_o)$.

Since $\mathbb{C} = \mathbb{C}_o$ and $\mathbb{C}_c \cong \mathbb{C}$, with conjugation serving as the completely isometric isomorphism, we have as a special case that $(X_c)^* \cong (X^*)_c$ and $(X_o)^* = (X^*)_o$. Moreover, we can see the assignments $(\cdot)_c$: **FdOS** $\rightarrow$ **FdOS** and $(\cdot)_o$: **FdOS** $\rightarrow$ **FdOS** as (covariant) functors by simply defining $f_o \triangleq f$ and $f_c \triangleq f$. It is easy to see that $(\cdot)_c \circ (\cdot)_c = \mathrm{Id} = (\cdot)_o \circ (\cdot)_o$ and $(\cdot)_c \circ (\cdot)_o = (\cdot)_o \circ (\cdot)_c$ and therefore we have a pair of commuting functorial involutions on **FdOS** which are (set-theoretic) identities on morphisms, but not on objects. Even more, these functors behave well with other functorial constructions we have seen.

**Proposition 26.** Given operator spaces X, Y ∈ **FdOS**, we have:

$$ - (X \hat{\otimes} Y)_c = X_c \hat{\otimes} Y_c \quad - (X \oplus Y)_c = X_c \oplus Y_c \quad - (X \hat{\otimes} Y)_c = X_c \hat{\otimes} Y_c $$

$$ - (X \tilde{\otimes} Y)_c = X_c \tilde{\otimes} Y_c \quad - (X \oplus Y)_c = X_c \oplus Y_c \quad - \text{a c.i.i., called } \gamma, $$

$$ - (X \hat{\otimes} Y)_o = X_o \hat{\otimes} Y_o \quad - (X \oplus Y)_o = X_o \oplus Y_o \quad (X \hat{\otimes} Y)_o \cong Y_o \hat{\otimes} X_o $$

$$ - (X \tilde{\otimes} Y)_o = X_o \tilde{\otimes} Y_o \quad - (X \oplus Y)_o = X_o \oplus Y_o \quad \gamma(x \otimes y) \triangleq (y \otimes x). $$

*Proof.* Some of these facts are already known and the rest are in Appendix A.

Therefore, even though the Haagerup tensor is not symmetric in **FdOS**, we can still talk about the usual symmetry $\gamma$, provided we use the involution $(\cdot)_o$. We also use $\gamma$ to denote its own inverse, i.e. $\gamma: Y_o \hat{\otimes} X_o \cong (X \hat{\otimes} Y)_o$, since both $\gamma$ and $\gamma^{-1}$ are defined in the same way, namely, via swapping.

We also briefly use the category **FdOS**<sub>cb</sub> which has f.d. operator spaces as objects, but whose morphisms are the c.b. maps between them. All the categorical properties from this section hold verbatim for **FdOS**<sub>cb</sub> and the subcategory inclusion **FdOS** $\rightarrow$ **FdOS**<sub>cb</sub> preserves this data. However, as we already explained, the linear and c.b. maps coincide for f.d. operator spaces, so it follows that the forgetful functor $V: \mathrm{FdOS}_\mathrm{cb} \rightarrow \mathrm{FdVect}$ gives an equivalence of categories **FdOS**<sub>cb</sub> $\simeq$ **FdVect**. Because of this, **FdOS**<sub>cb</sub> is a less interesting, and more degenerate (from a MALL perspective) category compared to **FdOS**, e.g. the three tensors $\hat{\otimes}, \oplus, \tilde{\otimes}$ are naturally isomorphic to each other, via the identity map, in **FdOS**<sub>cb</sub> and each of them gives a compact closed structure on **FdOS**<sub>cb</sub>. Of course, this is not true in **FdOS**, which is the primary category of interest.

# 4 Novel Categorical Approach to von Neumann Algebras

Operator spaces alone do not have sufficient structure for quantum computation. In the Heisenberg picture, we can use von Neumann algebras (vN-algebras) to define the relevant quantum operations. Every such algebra can be equipped with a canonical o.s.s., so we may think of them as operator spaces with additional structure. In this section, we consider f.d. vN-algebras and show that they may be equivalently defined as certain kinds of involutive monoid objects in
commute, and for every complete isometry (i.e. strong mono) $a : \mathbb{C} \to A$, the following composite in FdOS is also a complete isometry (i.e. strong mono)

$$ \mathbb{C} \cong \mathbb{C}_o^{\mathfrak{h}} \mathbb{C}_c \xrightarrow{a \otimes a} A_o \otimes A_c \xrightarrow{A_o \otimes i} A_o \otimes A_o \xrightarrow{\gamma} (A \otimes A)_o \xrightarrow{\mu} A_o. \quad (2) $$

**Theorem 30.** Definitions 27 and 29 are equivalent, i.e. the data in one definition uniquely determines the data in the other.

Our theorem is proven in Appendix B and its proof shows that the two (equivalent) notions may be used interchangeably. We do so in the sequel.

**Definition 31.** Let vNAlg denote the subcategory of FdOS whose objects are f.d. vN-algebras and whose morphisms $\varphi: A \to B$ are multiplicative, unital, and involutive, i.e. the following diagrams in FdOS commute<fnref n="1" />.

<figure><img src="image_2.png" /></figure>

In the literature, these morphisms are known as “unital *-homomorphisms”, but we simply call them vN-algebra morphisms. It is already known that (possibly infinite-dimensional) vN-algebras form a symmetric monoidal category with small products [28], so the next propositions follow easily. However, we present them in our categorical style, as this helps for the development in §5.

**Proposition 32.** Given $A, B \in \text{vNAlg}$, then the operator space $A \tilde{\otimes} B$ with

$$ - \text{unit } \eta \triangleq (\mathbb{C} \cong \mathbb{C} \tilde{\otimes} \mathbb{C} \xrightarrow{\eta_A \otimes \eta_B} A \tilde{\otimes} B) $$

$$ - \text{involution } i \triangleq ((A \tilde{\otimes} B)_c = A_c \tilde{\otimes} B_c \xrightarrow{i_A \otimes i_B} A_o \tilde{\otimes} B_o = (A \tilde{\otimes} B)_o) $$

$$ - \text{multiplication } \mu \triangleq ((A \tilde{\otimes} B) \otimes (A \tilde{\otimes} B) \xrightarrow{v} (A \otimes A) \tilde{\otimes} (B \otimes B) \xrightarrow{\mu_A \otimes \mu_B} A \tilde{\otimes} B) $$

is also a vN-algebra. Also, the operator space $A \tilde{\oplus} B$ is a vN-algebra with

$$ - \text{unit } \eta \triangleq (\mathbb{C} \xrightarrow{\langle \eta_A, \eta_B \rangle} A \tilde{\oplus} B) $$

$$ - \text{involution } i \triangleq ((A \tilde{\oplus} B)_c = A_c \tilde{\oplus} B_c \xrightarrow{i_A \oplus i_B} A_o \tilde{\oplus} B_o = (A \tilde{\oplus} B)_o) $$

$$ - \text{multiplication } \mu \triangleq ((A \tilde{\oplus} B) \otimes (A \tilde{\oplus} B) \xrightarrow{\mu_A \circ (\pi_A \otimes \pi_A), \mu_B \circ (\pi_B \otimes \pi_B)} A \tilde{\oplus} B) $$

**Proposition 33.** The category vNAlg is symmetric monoidal and has finite products with the action of $\tilde{\otimes}$ and $\tilde{\oplus}$ extended as in Proposition 32.

<footnote n="1">¹ It suffices to assume $\varphi$ is linear and the three diagrams imply it is also c.c.</footnote>
commute, and for every complete quotient map (i.e. strong epi) $e: C \to \mathbb{C}$, the following composite in FdOS is a complete quotient map (i.e. strong epi)

$$C_o \xrightarrow{\delta} (C \otimes^h C)_o \xrightarrow{\gamma} C_o \otimes^h C_o \xrightarrow{C_o \otimes^h j} C_o \otimes^h C_c \xrightarrow{e \otimes e} C_o \otimes^h C_c \cong \mathbb{C}. \quad (5)$$

We dualise Definition 31 in order to obtain the next one.

**Definition 37.** Let vNCoalg denote the subcategory of FdOS whose objects are f.d. vN-coalgebras and whose morphisms $\varphi: C \to D$ are comultiplicative, counital, and involutive, i.e. the following diagrams in FdOS commute.

<figure><img src="image_3.png" /></figure>

We also dualise the construction of tensors and direct sums from Proposition 32 and we now use the remaining shuffle map from Proposition 19.

**Proposition 38.** Given $C, D \in \text{vNCoalg}$, then the operator space $C \hat{\otimes} D$ with

$$- \text{ counit } \varepsilon \triangleq (C \hat{\otimes} D \xrightarrow{\varepsilon_C \otimes \varepsilon_D} C \hat{\otimes} C \cong C)$$

$$- \text{ involution } j \triangleq ((C \hat{\otimes} D)_o = C_o \hat{\otimes} D_o \xrightarrow{j_C \otimes j_D} C_c \hat{\otimes} D_c = (C \hat{\otimes} D)_c)$$

$$- \text{ comult. } \delta \triangleq (C \hat{\otimes} D \xrightarrow{\delta_C \otimes \delta_D} (C \hat{\otimes} C) \hat{\otimes} (D \hat{\otimes} D) \xrightarrow{w} (C \hat{\otimes} D) \hat{\otimes} (C \hat{\otimes} D))$$

is also a vN-coalgebra. Also, the operator space $C \hat{\oplus} D$ is a vN-coalgebra with

$$- \text{ counit } \varepsilon \triangleq (C \hat{\oplus} D \xrightarrow{[\varepsilon_C, \varepsilon_D]} C)$$

$$- \text{ involution } j \triangleq ((C \hat{\oplus} D)_o = C_o \hat{\oplus} D_o \xrightarrow{j_C \oplus j_D} C_c \hat{\oplus} D_c = (C \hat{\oplus} D)_c)$$

$$- \text{ comultiplication } \delta \triangleq (C \hat{\oplus} D \xrightarrow{[(\iota_C \otimes \iota_C)^{\circ \delta_C}, (\iota_D \otimes \iota_D)^{\circ \delta_D}]} (C \hat{\oplus} D) \hat{\otimes} (C \hat{\oplus} D))$$

**Proposition 39.** The category vNCoalg is symmetric monoidal and has finite coproducts with the action of $\hat{\otimes}$ and $\hat{\oplus}$ extended as in Proposition 38.

**Example 40.** The operator space $T_n$ is a vN-coalgebra with: (1) counit given by the trace functional $\varepsilon \triangleq \text{tr}: T_n \to \mathbb{C} \;::\; t \mapsto \text{tr}(t)$; (2) involution given by the conjugate transpose $j \triangleq (\cdot)^{\dagger}: T_n^o \to T_n^c$; (3) comultiplication given by $\delta: T_n \to T_n \hat{\otimes} T_n \;::\; e_{ij} \mapsto \sum_k (e_{kj} \otimes e_{ik})$, where $e_{ij} \in T_n$ is the matrix that has 1 in position $(i, j)$ and 0 elsewhere. Using Proposition 38, we see that $\hat{\oplus}_{1 \le i \le n} T_{k_i}$ is also a vN-coalgebra and using Theorem 44 and Example 28 it follows that each f.d. vN-coalgebra is of this form, modulo vN-coalgebra isomorphism.
*Quantum Operations.* We use f.d. vN-coalgebras to define the quantum operations in the Schrödinger picture, i.e. the CPTP maps. This can be done by dualising the CPU maps from §4.

The dual of a unital map is obviously a counital one. Unfortunately, in quantum information theory, the term “trace-preserving” is standard, so we simply say that a linear map $\varphi: C \to D$ between two vN-coalgebras is trace-preserving if it is counital. We explain why this is justified. Note that if $C = T_n$ and $D = T_m$, then from Example 40, we see that a counital map is precisely a trace-preserving one. More generally, the direct sum $\bigoplus_i T_{k_i}$ can be linearly embedded block-diagonally in a sufficiently large matrix space $M_m$ and then counitality becomes equivalent to trace-preservation with respect to this view. By the final remarks in Example 40, this covers all f.d. vN-coalgebras, modulo isomorphism, and this completes our justification.

We proceed with the (dual) definition of a completely positive map between two f.d. vN-coalgebras. We work again in the category **FdOS**<sub>cb</sub>. Given a vN-coalgebra C, we say that a linear functional p: C → C is positive if there exists a linear functional a: C → C, such that the following diagram:

$$ \begin{tikzcd} (C \otimes C)_o \arrow[r, "\gamma"] \arrow[d, "\delta"] & C_o \otimes C_o \arrow[r, "C_o \otimes j"] \arrow[d, "p"] & C_o \otimes C_c \arrow[r, "a \otimes a"] \arrow[d, "\cong"] & C_o \otimes C_c \arrow[r, "\cong"] & C \arrow[d, "\sim"] \\ C_o & & & & \end{tikzcd} \quad (6) $$

commutes. Note that (6) is dual to (3). Continuing in this fashion, we say that a linear map $\varphi: C \to D$ between vN-coalgebras is positive if it reflects positive linear functionals, i.e. $p \circ \varphi: C \to C$ is a positive linear functional on C for every positive linear functional p: D → C. Finally, we say $\varphi$ is completely positive if $T_n \otimes \varphi: T_n \otimes C \to T_n \otimes D$ is positive for every $n \in \mathbb{N}$. This notion of (complete) positivity is equivalent to the usual/concrete one (see Appendix C).

We write **S** for the category whose objects are f.d. vN-coalgebras and whose morphisms are the CPTP maps between them. Finally, we show that **S** is a subcategory of **FdOS** and summarise its structure relevant to MALL.

**Proposition 41.** Let $\varphi: C_1 \to C_2$ be a linear trace-preserving map between f.d. vN-coalgebras. Then, $\varphi$ is completely positive iff $\varphi$ is completely contractive.

**Proposition 42.** The category **S** is symmetric monoidal (w.r.t. $\hat{\otimes}$), has finite coproducts (w.r.t. $\oplus^1$), both preserved via inclusions $vNCoalg \hookrightarrow S \hookrightarrow FdOS$.

# 6 Heisenberg-Schrödinger Duality and vN-(co)algebras

In f.d. quantum theory, the quantum operations in the Heisenberg picture are precisely the CPU maps, i.e. the morphisms of **H**, whereas the quantum operations in the Schrödinger picture are precisely the CPTP maps, i.e. the morphisms of **S**. Both **H** and **S** are subcategories of **FdOS** and the dual functor $(-)^*: FdOS \to FdOS$ can be defined also on vN-(co)algebras, as we show next.
**Proposition 43.** If $A \in \text{vNAlg}$, then the o.s. dual $A^*$ is a vN-coalgebra with counit $\varepsilon \triangleq (A^* \xrightarrow{\eta^*} \mathbb{C}^* \cong \mathbb{C})$, involution $j \triangleq ((A^*)_o = (A_o)^* \xrightarrow{i^*} (A_c)^* \cong (A^*_c))$, and comultiplication $\delta \triangleq (A^* \xrightarrow{\mu^*} (A \otimes^h A)^* \cong A^* \otimes^h A^*)$. Conversely, if $C \in \text{vNCoalg}$, then the o.s. dual $C^*$ is a vN-algebra with unit $\eta \triangleq (\mathbb{C} \cong \mathbb{C}^* \xrightarrow{\varepsilon^*} C^*)$, involution $i \triangleq ((C^*_c) \cong (C_c)^* \xrightarrow{j^*} (C_o)^* = (C^*_o))$, and multiplication given by $\mu \triangleq (C^* \otimes^h C^* \cong (C \otimes^h C)^* \xrightarrow{\delta^*} C^*)$.

**Theorem 44.** *We have equivalences of categories $(\cdot)^* : \mathbf{S} \simeq \mathbf{H}^{\text{OP}} : (\cdot)^* \text{ and also } (\cdot)^* : \text{vNCoalg} \simeq \text{vAlg}^{\text{OP}} : (\cdot)^*$, where the action on objects of $(\cdot)^*$ is defined as in Proposition 43 and on morphisms in the usual way, i.e. as in FdOS. Moreover, these equivalences are strong monoidal (and (co)product preserving).*

*Example 45.* The c.i.i. from Example 10 is a vN-coalgebra isomorphism $T_n \cong M_n^*$, which determines a vN-algebra isomorphism $T_n^* \cong M_n$ by duality.

Theorem 44 gives a categorical formulation of the Heisenberg-Schrödinger duality and shows precisely how the MALL related structure of **vNCoalg** and **S** from §5 is dual to that of **vNAlg** and **H** from §4, and vice-versa. The duality **vNCoalg** ≃ **vNAlg**<sup>OP</sup> shows that vN-(co)algebras are dual to each other in a strong mathematical sense, whereas the duality **S** ≃ **H**<sup>OP</sup> is more relevant for quantum computation and we use it in the next section.

# 7 Revisiting Quantum Coherence Spaces

The preparatory categorical work in Sections 3–6 shows that the (sub)categories **S** $\hookrightarrow$ **FdOS** $\hookrightarrow$ **H** have the right categorical structure for a model of MALL whose duality is induced by the Heisenberg-Schrödinger duality. However, there is one problem: neither **S**, nor **H**, is a full subcategory of **FdOS**. Luckily, there is a simple solution: a semantic technique of Hyland and Schalk [24], based on *gluing and orthogonality*, allows us to carve out a category **Q** from **FdOS**, such that we get fully faithful inclusions **S** $\hookrightarrow$ **Q** $\hookrightarrow$ **H**, while preserving all the MALL structure. The main idea, in a nutshell, is to ensure trace-preservation (unitality) for morphisms between vN-coalgebras (vN-algebras) and then fullness is guaranteed by Propositions 34 and 41. This is precisely what we do by defining a suitable notion of orthogonality/polarity based on ideas from [24]. The resulting model has an obvious resemblance to (probabilistic) coherence spaces.

More specifically, we are interested in pairs $(X, S)$, where $X$ is an operator space and $S \subseteq \text{Ball}(X)$ is a subset of the unit ball of $X$, which satisfies an additional condition that we proceed to explain. Our next definition has similarities with *polar sets* from functional and convex analysis, which motivates its name.

**Definition 46 (Polar).** Given an operator space $X$ and a subset $S \subseteq \text{Ball}(X)$, we define the polar of $S$ to be $S^\circ \triangleq \{f \in \text{Ball}(X^*) | \forall s \in S. f(s) = 1\}$. We say that $S$ is bipolar if $S = d^{-1}[S^{\circ\circ}]$, where $d: X \cong X^{**}$ is the canonical c.i.i.
To avoid notational overhead, when dealing with multiple applications of the polar construction, we implicitly consider the image of such polar sets to be taken under $d^{-1}$. With this convention, we simply say that $S$ is bipolar if $S = S^{\circ\circ}$. As expected, the polar introduces a Galois connection [24, §5.1] and we have: for $R \subseteq S \subseteq \text{Ball}(X)$, it follows that $S^{\circ} \subseteq R^{\circ}$, $S \subseteq S^{\circ\circ}$, and $S^{\circ\circ\circ} = S^{\circ}$. We can now introduce the main category that is computationally relevant.

**Definition 47.** Let $\mathbf{Q}$ be the category whose objects are pairs $(X, S)$ with $X \in \text{FdOS}$ and $S \subseteq \text{Ball}(X)$ a bipolar set, i.e. $S = S^{\circ\circ}$, and whose morphisms $f: (X, S) \to (Y, R)$ are complete contractions $f: X \to Y$ such that $f[S] \subseteq R$.

Our notion of polarity is instead called “orthogonality” in [24] and it enjoys useful properties, e.g. it is *tight*, *stable*, *focused*, *precise*, in the sense of [24]. By using results from [24], it follows that the MALL structure of **FdOS** is preserved.

**Theorem 48.** The category **Q** is \*-autonomous and has finite (co)products: duals are given by $(X, S)^* \triangleq (X^*, S^\circ)$; the initial object by $(0, \emptyset)$; the terminal object by $(0, \{0\})$; binary products by $(X, S) \underset{\sim}{\oplus} (Y, R) \triangleq (X \underset{\sim}{\oplus} Y, S \times R)$; binary coproducts by $(X, S) \overset{\oplus}{\oplus} (Y, R) \triangleq (X \overset{\oplus}{\oplus} Y, (S^\circ + R^\circ)^\circ)$, where

$$S^\circ + R^\circ \triangleq \{[s', r'] : X \overset{\oplus}{\oplus} Y \to \mathbb{C} \mid s' \in S^\circ, r' \in R^\circ\} \subseteq \text{Ball}((X \overset{\oplus}{\oplus} Y)^*);$$

multiplicative conjunction is given by $(X, S) \hat{\otimes}(Y, R) \triangleq (X \hat{\otimes} Y, (S \otimes R)^{\circ\circ})$, where $S \otimes R \triangleq \{s \otimes r \mid s \in S, r \in R\}$; multiplicative disjunction by $(X, S) \tilde{\otimes}(Y, R) \triangleq (X \tilde{\otimes} Y, (S^\circ \otimes R^\circ)^\circ)$; tensor unit for both tensors is given by $(\mathbb{C}, \{1\})$.

The polarity that we introduced behaves very well w.r.t. vN-(co)algebras. In order to see this, we first introduce some notation. For a vN-coalgebra $C$, we say that an element $p \in C$ is *positive* if the map $1 \mapsto p: \mathbb{C} \to C$ is positive and we write $p \ge 0$ to indicate this. The *density operators* on $C$ are defined to be the set $P_C \triangleq \{p \in C \mid \varepsilon(p) = 1 \text{ and } p \ge 0\}$. Note that if $C = T_n$, then $P_C$ is precisely the set of density matrices, i.e. $n \times n$ positive matrices whose trace is 1.

**Theorem 49.** If $A \in \text{vNAlg}$ is a vN-algebra, then $(A, \{1_A\}) \in \text{Ob}(\mathbf{Q})$. Moreover, the functor $H: \mathbf{H} \to \mathbf{Q}$ defined by $H(A) \triangleq (A, \{1_A\})$ and $H(f) \triangleq f$, is fully faithful, strict monoidal w.r.t $\tilde{\otimes}$, and it strictly preserves finite products.

**Theorem 50.** If $C \in \text{vNCoalg}$ is a vN-coalgebra, then $(C, P_C) \in \text{Ob}(\mathbf{Q})$. Moreover, the functor $S: \mathbf{S} \to \mathbf{Q}$ defined by $S(C) \triangleq (C, P_C)$ and $S(f) \triangleq f$, is fully faithful, strict monoidal w.r.t $\hat{\otimes}$, and it strictly preserves finite coproducts.

We are now justified in presenting the summary provided by Figure 1, on which we elaborate. In our model **Q**, formulas in MALL admit interpretations as objects of **Q** and proofs are interpreted as morphisms of **Q**. Formulas with positive (negative) logical polarities admit natural interpretations as vN-coalgebras $C, D$ (vN-algebras $A, B$) and proofs between such formulas correspond precisely to the CPTP (CPU) maps, i.e. the quantum operations in the Schrödinger
<table><thead><tr><th>Schrödinger Picture</th><th>S<sub>full</sub>→Q</th><th>LL<sub>+</sub></th></tr></thead><tbody><tr><td>System description</td><td>C, D</td><td>P, R</td></tr><tr><td>Quantum composition</td><td>C ⊗ D</td><td>P ⊗ R</td></tr><tr><td>Classical composition</td><td>C ⊕ D</td><td>P ⊕ R</td></tr><tr><td>Quantum operation</td><td>C CPTP→D</td><td>P ⊢ R</td></tr></tbody></table>

<table><caption>**Fig. 1.** Schrödinger/Heisenberg picture and positive/negative logical polarity.</caption><thead><tr><th>Heisenberg Picture</th><th>H<sub>full</sub>→Q</th><th>LL<sub>-</sub></th></tr></thead><tbody><tr><td>System description</td><td>A, B</td><td>N, M</td></tr><tr><td>Quantum composition</td><td>A ⊗ B</td><td>N, M</td></tr><tr><td>Classical composition</td><td>A ⊕ B</td><td>N, M</td></tr><tr><td>Quantum operation</td><td>B CPU→A</td><td>M, N</td></tr></tbody></table>

(Heisenberg) picture. Moreover, this correspondence is preserved by classical composition (spacewise composition of systems where only a limited amount of classical interactions are possible) and by quantum composition (spacewise composition with the full range of quantum interactions possible, e.g. entanglement), in both pictures, whose interpretations are provided by the respective tables.

**Pure Quantum Computation.** We showed that **Q** captures mixed state quantum computation in both pictures. In fact, **Q** also has interesting properties that are relevant to pure state quantum computation, where unitarity is very important. If $M$ is a f.d. vN-algebra and $u \in M$ a unitary element, i.e. $uu^* = 1_M = u^*u$, then $(M, \{u\}) \in \text{Ob}(\mathbf{Q})$, i.e. the set $\{u\}$ is bipolar in $M$. We can reason in $\mathbf{Q}$ about interesting higher-order (pure state) maps, such as the pure state quantum switch. This is the linear map qsw: $M_n \hat{\otimes} M_n \to M_{2n}$ defined by qsw$(a \otimes b) \triangleq (|0\rangle\langle 0| \otimes (ab)) + (|1\rangle\langle 1| \otimes (ba))$ and it is a complete contraction (special case of [30, §IV]). For any $n \times n$ unitary matrices $u$ and $v$, we have object equalities $(M_n, \{u\}) \hat{\otimes} (M_n, \{v\}) = (M_n \hat{\otimes} M_n, \{u \otimes v\}^{\circ\circ}) = (M_n \hat{\otimes} M_n, \{u \otimes v\})$ and qsw becomes a morphism of our model $\mathbf{Q}$ (see Appendix E.3 for a proof) with type qsw: $(M_n, \{u\}) \hat{\otimes} (M_n, \{v\}) \to (M_{2n}, \{|0\rangle\langle 0| \otimes (uv) + |1\rangle\langle 1| \otimes (vu)\})$, where all three bipolar subsets consist of unitary matrices. Therefore qsw may be recognised as a valid (unitary-preserving) higher-order map in $\mathbf{Q}$. The “higher-order” description of qsw is justified by the fact that in pure state computation, we think of the unitary elements $u, v \in M_n$ as first-order (reversible) functions.

One of the reasons qsw is interesting, is that it does not admit any reasonable multi-linear decomposition, i.e. there are no complete contractions $\varphi_1: M_n \hat{\otimes} M_n \to M_{2n,m}$ and $\varphi_2: M_n \hat{\otimes} M_n \to M_{m,2n}$ such that qsw$(a \otimes b) = \varphi_1(a)\varphi_2(b)$. In fact, this can be determined by replacing $\hat{\otimes}$ with $\otimes^h$ and checking if qsw is still a complete contraction (see [30, §IV] for more details). The main reason why this is the case is that the Haagerup tensor actually characterises maps that do admit similar multi-linear decompositions [15, §9.4]. Such reasoning can also be done in $\mathbf{Q}$, because we have a valid object $(M_n \hat{\otimes} M_n, \{u \otimes v\}) \in \text{Ob}(\mathbf{Q})$, but qsw: $(M_n \hat{\otimes} M_n, \{u \otimes v\}) \to (M_{2n}, \{|0\rangle\langle 0| \otimes (uv) + |1\rangle\langle 1| \otimes (vu)\})$ is not a complete contraction, so we can conclude following the same arguments as in [30, §IV]. This shows our model $\mathbf{Q}$ can be used to reason about complex higher-order behaviour via the Haagerup tensor, which is outside the scope of Linear Logic, and suggests a new direction for future work.
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# A The category FdOS

In this appendix we provide some omitted proofs about the categorical properties of FdOS.

## A.1 Model of MALL

The definition of the completely projective tensor that we provided in §3 is not the standard one that may be found in the literature. For finite-dimensional operator spaces, it is equivalent to the standard one and we prove this here. The reason for presenting the simpler definition in §3 is that it has a more logical/categorical character and we believe it makes the paper more accessible to experts in these topics.

We begin the development by recalling the standard definition for the completely projective tensor product $X \hat{\otimes} Y$ and we prove at the end of this subsection that this determines the same o.s.s on $X \otimes Y$ as the one in Definition 15.

**Definition 51 (Completely projective tensor [15, §7.1]).** Let $X, Y \in \text{FdOS}$. The completely projective tensor product, written $X \otimes Y$, is the vector space $X \otimes Y$ together with the o.s.s defined for $v \in \mathbb{M}_n(X \otimes Y)$ by:

$$ \|v\|_{\Lambda} \triangleq \inf\{\|\alpha\|\|x\|\|y\|\|\beta\| : p \in \mathbb{N}, q \in \mathbb{N}, x \in M_p(X), y \in M_q(Y), \\ \alpha \in M_{n,pq}, \beta \in M_{pq,n}, \text{ and } v = \alpha(x \otimes y)\beta\}. $$

Here $x \otimes y \in M_{pq}(X \otimes Y)$ stands for the “tensor product of matrices” given by $x \otimes y \triangleq [x_{ij} \otimes y_{kl}]_{(i,k),(j,l)}$, which generalises the usual Kronecker product.

**Proposition 52 ([15, Sec. 7-9]).** The tensor product $\otimes^{\alpha}$ (for $\alpha = \hat{,}, \tilde{,} h$) is

- *Functorial wrt c.b maps and c.c maps*

- *Associative:* $(A \hat{\otimes} B) \hat{\otimes} C \cong A \hat{\otimes}(B \hat{\otimes} C) : (a \otimes b) \otimes c \mapsto a \otimes (b \otimes c)$

- *Has unitors:* $\mathbb{C} \hat{\otimes} A \cong A \cong A \hat{\otimes} \mathbb{C} : \lambda \otimes a \mapsto \lambda a \leftrightarrow a \otimes \lambda$

and for $\alpha = \hat{,}, \tilde{,}$

- *Symmetric:* $A \hat{\otimes} B \cong B \hat{\otimes} A : a \otimes b \mapsto b \otimes a$

**Proposition 23.** The category FdOS is a monoidal category when equipped with any of the tensors $\hat{\otimes} / \tilde{\otimes} / \hat{\otimes}^h$ as monoidal product and $\mathbb{C}$ as monoidal unit. The first two monoidal structures are symmetric and in all cases the (symmetric) monoidal natural isomorphisms coincide with those in FdVect, i.e. the forgetful functor $V: \text{FdOS} \to \text{FdVect}$ is a strict (symmetric) monoidal functor.

*Proof.* All three tensors can be extended to bifunctors and the associators, unitalities and symmetries are the ones given in Proposition 52. The coherence diagrams are straightforward to verify. To see that the functor $V: \text{FdOS} \to \text{FdVect}$ is a strict monoidal, note that the underlying vector space of $A \hat{\otimes} B$ (for $\alpha = \hat{,}, \tilde{,} h$) is $A \otimes B$ for any f.d. operator spaces $A$ and $B$ and that (symmetric) monoidal natural isomorphisms in Proposition 52 coincide with the ones for FdVect. □
**Lemma 53.** The symmetric monoidal category (FdOS, C, ⊗) is closed w.r.t. CB(-, -) : FdOS<sup>OP</sup> × FdOS → FdOS.

*Proof.* Follows from restricting [30, Prop. III.33] to FdOS. □

**Proposition 54.** The category FdOS is *-autonomous with C as dualizing object.

*Proof.* Because of Lemma 53, it suffices to prove that C is a global dualizing object. Let A ∈ FdOS, currying the evaluation map ev<sub>A,C</sub> : CB(A, C) ⊗ A → C, gives us the canonical embedding of A into its double dual

$$d_A : A \to CB(CB(A, C), C) \\ a \mapsto (f \mapsto f(a))$$

which by [15, Prop. 3.1.2] is a c.i. inclusion. As A is f.d, we have by the rank-nullity argument that this is a isomorphism. □

**Theorem 24.** The category FdOS has finite (co)products, it is *-autonomous with relevant data given in Table 1, and is therefore a model of MALL.

*Proof.* Follows by combining Proposition 54 with results about (co)products from Section 2.3. □

We can now easily prove that the two definitions of the completely projective tensor are equivalent on finite-dimensional operator spaces, i.e. they define the same o.s.s. on X ⊗ Y.

**Proposition 55.** Let X and Y be two f.d. operator spaces. Then the o.s.s on X ⊗ Y given by Definition 15 coincides with the o.s.s of Definition 51.

*Proof.* From Theorem 24, we know that FdOS is *-autonomous and X ⊗ Y ≅ (X* ⊗ Y*)* when ⊗ is defined in the standard way, i.e. as in Definition 51. But this c.i.i is exactly how the o.s.s. of X ⊗ Y is defined in Definition 15, so the two operator space structures are exactly equal. □

## A.2 BV-categorical structure

**Definition 56.** Let C be a category with two monoidal structure (C, I, ⊗) and (C, J, ≺), then (J, ≺) is normal duoidal to (I, ⊗) if we have a natural transformation:

$$w : (A \triangleleft B) \otimes (C \triangleleft D) \to (A \otimes C) \triangleleft (B \otimes D)$$

(the morphism w is often called weak interchange) and and morphisms

$$i : I \cong J \quad \delta_I : I \to I \triangleleft I \quad \mu_J : J \otimes J \to J$$
Along with coherence conditions for associativity,

$$ \begin{align} & ((A \triangleleft B) \otimes (C \triangleleft D)) \otimes (E \triangleleft F) \xrightarrow{\alpha} (A \triangleleft B) \otimes ((C \triangleleft D) \otimes (E \triangleleft F)) \nonumber \\ & \xleftarrow{w \otimes \mathrm{id}} \xrightarrow{\mathrm{id} \otimes w} ((A \otimes C) \triangleleft (B \otimes D)) \otimes (E \triangleleft F) \nonumber \\ & \xrightarrow{w} ((A \otimes C) \otimes E) \triangleleft ((B \otimes D) \otimes F) \xrightarrow{\alpha \triangleleft \alpha} (A \otimes (C \otimes E)) \triangleleft (B \otimes (D \otimes F)) \tag{7} \end{align} $$

$$ \begin{align} & ((A \triangleleft C) \triangleleft E) \otimes ((B \triangleleft D) \triangleleft F) \xrightarrow{\alpha \otimes \alpha} (A \triangleleft (C \triangleleft E)) \otimes (B \triangleleft (D \triangleleft F)) \nonumber \\ & \xrightarrow{w} ((A \triangleleft C) \otimes (B \triangleleft D)) \triangleleft (E \otimes F) \nonumber \\ & \xrightarrow{w \triangleleft \mathrm{id}} ((A \otimes B) \triangleleft ((C \triangleleft E) \otimes (D \triangleleft F)) \nonumber \\ & \xrightarrow{\alpha} (A \otimes B) \triangleleft ((C \otimes D) \triangleleft (E \otimes F)) \tag{8} \end{align} $$

and unitality,

$$ \begin{align} I \otimes (A \triangleleft B) \xrightarrow{\delta_I \otimes \mathrm{id}} (I \triangleleft I) \otimes (A \triangleleft B) \qquad &(A \triangleleft B) \otimes I \xrightarrow{\mathrm{id} \otimes \delta_I} (A \triangleleft B) \otimes (I \triangleleft I) \nonumber \\ \lambda_{A \triangleleft B} \uparrow \qquad &(\rho_{A \triangleleft B} \uparrow \qquad \uparrow w \nonumber \\ A \triangleleft B \xrightarrow{\lambda_{A \triangleleft \lambda_B}} (I \otimes A) \triangleleft (I \otimes B) \qquad &(A \triangleleft B) \xrightarrow{\rho_{A \triangleleft \rho_B}} (A \otimes I) \triangleleft (B \otimes I) \tag{9} \end{align} $$

$$ \begin{align} J \triangleleft (A \otimes B) \xleftarrow{\mu_J \triangleleft \mathrm{id}} (J \otimes J) \triangleleft (A \otimes B) \qquad &(A \otimes B) \triangleleft J \xleftarrow{\mathrm{id} \triangleleft \mu_J} (A \otimes B) \triangleleft (J \otimes J) \nonumber \\ \lambda_{A \otimes B} \uparrow \qquad &(\rho_{A \otimes B} \uparrow \qquad \uparrow w \nonumber \\ A \otimes B \xrightarrow{\lambda_{A \otimes \lambda_B}} (J \triangleleft A) \otimes (J \triangleleft B) \qquad &(A \otimes B) \xrightarrow{\rho_{A \otimes \rho_B}} (A \triangleleft J) \otimes (B \triangleleft J) \tag{10} \end{align} $$

as well as diagrams saying that $I$ is a comonoid in $(C, J, \triangleleft)$ and that $J$ is a monoid in $(C, I, \otimes)$.

**Definition 57.** [9, Sec. 3] A BV-category with negation is a \*-autonomous category $(C, I, \otimes, \neg)$ with an additional monoidal structure $(C, J, \triangleleft)$ that is normal
duoidal to $(\mathbf{C}, I, \otimes)$, such that the interchange commutes with the symmetry of $\otimes$ in the following way:

$$ \begin{gathered} (A \triangleleft B) \otimes (C \triangleleft D) \xrightarrow{\cong} (C \triangleleft D) \otimes (A \triangleleft B) \\ w \downarrow \qquad \downarrow w \qquad (11) \\ (A \otimes C) \triangleleft (B \otimes D) \xrightarrow{\cong} (C \otimes A) \triangleleft (D \otimes B) \end{gathered} $$

**Remark 58.** Note that in a BV-category with negation we also have a duoidal structure between $(\mathbf{C}, \perp, \mathcal{N})$ and $(\mathbf{C}, J, \triangleleft)$, where $(-) \mathcal{N} (-) \triangleq (-)^* \to (-)$.

**Proposition 59.** The category FdVect is a BV-category with negation w.r.t. $(\mathrm{FdVect}, \mathcal{C}, \otimes, L(-, -))$ and $(\mathrm{FdVect}, \mathcal{C}, \otimes)$.

*Proof.* FdVect is compact closed, hence *-autonomous, and any symmetric monoidal structure is normal duoidal to itself (special case of [2, Prop. 6.10.]).* $\square$

<figure><img src="image_4.png" /></figure>

**Proposition 60.** The category FdOS is a BV-category with negation w.r.t. $(\mathrm{FdOS}, \mathcal{C}, \hat{\otimes}, \mathrm{CB}(-, -))$ and $(\mathrm{FdOS}, \mathcal{C}, \otimes^h)$.

*Proof.* We know by Proposition 54 that FdOS is *-autonomous. By Proposition 19 we have natural weak interchange maps in FdOS and as we have a common unit for all tensors, the result now follows from Proposition 23 as the appropriate morphisms are mapped to the BV-structure on FdVect. $\square$*

<figure><img src="image_5.png" /></figure>

### A.3 The conjugate/opposite functorial involutions

We are going to use the following proposition multiple times, so we recall it here for convenience.

**Proposition 25.** Given operator spaces $X$ and $Y$, we have operator space equalities $\mathrm{CB}(X, Y)_c = \mathrm{CB}(X_c, Y_c)$ and $\mathrm{CB}(X, Y)_o = \mathrm{CB}(X_o, Y_o)$.

In particular, we use Proposition 25 to easily prove many of the following results.

**Proposition 26.** Given operator spaces $X, Y \in \mathrm{FdOS}$, we have:

$$ \begin{align*} -(X \hat{\otimes} Y)_c &= X_c \hat{\otimes} Y_c & -(X \oplus^1 Y)_c &= X_c \oplus^1 Y_c & -(X \otimes^h Y)_c &= X_c \otimes^h Y_c \\ -(X \tilde{\otimes} Y)_c &= X_c \tilde{\otimes} Y_c & -(X \oplus^\infty Y)_c &= X_c \oplus^\infty Y_c & -a \text{ c.i.i., called } \gamma, \\ -(X \hat{\otimes} Y)_o &= X_o \hat{\otimes} Y_o & -(X \oplus^1 Y)_o &= X_o \oplus^1 Y_o & (X \otimes^h Y)_o &\cong Y_o \otimes^h X_o \\ -(X \tilde{\otimes} Y)_o &= X_o \tilde{\otimes} Y_o & -(X \oplus^\infty Y)_o &= X_o \oplus^\infty Y_o & \gamma(x \otimes y) &\triangleq (y \otimes x). \end{align*} $$

*Proof.* The last two facts are already known, so we provide references.
**Case** $(X \overset{h}{\otimes} Y)_c = X_c \overset{h}{\otimes} Y_c$. It is claimed in [31, p. 97] that this is a completely isometric isomorphism. In our presentation, $X_c$ has the same underlying set as $X$, so this is actually an equality.

**Case** $(X \overset{h}{\otimes} Y)_o \cong Y_o \overset{h}{\otimes} X_o$. This is claimed in [31, pp. 96–97].

**Case** $(X \tilde{\otimes} Y)_c = X_c \tilde{\otimes} Y_c$. This is easy to prove using Proposition 25. Let $\iota_{X,Y}: X \tilde{\otimes} Y \cong \text{CB}(Y^*, X)$ be the canonical completely isometric isomorphism from Definition 14. More specifically, $\iota_{X,Y}(x \otimes y) \triangleq (\varphi \mapsto \varphi(y)x)$. The following composite

$$ (X \tilde{\otimes} Y)_c \xrightarrow{\iota_c} \text{CB}(Y^*, X)_c = \text{CB}((Y^*)_c, X_c) \cong \text{CB}((Y_c)^*, X_c) \xrightarrow{\iota^{-1}} X_c \tilde{\otimes} Y_c $$

consists of completely isometric isomorphisms. Evaluating on elementary tensors

$$ \begin{align*} x \otimes y &\xrightarrow{\iota_c} (\varphi \mapsto \varphi(y)x) \\ &= (\varphi \mapsto \overline{\varphi(y)} \star x) \\ &\xrightarrow{\cong} (\varphi \mapsto \varphi(y) \star x) \\ &\xrightarrow{\iota^{-1}} x \otimes y \end{align*} $$

shows the above composite is the identity map. Here we have written $\star$ for the scalar multiplication in $X_c$ and the vertically aligned symbols indicate the action of the isomorphisms. Since this c.i.i. is the identity map, the equality follows.

**Case** $(X \tilde{\otimes} Y)_o = X_o \tilde{\otimes} Y_o$. The proof is analogous to the previous case. The following composite

$$ (X \tilde{\otimes} Y)_o \xrightarrow{\iota_o} \text{CB}(Y^*, X)_o = \text{CB}((Y^*)_o, X_o) = \text{CB}((Y_o)^*, X_o) \xrightarrow{\iota^{-1}} X_o \tilde{\otimes} Y_o $$

consists of completely isometric isomorphisms. Evaluating on elementary tensors, we see that this c.i.i. is the identity map and the equality follows.

**Case** $(X \hat{\otimes} Y)_c = X_c \hat{\otimes} Y_c$. This can be proven by using the fact that $(X \tilde{\otimes} Y)_c = X_c \tilde{\otimes} Y_c$ and adapting the proof of the latter fact. Let $\iota_{X,Y}: X \hat{\otimes} Y \cong (X^* \tilde{\otimes} Y^*)^*$ be the canonical c.i.i. defined by $\iota_{X,Y}(x \otimes y) \triangleq (\varphi \otimes \psi) \mapsto \varphi(x)\psi(y)$. The following composite

$$ (X \hat{\otimes} Y)_c \xrightarrow{\iota_c} ((X^* \tilde{\otimes} Y^*)^*)_c \cong ((X^* \tilde{\otimes} Y^*)_c)^* = ((X^*)_c \tilde{\otimes} (Y^*)_c)^* \cong ((X_c)^* \tilde{\otimes} (Y_c)^*)^* \xrightarrow{\iota^{-1}} X_c \hat{\otimes} Y_c $$

consists of completely isometric isomorphisms. Evaluating on elementary tensors

$$ \begin{align*} (x \otimes y) &\xrightarrow{\iota_c} ((\varphi \otimes \psi) \mapsto \varphi(x) \cdot \psi(y)) \\ &\xrightarrow{\cong} ((\varphi \otimes \psi) \mapsto \overline{\varphi(x) \cdot \psi(y)}) \\ &\xrightarrow{\cong} ((\varphi \otimes \psi) \mapsto \overline{\overline{\varphi(x)} \cdot \overline{\psi(y)}}) \\ &= ((\varphi \otimes \psi) \mapsto \varphi(x) \cdot \psi(y)) \\ &\xrightarrow{\iota^{-1}} (x \otimes y) \end{align*} $$
shows that this composite is the identity map and the equality follows.

**Case** $(X \hat{\otimes} Y)_o = X_o \hat{\otimes} Y_o$. This can be proven analogously to the previous case. The following composite

$$ (X \hat{\otimes} Y)_o \xrightarrow{\iota_o} ((X^* \hat{\otimes} Y^*)^)_o = ((X_o)^* \hat{\otimes}(Y_o)^*)^* \xrightarrow{\iota^{-1}} X_o \hat{\otimes} Y_o $$

consists of completely isometric isomorphisms. After evaluating on elementary tensors we see that the composite is the identity map and the equality follows.

**Case** $(X \hat{\oplus} Y)_o = X_o \hat{\oplus} Y_o$. Clearly, the identity function gives a linear isomorphism id: $(X \hat{\oplus} Y)_o \cong X_o \hat{\oplus} Y_o$. To show that it is completely isometric, let $[(x_{ij}, y_{ij})] \in M_n((X \hat{\oplus} Y)_o)$. We verify:

$$ \begin{aligned} \|[(x_{ij}, y_{ij})]\|_{M_n((X \hat{\oplus} Y)_o)} &= \|[(x_{ji}, y_{ji})]\|_{M_n((X \hat{\oplus} Y))} \\ &= \|([x_{ji}], [y_{ji}])\|_{M_n(X) \hat{\oplus} M_n(Y)} \\ &= \max(\|[x_{ji}]\|_{M_n(X)}, \|[y_{ji}]\|_{M_n(Y)}) \\ &= \max(\|[x_{ij}]\|_{M_n(X_o)}, \|[y_{ij}]\|_{M_n(Y_o)}) \\ &= \|([x_{ij}], [y_{ij}])\|_{M_n(X_o) \hat{\oplus} M_n(Y_o)} \\ &= \|[(x_{ij}, y_{ij})]\|_{M_n(X_o \hat{\oplus} Y_o)} \end{aligned} $$

where each equality holds by definition of the relevant (matrix) norm.

**Case** $(X \hat{\oplus} Y)_c = X_c \hat{\oplus} Y_c$. The proof is fully analogous to the previous case. The identity function a linear isomorphism id: $(X \hat{\oplus} Y)_c \cong X_c \hat{\oplus} Y_c$. To show that it is completely isometric, one can simply follow the same steps as in the previous proof while using the o.s.s of the conjugated spaces.

**Case** $(X \hat{\oplus} Y)_c = X_c \hat{\oplus} Y_c$. Let $\iota_{X,Y}: X \hat{\oplus} Y \cong (X^* \hat{\oplus} Y^*)^*$ be the canonical completely isometric isomorphism that defines the o.s.s of $X \hat{\oplus} Y$. It is given by $\iota_{X,Y}(x, y) = ((\varphi, \psi) \mapsto \varphi(x) + \psi(y))$. The following composite

$$ (X \hat{\oplus} Y)_c \xrightarrow{\iota_c} ((X^* \hat{\oplus} Y^*)^)_c \cong ((X^* \hat{\oplus} Y^*)^)_c^* \cong ((X_c)^* \hat{\oplus} (Y_c)^*)^* \xrightarrow{\iota^{-1}} X_c \hat{\oplus} Y_c $$

consists of completely isometric isomorphisms. Evaluating on elements

$$ \begin{aligned} (x, y) &\xrightarrow{\iota_c} ((\varphi, \psi) \mapsto \varphi(x) + \psi(y)) \\ &\xrightarrow{\cong} ((\varphi, \psi) \mapsto \overline{\varphi(x) + \psi(y)}) \\ &\xrightarrow{\cong} ((\varphi, \psi) \mapsto \overline{\varphi(x) + \overline{\psi(y)}}) \\ &= ((\varphi, \psi) \mapsto \varphi(x) + \psi(y)) \\ &\xrightarrow{\iota^{-1}} (x, y) \end{aligned} $$

shows that this composite is the identity map and the equality follows.
$\mu: A \otimes A \to A$. This linear map is actually a complete contraction with respect to the Haagerup tensor, i.e. $\mu: A \otimes A \to A$ is a complete contraction [15, Proof of Theorem 17.1.2]. The involution $i: A_c \to A_o$ is defined in the obvious way $i(a) \triangleq a^*$ and this is a completely isometric isomorphism [31, p. 65]. The commutativity of diagrams (involution) and (reverse multiplicative) follows easily from Definition 27 (3.). From Definition 27 (1.) and (2.) it follows that $(A, \eta, \mu)$ is a monoid object. Finally, any (complete) isometry $a: C \to A$ may be identified with an element $a \in A$ with $\|a\| = 1$ and then the composite in $(C^*-\text{id})$ is identified with the element $a^*a \in A$. The requirement that this composite is a complete isometry means that $\|a^*a\| = 1$ which follows from the $C^*-\text{id}$-identity of $A$, i.e. Definition 27 (4.).

($\Leftarrow$). This direction is easier. It is obvious how to define the unit, multiplication and involution from $\eta$, $\mu$ and $i$. Conditions (1.) – (3.) of Definition 27 now follow easily. The $C^*$-norm of $A$ is simply the ground norm of $A$. That this norm is submultiplicative follows from [15, Proof of Theorem 17.1.2]. It only remains to show that the $C^*$-identity holds. Let $a \in A$. If $a = 0$, then obviously $\|a^*a\| = \|\mu(0 \otimes 0)\| = 0 = \|a\|^2$, so let us assume that $a \neq 0$. We can now use a simple renormalisation argument to complete the proof. Let $b = \frac{a}{\|a\|}$ so that $\|b\| = 1$. We can now identify $b$ with a (complete) isometry $b: C \to A$ and we know that the composite

$$C \cong C_o^h \otimes C_c \xrightarrow{b \otimes b} A_o^h \otimes A_c \xrightarrow{A_o \otimes i} A_o^h \otimes A_o \xrightarrow{\gamma} (A^h \otimes A)_o \xrightarrow{\mu} A_o$$

is also a complete isometry. But this composite may be identified with the element $b^*b$ and therefore $\|b^*b\| = 1$. The last equation now implies $\|a^*a\| = \|a\|^2$, as required. $\square$

**Proposition 32.** Given $A, B \in \text{vNAlg}$, then the operator space $A \bar{\otimes} B$ with

$$- \text{unit } \eta \triangleq (C \cong C \bar{\otimes} C \xrightarrow{\eta_A \otimes \eta_B} A \bar{\otimes} B)$$

$$- \text{involution } i \triangleq ((A \bar{\otimes} B)_c = A_c \bar{\otimes} B_c \xrightarrow{i_A \otimes i_B} A_o \bar{\otimes} B_o = (A \bar{\otimes} B)_o)$$

$$- \text{multiplication } \mu \triangleq ((A \bar{\otimes} B)^h \bar{\otimes} (A \bar{\otimes} B) \xrightarrow{v} (A^h \bar{\otimes} A) \bar{\otimes} (B^h \bar{\otimes} B) \xrightarrow{\mu_A \otimes \mu_B} A \bar{\otimes} B)$$

is also a $\text{vN-algebra}$. Also, the operator space $A \bar{\otimes} B$ is a $\text{vN-algebra}$ with

$$- \text{unit } \eta \triangleq (C \xrightarrow{\langle \eta_A, \eta_B \rangle} A \bar{\otimes} B)$$

$$- \text{involution } i \triangleq ((A \bar{\otimes} B)_c = A_c \bar{\otimes} B_c \xrightarrow{i_A \oplus i_B} A_o \bar{\otimes} B_o = (A \bar{\otimes} B)_o)$$

$$- \text{multiplication } \mu \triangleq ((A \bar{\otimes} B)^h \bar{\otimes} (A \bar{\otimes} B) \xrightarrow{\langle \mu_A \circ (\pi_A \bar{\otimes} \pi_A), \mu_B \circ (\pi_B \bar{\otimes} \pi_B) \rangle} A \bar{\otimes} B)$$

*Proof.* We prove both statements simultaneously, saving the $C^*$-identities for last. To prove the first statement, we start by ascertaining that $(A \bar{\otimes} B, \eta, \mu)$ is indeed a monoid. To do so we keep Remark 58 in mind.
We can verify the left and right unital laws:

<figure><img src="image_6.png" /></figure>

The two inner squares commute by naturality of $v$, the lower triangles by assumption and outer squares by duoidity axioms (see (9)).
The multiplication is associative:

<figure><img src="image_7.png" /></figure>

The upper square commutes because FdOS is a BV-category (see (7)). The two
diamonds on the sides commute by naturality of $v$ and the pentagon at the bot-
tom by assumption on $A$ and $B$ being von Neuman algebras.

Now for the vN-algebra specific coherences. We have that $i$ satisfies the involution
condition:

<figure><img src="image_8.png" /></figure>

The involution $i$ is also reverse multiplicative:

<figure><img src="image_9.png" /></figure>
The coherence diagrams for $A \underset{\sim}{\oplus} B$ are analogous as the binary product is automatically duoidal to any monoidal structure (see [2, Example 6.19]). To give an indication of the proof, note that the interchange law for this duoidal structure is given by the universal property of the product:

$$v' \triangleq \langle \pi_A \otimes \pi_C, \pi_B \otimes \pi_D \rangle : (A \underset{\sim}{\oplus} B) \overset{h}{\otimes} (C \underset{\sim}{\oplus} D) \to (A \overset{h}{\otimes} C) \underset{\sim}{\oplus} (B \overset{h}{\otimes} D)$$

and that the unit and multiplication for $A \underset{\sim}{\oplus} B$ can be factored in the following way:

<figure><img src="image_10.png" /></figure>

For the $C^*$-identity of the product, note that given f.d operator spaces $A$ and $B$ and a c.c map $u = \langle a, b \rangle : \mathbb{C} \to A \underset{\sim}{\oplus} B$ then $u$ is a c.i morphism if and only if $a$ or $b$ is a c.i map. Let $u = \langle a, b \rangle : \mathbb{C} \to A \underset{\sim}{\oplus} B$ be a c.i. map, we know by naturality and the universal property of the product that the following diagram commutes:

<figure><img src="image_11.png" /></figure>

Hence it suffices for either

$$C \cong C_o \overset{h}{\otimes} C_c \xrightarrow{a \otimes a} A_o \overset{h}{\otimes} A_c \xrightarrow{A_o \otimes i} A_o \overset{h}{\otimes} A_o \xrightarrow{\gamma} (A \overset{h}{\otimes} A)_o \xrightarrow{\mu} A_o$$

or

$$C \cong C_o \overset{h}{\otimes} C_c \xrightarrow{b \otimes b} B_o \overset{h}{\otimes} B_c \xrightarrow{B_o \otimes i} B_o \overset{h}{\otimes} B_o \xrightarrow{\gamma} (B \overset{h}{\otimes} B)_o \xrightarrow{\mu} B_o$$
to be a c.i map. We know that either *a* or *b* is c.i, hence by assumption we have
that the $C^*$-identity holds for $A \overset{\infty}{\oplus} B$.

For $C^*$-identity of the tensor, recall that any f.d vN-algebra is isomorphic to
a direct sum of the following form: $\bigoplus_{1 \le i \le n} M_{k_i}$. Now since $\tilde{\otimes}$ distributes over $\bigoplus$, it is sufficient to prove that the $C^*$-identity holds on $M_n \tilde{\otimes} M_m \cong M_{nm}$. It is straightforward to check that the c.i.i. $M_n \tilde{\otimes} M_m \cong M_{nm}$ is multiplicative, unital and preserves the involution. Suppose that $u : \mathbb{C} \to M_n \tilde{\otimes} M_m$ is a c.i. map. Then $r \triangleq (\mathbb{C}^u \to M_n \tilde{\otimes} M_m \cong M_{nm})$ is a c.i. map too. The following diagram commutes:

<figure><img src="image_12.png" /></figure>

We know that $M_{nm}$ is a vN-algebra, hence the right side of the diagram is a c.i map. Therefore the left side must be c.i too, because the horizontal arrows are c.i.i. Therefore $M_n \tilde{\otimes} M_m$ also satisfies the $C^*$-identity.

□

**Proposition 33.** The category vNAlg is symmetric monoidal and has finite products with the action of $\tilde{\otimes}$ and $\bigoplus$ extended as in Proposition 32.

*Proof.* We know from Proposition 32 that the objects in vNAlg are closed under the injective tensor. Suppose that $f : A \to C, g : B \to D$ are morphisms in vNAlg. The following diagrams:

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>
<figure><img src="image_15.png" /></figure>

commute, so the functor $\tilde{\mathfrak{A}} : \mathfrak{vNAlg} \times \mathfrak{vNAlg} \to \mathfrak{vNAlg}$ is well-defined. Next, we need to verify that the associator and unitors of $\tilde{\mathfrak{A}}$ are indeed morphisms in $\mathfrak{vNAlg}$.

We begin with the associator. Let $A$, $B$ and $C$ be f.d vN-algebras and $\alpha$ be the associator of $\tilde{\mathfrak{A}}$. To show that $\alpha$ is multiplicative, consider the following diagram.

<figure><img src="image_16.png" /></figure>

The upper square commutes as FdOS is a BV-category and the lower one by naturality of $\alpha$. The two diagrams below commute by naturality of $\alpha$ (modulo rewriting some equalities).

<figure><img src="image_17.png" /></figure>

This shows that the associator $\alpha$ is also a natural transformation in $\mathfrak{vNAlg}$. We proceed similarly for the left unitor $\lambda$.

<figure><img src="image_18.png" /></figure>
The upper square commutes by the BV-assumption, and the lower one by nat-
urity of $\lambda$. Note that $\mu_C = \lambda_C$. We proceed similarly for unitality and involu-
tivity:

<figure><img src="image_19.png" /></figure>

This shows that the left unitor $\lambda$ is also a natural transformation in $\mathbf{vNAlg}$. We
can show that the right unitor $\rho$ is also a natural transformation in $\mathbf{vNAlg}$ using
similar arguments. Clearly the symmetries are in $\mathbf{vNAlg}$ as they are natural and
commute with $v$.

Finally, in order to see that $\mathbf{vNAlg}$ is symmetric monoidal, we note that the inclusion functor $\mathbf{vNAlg} \hookrightarrow \mathbf{FdOS}$ is faithful and preserves the tensor $\tilde{\otimes}$ strictly, so the necessary coherence diagrams also commute in $\mathbf{vNAlg}$. $\square$

**Lemma 65.** Let $A, B \in \text{Ob}(\mathbf{vNAlg})$ and let $f: A \to B$ be a morphism in $\mathbf{vNAlg}$. Then $f$ is a CPU map in the sense of Section 4.

*Proof.* Let $f: A \to B$ be a morphism of $\mathbf{vNAlg}$. This map is unital by assumption, so we only have to show it is completely positive. Suppose that $p: \mathbb{C} \to M_n \tilde{\otimes} A$ is a positive element of $M_n \tilde{\otimes} A$, and let $a: \mathbb{C} \to M_n \tilde{\otimes} A$ be such that

<figure><img src="image_20.png" /></figure>

Recall that $M_n$ is a vN-algebra, so by Proposition 33, the map $M_n \otimes f: M_n \tilde{\otimes} A \to M_n \tilde{\otimes} B$ is multiplicative and preserves the involution, hence the following diagram commutes in FdOS :

<figure><img src="image_21.png" /></figure>
The outer edges of the diagram give us a factorization of $(M_n \otimes f) \circ p$ into the product of $(M_n \otimes f) \circ a$ and its involution. □

**Proposition 35.** The category **H** is symmetric monoidal (w.r.t. $\tilde{\otimes}$), has finite products (w.r.t. $\oplus$), both preserved via the inclusions $\mathbf{vNAlg} \hookrightarrow \mathbf{H} \hookrightarrow \mathbf{FdOS}$.

*Proof.* We know from Proposition 32 that the objects in **H** are closed under the injective tensor, and that the tensor of unital maps is unital. By functoriality of the injective tensor with respect to c.c. maps and as c.p.u maps are equivalent to c.c.u maps by Proposition 34, the injective tensor product is indeed a functor $\tilde{\otimes} : \mathbf{H} \times \mathbf{H} \to \mathbf{H}$. The proof of Proposition 33 shows that that the associator, left/right unitors, and symmetry of $\tilde{\otimes}$ are unital maps, and as they are c.c., they are also c.p. maps.

To see that $\mathbf{H}$ is symmetric monoidal w.r.t $\tilde{\otimes}$, note that the inclusion functor $\mathbf{H} \hookrightarrow \mathbf{FdOS}$ is faithful and preserves $\tilde{\otimes}$ strictly, so the necessary coherence diagrams also commute in $\mathbf{H}$.

Finally, it follows by construction that the inclusion functor strictly preserves the symmetric monoidal structure and products. □

## B.2 Duality between von Neumann algebras and von Neumann coalgebras

Let us recall the definition of a vN-coalgebra.

**Definition 66.** A f.d. vN-coalgebra is an operator space $C$ together with

1. a complete contraction $\delta: C \to C \otimes_C^h C$, called comultiplication;

2. a complete contraction $\varepsilon: C \to \mathbb{C}$, called counit;

3. a complete contraction $j: C_o \to C_c$, called involution,

such that

- $(C, \varepsilon, \delta)$ is a comonoid object in $(\mathbf{FdOS}, \mathbb{C}, \otimes_C^h)$;

- $j$ is an involution, meaning that the following diagram commutes:

$$ \begin{tikzcd} C \arrow[r, "C"] \arrow[d, "\parallel"] & C \arrow[d, "\parallel"] \\ C_{oo} \arrow[r, "\xrightarrow{j_o} C_{co} = C_{oc} \xrightarrow{j_c} C_{cc}"] & C_{cc} \arrow[r, "C_{cc}", "(\text{involution})"] \arrow[d, "\parallel"] \end{tikzcd} $$

- $j$ is reverse comultiplicative, meaning that the following diagram commutes:

$$ \begin{tikzcd} C_o \arrow[r, "j"] \arrow[d, "\delta"] & C_c \arrow[r, "\delta"] \arrow[d, "(C \otimes_C^h C)_c"] & (\text{reverse comultiplicative}) \arrow[d, "\parallel"] \\ (C \otimes_C^h C)_o \arrow[r, "\gamma", "(\text{reverse comultiplicative})"] & C_o \otimes_C^h C_o \arrow[r, "j \otimes j"] & C_c \otimes_C^h C_c \arrow[r, "C_{cc}", "(\text{reverse comultiplicative})"] \arrow[d, "\parallel"] \end{tikzcd} $$
- for every complete quotient map (i.e. strong epi) $e: C \to \mathbb{C}$, the following composite in **FdOS** is a complete quotient map (i.e. strong epi)

$$C_o \xrightarrow{\delta} (C \otimes_C C)_o \xrightarrow{\gamma} C_o \otimes_C C_o \xrightarrow{C_o \otimes j} C_o \otimes_C C_c \xrightarrow{e \otimes e} C_o \otimes_C C_c \cong C. \quad (\text{co-C}^*-\text{id})$$

**Remark 67.** The definition above is the same as the one in Section 5 only slightly expanded to give the coherence conditions names.

**Proposition 43.** If $A \in \mathbf{vNAlg}$, then the o.s. dual $A^*$ is a vN-coalgebra with counit $\varepsilon \triangleq (A^* \xrightarrow{\eta^*} \mathbb{C}^* \cong \mathbb{C})$, involution $j \triangleq ((A^*)_o = (A_o)^* \xrightarrow{i^*} (A_c)^* \cong (A^*)_c)$, and comultiplication $\delta \triangleq (A^* \xrightarrow{\mu^*} (A \otimes_A A)^* \cong A^* \otimes_A A^*)$. Conversely, if $C \in \mathbf{vNCoalg}$, then the o.s. dual $C^*$ is a vN-algebra with unit $\eta \triangleq (\mathbb{C} \cong \mathbb{C}^* \xrightarrow{\varepsilon^*} C^*)$, involution $i \triangleq ((C^*)_c \cong (C_c)^* \xrightarrow{j^*} (C_o)^* = (C^*)_o)$, and multiplication given by $\mu \triangleq (C^* \otimes_A C^* \cong (C \otimes_A C)^* \xrightarrow{\delta^*} C^*)$.

**Proof.** Given a $A: \mathbf{vNAlg}$, clearly $A^*$ is a comonoid. For the remaining structure we can easily check that the following diagrams commute. We have that $j$ satisfies the involution coherence diagram:

<figure><img src="image_22.png" /></figure>

It is also reverse comultiplicative:

<figure><img src="image_23.png" /></figure>
C*-identity: Let $a : A^* \to \mathbb{C}$ be a c.q. map. By Remark 11, $a^* : \mathbb{C} \to A$ is a c.i. map. We know that the following diagram commutes:

<figure><img src="image_24.png" /></figure>

As $A$ is a $vN$-algebra, the top morphism is the dual of a c.i. map, i.e. a c.q. map, thus the bottom morphism is a c.q. map too.

We can prove that the dual of a $vN$-coalgebra is a $vN$-algebra in a similar way. $\square$

**Proposition 68.** We have the following duality between the morphisms in $vNAlg$ and $vNCoalg$ (dual taken in $FdOS$):

- Suppose that $f : A \to B \in vNAlg$ then $f^* : B^* \to A^* \in vNCoalg$;

- Suppose that $g : C \to D \in vNCoalg$ then $g^* : D^* \to C^* \in vNAlg$.

*Proof.* We prove the first statement as the second one is proved completely analogously. Let $f : A \to B \in vNAlg$, then the following diagrams commute

<figure><img src="image_25.png" /></figure>

The lower squares all commute by assumption, and upper ones by naturality. $\square$

**Proposition 69.** The dual on $vN$-algebras and $vN$-coalgebras as defined in Proposition 43 extends to functors

$$ (-)^* : vNCoalg \rightleftharpoons vNAlg^{\text{OP}} : (-)^* . $$

*Proof.* Proposition 68 assures us that both functors are well-typed. $\square$

**Lemma 70.** The functors $(-)^* : vNCoalg \rightleftharpoons vNAlg^{\text{OP}} : (-)^*$ give rise to:

- a natural isomorphism $d : \text{Id} \cong (-)^{**} : vNAlg \to vNAlg$;

- a natural isomorphism $d : \text{Id} \cong (-)^{**} : vNCoalg \to vNCoalg$;

and both of these isomorphisms coincide with the double-dual natural isomorphism $d : \text{Id} \cong (-)^{**} : \text{FdOS} \to \text{FdOS}$ under the obvious forgetful functors.
For the second statement, note that the following diagrams commute:

$$ \begin{tikzcd} C \underset{d \otimes d}{\overset{h}{\otimes}} C \arrow[r, "d"] & (C \underset{d \otimes d}{\overset{h}{\otimes}} C)^{**} \arrow[d, "\cong"] \\ C^{**} \underset{\cong}{\overset{h}{\otimes}} C^{**} \arrow[r, "\cong"] & (C^* \underset{\cong}{\overset{h}{\otimes}} C^*)^* \arrow[d, "\cong"] \arrow[dl, "(d)_c"] \arrow[dr, "(C^{**})_c"] \arrow[dl, "\cong"] \arrow[dr, "((C^*)_c)^*"'] \end{tikzcd} \quad (13) $$

The rest of the proof is analogous. □

**Theorem 71.** We have an equivalence of categories $(-)^*$: vNCoalg $\simeq$ vNAlg$^{OP}$: $(-)^*$, where the action on objects of $(-)^*$ is defined as in Proposition 43 and on morphisms in the usual way, i.e. as in FdOS.

*Proof.* Follows by combining Proposition 69 and Lemma 70. □

## B.3 von Neumann coalgebras

Most of this section will be dedicated to prove the following proposition:

**Proposition 38.** Given $C, D \in \text{vNCoalg}$, then the operator space $C \hat{\otimes} D$ with

$$ - \text{ counit } \varepsilon \triangleq (C \hat{\otimes} D \xrightarrow{\varepsilon_C \hat{\otimes} \varepsilon_D} C \hat{\otimes} C \cong C) $$

$$ - \text{ involution } j \triangleq ((C \hat{\otimes} D)_o = C_o \hat{\otimes} D_o \xrightarrow{j_C \hat{\otimes} j_D} C_c \hat{\otimes} D_c = (C \hat{\otimes} D)_c) $$

$$ - \text{ comult. } \delta \triangleq (C \hat{\otimes} D \xrightarrow{\delta_C \hat{\otimes} \delta_D} (C \hat{\otimes} C) \hat{\otimes} (D \hat{\otimes} D) \xrightarrow{w} (C \hat{\otimes} D) \hat{\otimes} (C \hat{\otimes} D)) $$

is also a vN-coalgebra. Also, the operator space $C \overset{1}{\oplus} D$ is a vN-coalgebra with

$$ - \text{ counit } \varepsilon \triangleq (C \overset{1}{\oplus} D \xrightarrow{[\varepsilon_C, \varepsilon_D]} C) $$

$$ - \text{ involution } j \triangleq ((C \overset{1}{\oplus} D)_o = C_o \overset{1}{\oplus} D_o \xrightarrow{j_C \overset{1}{\oplus} j_D} C_c \overset{1}{\oplus} D_c = (C \overset{1}{\oplus} D)_c) $$

$$ - \text{ comultiplication } \delta \triangleq (C \overset{1}{\oplus} D \xrightarrow{[(\iota_C \hat{\otimes} \iota_C)^* \circ \delta_C, (\iota_D \hat{\otimes} \iota_D)^* \circ \delta_D]} (C \overset{1}{\oplus} D)^h \hat{\otimes} (C \overset{1}{\oplus} D)) $$

In order to prove this we want to leverage the equivalence between vNCoalg $\simeq$ vNAlg$^{OP}$ proved in the previous section, thus we need the following two lemmas.

**Lemma 72.** The c.i. natural isomorphism:

$$ \theta : C \hat{\otimes} D \cong (C^* \hat{\otimes} D^*)^* \quad (14) $$

in FdOS, preserves the comonoid structure and the involution as defined above for $C \hat{\otimes} D$.
*Proof.* We can verify that the following diagrams commute.

The c.i. natural isomorphism $\theta$ is comultiplicative:

<figure><img src="image_26.png" /></figure>

The upper square commutes by naturality, and the lower one by the duality between the two interchange maps $v$ and $w$ in a BV-category with negation.

It is counital:

<figure><img src="image_27.png" /></figure>
And it preserves the involution:

<figure><img src="image_28.png" /></figure>

We can check that the lower square commutes by reducing to elementary tensors and leveraging that the double dual commutes with conjugation (see (13)). □

**Lemma 73.** *The c.i. natural isomorphism:*

$$C \overset{1}{\oplus} D \cong (C^* \overset{\infty}{\oplus} D^*)^* \qquad (15)$$

in FdOS, preserves the comonoid structure and the involution as defined above for $C \overset{1}{\oplus} D$.

*Proof.* The proof of this is analogous to the proof of Lemma 72. □

*Proof (Proposition 38).* By Proposition 43 we have that $C^*$ and $D^*$ are both vN-algebras, and by Proposition 32 we know $C^* \tilde{\otimes} D^*$ is a vN-algebra. Applying Proposition 43 gives us that $(C^* \tilde{\otimes} D^*)^*$ is a vN-coalgebra. Now, by Lemma 72 $(C \hat{\otimes} D, \varepsilon, j, \delta)$ is a comonoid $(C \hat{\otimes} D, \varepsilon, \delta)$, where $j$ is an involution that is reverse comultiplicative. It remains to show that the co-C$^*$-identity is satisfied. To do so, suppose that $e: C \hat{\otimes} D \to \mathbb{C}$ is a c.q. map, then $r \triangleq ( (C^* \tilde{\otimes} D^*)^* \xrightarrow{\theta^{-1}} C \hat{\otimes} D \xrightarrow{e} \mathbb{C} )$ is also a c.q. map. Now the following diagram commutes:
<figure><img src="image_29.png" /></figure>

As $(C^* \tilde{\otimes} D^*)^*$ is a vN-coalgebra the right side is c.q, hence the left one is too. The result for the coproduct follows similarly, using Lemma 73. $\square$

**Lemma 74.** Let $C, D \in \text{vNCoalg}$ and let $g : C \to D$ be a morphism in $\text{vNCoalg}$. Then $g$ is a CPTP map in the sense of Section 5.

*Proof.* This proof is analogous to the proof of Lemma 65. $\square$

<figure><img src="image_30.png" /></figure>

## B.4 Duality between H and S

**Lemma 75.** Let $A \in \text{vNAlg}$ and $p : \mathbb{C} \to A$ be a c.b map, then the following are equivalent:

1. $p$ is a positive element of $A$;

2. $p^*$ is a positive linear functional on $A^*$.

*Proof.* We prove that 1. implies 2. (the other direction is proved completely analogously). Let $p : \mathbb{C} \to A$ be a positive element of $A$, this means that we have some $a : \mathbb{C} \to A$ such that the following diagram

<figure><img src="image_31.png" /></figure>
commutes. The following diagram shows that $p^*$ is factorized as required.

<figure><img src="image_32.png" /></figure>

□

**Proposition 76.** We have the following duality between c.p. maps on $vN$-algebras and c.p. maps on $vN$-coalgebras:

- Suppose that $A, B \in \mathbf{vNAlg}$ and that $f: A \to B$ is a completely positive map. Then $f^*: B^* \to A^*$ is a c.p. map between $vN$-coalgebras.

- Suppose that $C, D \in \mathbf{vNCoalg}$ and that $g: C \to D$ is a completely positive map. Then $g^*: D^* \to C^*$ is a c.p. map between $vN$-algebras.

*Proof.* We prove the first statement as the second one is proved completely analogously. Let $q: T_n \hat{\otimes} A^* \to \mathbb{C}$ be a positive linear functional. By Lemma 75, $q$ is the dual of a positive element $p: \mathbb{C} \to (T_n \hat{\otimes} A^*)^* \cong M_n \hat{\otimes} A$. By assumption, $(M_n \otimes f) \circ p: \mathbb{C} \to M_n \hat{\otimes} B$ is positive as well, i.e. it can be factorised using an element $b: \mathbb{C} \to M_n \hat{\otimes} B$ and its involution. Let $r$ denote the linear functional $r \triangleq (T_n \hat{\otimes} B^* \cong (M_n \hat{\otimes} B)^* \xrightarrow{b^*} \mathbb{C})$, then the following diagram commutes:

<figure><img src="image_33.png" /></figure>

Following the diagram above the linear functional $q \circ (T_n \otimes f^*) : T_n \hat{\otimes} B^* \to \mathbb{C}$ is factorized by the comultiplication into the linear functional $r$ and its involution.

□
**Proposition 77.** The dual on vN-algebras and vN-coalgebras as defined in Proposition 43 extends to functors

$$(-)^*: S \rightleftarrows H^{\text{op}}: (-)^*.$$

*Proof.* We know from Proposition 68 that the dual sends unital maps to counital/trace-preserving ones and vice-versa. Proposition 76 now assures us that the functors are well-typed. □

**Theorem 78.** We have an equivalence of categories $(-)^*: S \simeq H^{\text{op}}: (-)^*$ where the action on objects of $(-)^*$ is defined as in Proposition 43 and on morphisms in the usual way, i.e. as in FdOS.

*Proof.* As any vN-algebra morphism is also CPU (Lemma 65) and any vN-coalgebra morphism is also CPTP (Lemma 74), the double dual natural isomorphisms (Lemma 70) extend to natural isomorphisms:

$$-d: \text{Id} \cong (-)^{**}: H \to H;$$

$$-d: \text{Id} \cong (-)^{**}: S \to S.$$

□

**Proposition 39.** The category vNCoalg is symmetric monoidal and has finite coproducts with the action of $\otimes$ and $\oplus$ extended as in Proposition 38.

*Proof.* Recall that an equivalence of categories where one of the categories has a symmetric monoidal structure, induces a symmetric monoidal structure on the other category. We know that vNCoalg $\simeq$ vNAlg$^{\text{op}}$ by Theorem 71, and that vNAlg is monoidal by Proposition 33. Lemma 72 now assures us that the induced tensor is naturally isomorphic to the one given by $\otimes$ in Proposition 38. We can make a similar argument for the coproducts. □

**Proposition 42.** The category S is symmetric monoidal (w.r.t. $\hat{\otimes}$), has finite coproducts (w.r.t. $\oplus$), both preserved via inclusions vNCoalg $\hookrightarrow$ S $\hookrightarrow$ FdOS.

*Proof.* This follows by a similar argument to the proof of Proposition 39, using the equivalence S $\simeq$ H$^{\text{op}}$ from Theorem 78 and the monoidal structure on H (see Proposition 35) instead. The coproducts follows by the same argument too.

It now follows by construction that the inclusion functor preserves the monoidal structure and coproducts. □

**Proposition 79.** The equivalences $(-)^*: S \simeq H^{\text{op}}: (-)^*$ and $(-)^*: vNCoalg \simeq vNAlg^{\text{op}}: (-)^*$ are strong monoidal.

*Proof.* Follows by Lemma 72. □

*Example 40.* The operator space $T_n$ is a vN-coalgebra with: (1) counit given by the trace functional $\varepsilon \triangleq \text{tr}: T_n \to \mathbb{C}$; (2) involution given by the conjugate transpose $j \triangleq (\cdot)^{\dagger}: T_n^o \to T_n^c$; (3) comultiplication given by
$\delta: T_n \to T_n \otimes^h T_n :: e_{ij} \mapsto \sum_k (e_{kj} \otimes e_{ik})$, where $e_{ij} \in T_n$ is the matrix that has 1 in position $(i, j)$ and 0 elsewhere. Using Proposition 38, we see that $\oplus_{1 \le i \le n} T_{k_i}$ is also a vN-coalgebra and using Theorem 44 and Example 28 it follows that each f.d. vN-coalgebra is of this form, modulo vN-coalgebra isomorphism.

*Proof.* For the first part of the statement, recall that the canonical vN-algebra structure on $M_n$, has:

1. unit given by $\eta \triangleq 1_n : \mathbb{C} \to M_n :: \lambda \mapsto \lambda 1_n$

2. involution given by the conjugate transpose $i \triangleq (-)^{\dagger} : M_n^c \to M_n^o$

3. multiplication given by matrix multiplication $\mu : M_n \otimes^h M_n \to M_n :: a \otimes b \mapsto ab$

Recall also that we have the isomorphism $T_n \cong (M_n)^* :: a \mapsto \text{tr}(a(-))$. By Proposition 43, $(M_n)^*$ has a vN-coalgebra structure. Taking this structure modulo the isomorphism $T_n \cong (M_n)^*$ gives us

1. count:

$$ 
\begin{align*}
T_n &\cong (M_n)^* \xrightarrow{\eta^*} \mathbb{C}^* \cong \mathbb{C} \\
a &\xrightarrow{\cong} \text{tr}(a(-)) \\
\xrightarrow{\cong \circ \eta^*} \text{tr}(a1_n) &= \text{tr}(a)
\end{align*}
$$

2. involution:

$$ 
\begin{align*}
T_n^o &\cong ((M_n)^)_o = (M_n^o)^* \xrightarrow{((-)^\dagger)^*} (M_n^c)^* \cong ((M_n)^)_c \cong T_n^c \\
a &\xrightarrow{\cong} \text{tr}(a(-)) \\
\xrightarrow{\cong} ((-)^\dagger)^* &\to \text{tr}(a(-)^\dagger) \\
\xrightarrow{\cong} \xrightarrow{\cong} &\to \text{tr}(a(-)^\dagger) = \text{tr}((a(-)^\dagger)^\dagger) = \text{tr}((-)a^\dagger) = \text{tr}(a^\dagger(-)) \\
\xrightarrow{\cong} &\to a^\dagger
\end{align*}
$$

3. comultiplication

$$ 
\begin{align*}
T_n &\cong (M_n)^* \xrightarrow{\mu^*} (M_n \otimes^h M_n)^* \cong (M_n)^* \otimes^h (M_n)^* \cong T_n \otimes^h T_n \\
e_{ij} &\xrightarrow{\cong} \text{tr}(e_{ij}(-)) \\
\xrightarrow{\mu^*} &\to (a \otimes b \mapsto \text{tr}(e_{ij}ab) = \sum_k a_{jk}b_{ki} = \sum_k \text{tr}(e_{kj}a)\text{tr}(e_{ik}b)) \\
\xrightarrow{\cong} &\to \sum_k \text{tr}(e_{kj}(-)) \otimes \text{tr}(e_{ik}(-)) \\
\xrightarrow{\cong} &\to \sum_k (e_{kj} \otimes e_{ik})
\end{align*}
$$

For the second part of the statement, let $C \in \text{vNCoalg}$. By Proposition 43, $C^* \in \text{vNAlg}$ and we get the following chain of natural isomorphisms in $\text{vNCoalg}$

$$ C \cong C^{**} \qquad (\text{by Lemma 70}) $$

$$ \cong (\bigoplus_i M_{n_i})^* \qquad (\text{by Example 28}) $$

$$ \cong \bigoplus_i (M_{n_i})^* \qquad (\text{de Morgan duality}) $$

$$ \cong \bigoplus_i T_{n_i} \qquad (\text{by argument above}) $$

□
C von Neumann coalgebras and (complete) positivity

The purpose of this appendix is to explain that the definitions of positive linear functionals and (completely) positive maps between vN-coalgebras from §5 co-
incide with the usual concrete definitions of these notions on the spaces $\oplus_i T_{k_i}$,
which give all the f.d. vN-coalgebras modulo isomorphism. We refer to the vN-
coalgebras of the form $\oplus_i T_{k_i}$ as concrete vN-coalgebras.

Recall that an element $p \in \mathbb{M}_n$ is called positive iff $\langle h, ph \rangle \ge 0$ for every $h \in \mathbb{C}^n$ where we have written $\langle \cdot, \cdot \rangle: \mathbb{C}^n \times \mathbb{C}^n \to \mathbb{C}$ for the standard inner product on $\mathbb{C}^n$. Moreover, $p \in \mathbb{M}_n$ is positive iff there exists $a \in \mathbb{M}_n$ such that $p = a^\dagger a$. An element $(p_1, \dots, p_n) \in \oplus_{1 \le i \le n} \mathbb{M}_{k_i}$ is positive iff each $p_i \in \mathbb{M}_{k_i}$ is positive.

**Notation.** Throughout the remainder of this section, we simply write $V$ and $W$ to refer to two (potentially different) vector spaces of the form $\oplus_{1\le i\le n}\mathbb{M}_{k_i}$.

Note that the vector spaces $V$ and $W$ can be equipped with the structure of f.d. vN-(co)algebras, as we have already explained in the main body of the document. Moreover, all f.d. vN-(co)algebras have underlying vector spaces of this form, modulo isomorphism. We say that a linear map $f: V \to W$ is *positive* (in the concrete sense) if $f$ maps positive elements of $V$ to positive elements of $W$. A *positive linear functional* (in the concrete sense) is a positive map $f: V \to \mathbb{C}$. From the theory of $C^*$-algebras, we have the following proposition.

**Proposition 80.** Let $f: V \to W$ be a linear map. The following are equivalent:

- $f$ is positive;

- $f$ reflects positive linear functionals, i.e. for each positive linear functional $p: W \to \mathbb{C}$, the linear functional $p \circ f: V \to \mathbb{C}$ is also positive.

Proof. (⇒) This is obvious.

($\Leftarrow$) This follows easily from [27, Theorem 4.3.4]. In fact, the statement holds more generally for infinite-dimensional $C^*$-algebras.

Our first objective is to show that the positivity of a linear functional in the coalgebraic sense is equivalent to positivity in the concrete sense. We begin with a lemma.

**Lemma 81.** Let $\varphi: T_n \to \mathbb{C}$ be a linear functional on the indicated vN-coalgebra.
The following are equivalent:

- $\varphi$ is positive in the coalgebraic sense, i.e. there exists a linear functional $\psi: T_n \to \mathbb{C}$, such that the following diagram:

$$
\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
(T_n \otimes_T T_n)^o \arrow[r, "\gamma"] & T_n^o \otimes_T T_n^o \arrow[r, "T_n^o \otimes_j T_n^o"] & T_n^o \otimes_T T_n^c \arrow[r, "\psi \otimes \psi"] & \mathbb{C}_o \otimes_T \mathbb{C}_c \arrow[r, "\simeq"] & \mathbb{C} \\
\bigwedge^\delta \arrow[r, "T_n^o"] & \mathbb{C}_o & & & \\
\arrow[d, "\varphi"'] & & & &
\end{tikzcd}
\qquad (16)
$$

commutes.
– $\varphi$ is positive in the concrete sense.

*Proof.* (⇒) Let $\psi: T_n \to \mathbb{C}$ be a linear functional. It follows $\psi = \text{tr}(a\cdot)$ for some $a \in T_n$. A diagram chase along the upper leg of the diagram shows that $e_{ij} \in T_n$ is mapped to

$$ 
\begin{align*}
e_{ij} &\xrightarrow{\delta} \sum_k e_{kj} \otimes e_{ik} \\
&\xrightarrow{\gamma} \sum_k e_{ik} \otimes e_{kj} \\
&\xrightarrow{T_n^o \otimes j} \sum_k e_{ik} \otimes e_{jk} \\
&\xrightarrow{\psi \otimes \psi} \sum_k \text{tr}(e_{ik}a) \otimes \text{tr}(e_{jk}a) \\
&\xrightarrow{\cong} \sum_k \overline{\text{tr}(e_{ik}a)\text{tr}(e_{jk}a)} \\
&= \sum_k a_{ki} \overline{a_{kj}}
\end{align*}
$$

By taking the linear extension of the above composite and using the commutativity of the above diagram, it follows that

$$ \varphi(b) = \sum_{i,j,k} a_{ki} \overline{a_{kj}} b_{ij} $$

for any $b = [b_{ij}] \in T_n$. Straightforward verification now shows that

$$ \varphi(b) = \sum_{i,j,k} a_{ki} \overline{a_{kj}} b_{ij} = \text{tr}(a^\dagger b a) $$

and therefore $\varphi = \text{tr}((a^\dagger a) \cdot)$ is a positive linear functional in the concrete sense.

(←) Let $\varphi: T_n \to \mathbb{C}$ be a positive linear functional in the concrete sense. It follows that $\varphi = \text{tr}(p \cdot)$ for some positive element $p \in T_n$ in the concrete sense. This is equivalent to $\varphi = \text{tr}((a^\dagger a) \cdot)$ for some element $a \in T_n$. Now we can define $\psi: T_n \to \mathbb{C}$ by $\psi \triangleq \text{tr}(a \cdot)$ and the same reasoning as above shows that the required diagram commutes. $\square$

Therefore, for linear functionals $\varphi: T_n \to \mathbb{C}$, the abstract (coalgebraic) and concrete notions of positivity coincide, so we do not distinguish them any further.

Next, we extend this equivalence for linear functionals $\varphi: \bigoplus_{1 \le i \le n}^1 T_{k_i} \to \mathbb{C}$. Every such functional is necessarily of the form $\varphi = [\varphi_1, \dots, \varphi_n]$ where we have $\varphi_i: T_{k_i} \to \mathbb{C}$ and $\varphi(t_1, \dots, t_n) = \sum_i \varphi_i(t_i)$.

**Lemma 82.** Let $\varphi: \bigoplus_{1 \le i \le n}^1 T_{k_i} \to \mathbb{C}$ be a linear functional. Then $\varphi$ is positive in the coalgebraic sense iff each $\varphi_i: T_{k_i} \to \mathbb{C}$ (from the above decomposition) is positive.
**Proof.** For simplicity, let us assume that we have only two entries in the direct sum, i.e. we have a linear functional $\varphi: T_n \oplus^1 T_m \to \mathbb{C}$. The general case is analogous.

(⇒) Let $\varphi$ be positive in the coalgebraic sense. By performing a diagram chase in (6) on elements of the form $(t, 0) \in T_n \oplus T_m$ we see that $\varphi_1$ is positive. The argument for $\varphi_2$ is symmetric.

( ⇐ ) If both $\varphi_1$ and $\varphi_2$ are positive, then there exist $t_1 \in T_n$ and $t_2 \in T_m$ such that $\varphi_1 = \text{tr}(t_1^\dagger t_1)$ and $\varphi_2 = \text{tr}(t_2^\dagger t_2)$. We can now define a linear functional $a: T_n \oplus^1 T_m \to \mathbb{C}$ by $a(x_1, x_2) = \text{tr}(t_1 x_1) + \text{tr}(t_2 x_2)$. To verify that diagram (6) commutes, it suffices to perform the diagram chase on elements of the form $(e_{ij}, 0)$ and $(0, e_{lp})$. In both of these cases, the diagram chase easily reduces to the diagram chase in the proof of the preceding lemma and the commutativity holds using the same arguments.

**Proposition 83.** Let $\varphi: \bigoplus_{1 \le i \le n} T_{k_i} \to \mathbb{C}$ be a linear functional. Then, $\varphi$ is positive in the coalgebraic sense iff $\varphi$ is positive in the concrete sense.

**Proof.** By the previous lemma, it suffices to show that $\varphi$ is positive in the concrete sense iff each $\varphi_i: T_{k_i} \to \mathbb{C}$ is positive. This is obviously true. $\square$

Therefore, for concrete vN-coalgebras, the positivity of linear functionals in the abstract (coalgebraic) and concrete senses coincide.

**Theorem 84.** Let $f: C_1 \to C_2$ be a linear map between two concrete f.d. vN-coalgebras. Then:

1. $f$ is positive in the abstract (coalgebraic) sense iff $f$ is positive in the concrete sense;

2. $f$ is completely positive in the abstract (coalgebraic) sense iff $f$ is completely positive in the concrete sense.

**Proof.** The first statement follows by combining Propositions 83 and 80. The second statement now immediately follows from the first one, because the linear map $T_n \otimes \varphi: T_n \otimes C_1 \to T_n \otimes C_2$, from the coalgebraic definition of complete positivity, is precisely the same linear map as $M_n \otimes \varphi: M_n \otimes C_1 \to M_n \otimes C_2$ in the concrete definition of complete positivity. $\square$

It now easily follows that an element $p \in C$ of a concrete f.d. vN-coalgebra is positive in the coalgebraic sense (see §7) iff $p$ is positive in the concrete sense.

To summarise, all the notions of (complete) positivity that we have introduced in the coalgebraic sense coincide with the standard concrete definitions of (complete) positivity when we consider concrete f.d. vN-coalgebras.

Our next objective is to show that the set of density operators on a vN-coalgebra $C$ may be characterised in two equivalent ways:

$$ \{c \in C \mid \|c\| = 1 = \varepsilon(c)\} = \{\rho \in C \mid \varepsilon(\rho) = 1 \text{ and } \rho \ge 0\} = P_C. $$
Towards this end, we prove the following proposition. Its dual analogue for vN-algebras/C*-algebras is well-known, so we use a simple duality argument to establish our next proposition.

**Proposition 85.** Let $C$ be a f.d. vN-coalgebra and let $\rho \in C$ be an element with $\varepsilon(\rho) = 1$. The following are equivalent:

1. $\rho \ge 0$, i.e. $\rho$ is positive;

2. $\|\rho\| = 1$;

3. $\|\rho\| \le 1$.

*Proof.* Consider the linear map $p: \mathbb{C} \to C$ defined by $p(1) \triangleq \rho$. Since $\varepsilon(\rho) = 1$, the map $p$ is counital (i.e. trace-preserving).

(1) $\implies$ (2). The map $p$ is positive (by definition). The proof of Proposition 76 shows that $C^* \xrightarrow{p^*} C^* \cong \mathbb{C}$ is positive (simple special case) and it is clearly unital. From the theory of C*-algebras, we know that every positive unital functional $\varphi$ has norm $\|\varphi\| = 1 = \|\varphi\|_{cb}$ [27, Theorem 4.3.2] (and is also completely positive). The o.s. dual $(-)^*$ preserves the cb-norm of $p$, so $1 = \|p^*\|_{cb} = \|p\|_{cb} = \|p\| = \|\rho\|$.

(2) $\implies$ (3). Obvious.

(3) $\implies$ (1). The map $p$ is a (necessarily complete) contraction and also trace-preserving. By Proposition 41 it follows that $p$ is completely positive and therefore also positive, so $p(1) = \rho$ is positive (by definition). $\square$

**Corollary 86.** Let $C$ be a f.d. vN-coalgebra. Then the set of density operators $P_C$ may be defined in two equivalent ways:

$$P_C = \{\rho \in C \mid \varepsilon(\rho) = 1 \text{ and } \rho \ge 0\} = \{\rho \in C \mid \|\rho\| = 1 = \varepsilon(\rho)\} \quad (17)$$
# D Background on double gluing and omitted proofs from Section 7

In this section we revisit the hom-functor double gluing construction for classical linear logic from [24].

**Definition 87 (Glued category, [24, Sec. 4.3.2]).** Given a *-autonomous category $(\mathbf{C}, I, \otimes, [-, -])$ the (hom-functor) double glued category $G(\mathbf{C})$ has

- objects: $(c, S, V)$, with $c$ an object in $\mathbf{C}$ and $S \subseteq \mathbf{C}(I, c)$ and $R \subseteq \mathbf{C}(c, \perp)$

- morphisms: $f: (c, S, V) \to (d, R, W)$, where $f: c \to d$ in $\mathbf{C}$ and for all $s \in S$ we have $f \circ s \in R$ and for all $w \in W$ we have $w \circ f \in V$

*Remark 88.* When we do double gluing along the hom-functor, we can see the second argument in an object as a set of “proofs” of the formula $A$ and the third argument as a set of “proofs” of the formula $\neg A$ or “counter-proofs”.

**Proposition 89 (Glued category, [24, Prop. 41]).** The category $G(\mathbf{C})$ is
*-autonomous, with

- Negation: $(c, S, V)^* \triangleq (c^*, V, S)$

- Tensor: $(c, S, V) \otimes (d, R, W) \triangleq (c \otimes d, S \otimes R, G(\mathbf{C})((c, S, V), (d, R, W)^*))$
  where $S \otimes R \triangleq \{s \otimes u \mid s \in S, u \in U\}$, with unit: $(I, \{id_I\}, \mathbf{C}(I, \perp))$

This *-autonomous structure is not very meaningful, as we have no interaction between the “proofs” and “counter-proofs”, hence the dual is trivial. Note also that the monoidal structure adds a lot of extra data to the third argument. We thus need to carve out a subcategory of “meaningful” objects. To impose a restriction on objects we want the "proofs" to interact with the "counter-proofs", we thus use the notion of focussed orthogonality from [24].

**Definition 90 (focussed orthogonality).** Let $\mathbf{C}$ be a *-autonomous category, then a focussed orthogonality $\perp$ on $\mathbf{C}$ is a family of relations $\{\perp_c \in \mathbf{C}(I, c) \times \mathbf{C}(c, \perp)\}$ induced by $\perp \subseteq \mathbf{C}(I, \perp)$ meaning that

$$u \perp_c f \iff f \circ u \in \perp$$

We say that a focussed orthogonality induced by $\perp \subseteq \mathbf{C}(I, \perp)$ has focus on $\perp$.

**Proposition 91 (focussed precise, [24, Sec. 5.3]).** Let $\perp$ be a focussed orthogonality on $\mathbf{C}$, then $\perp$ is precise, meaning it satisfies

- For all $u: I \to c$, $v: I \to d$ and $f: c \otimes d \to \perp$

$$u \perp \langle v | f \rangle \quad \text{and} \quad v \perp \langle u | f \rangle \iff u \otimes v \perp f$$

where

$$\langle v | f \rangle \triangleq (c \cong c \otimes I \xrightarrow{v} c \otimes d \xrightarrow{f} \perp)$$

$$\langle u | f \rangle \triangleq (d \cong I \otimes d \xrightarrow{u} c \otimes d \xrightarrow{f} \perp)$$
For all $u: I \to c, v: d \to \perp$ and $f: c \to d$

$$f \circ u \perp v \quad \text{and} \quad u \perp v \circ f \iff f \perp u \to v$$

We introduce some notation, let $S \subseteq \mathbf{C}(I, c)$, $R \subseteq \mathbf{C}(I, d)$ and $V \subseteq \mathbf{C}(c, \perp)$, then

$$S \otimes R \triangleq \{s \otimes r \mid s \in S \text{ and } r \in R\} \subseteq \mathbf{C}(I, c \otimes d)$$

$$R \to V \triangleq \{r \to v \mid r \in R \text{ and } v \in V\} \subseteq \mathbf{C}(d \to c, \perp)$$

The polar set of a subset $S \subseteq \mathbf{C}(I, c)$ can now be defined as $S^\circ \triangleq \{f \in \mathbf{C}(c, \perp) \mid \forall s \in S \ s \perp f\}$. The set $S^\circ$ can be seen as the maximal subset where all elements are orthogonal to $S$. It has the following properties:

**Lemma 92 ([24, Sec. 5.1]).** For an $R \subseteq S \subseteq \mathbf{C}(I, c)$ we have that $S \subseteq S^{\circ\circ}$, $S^{\circ\circ\circ} = S^\circ$ and $S^\circ \subseteq R^\circ$

**Definition 93 (focussed orthogonal category, [24, Sec. 5.1]).** Let $\mathbf{C}$ be a symmetric monoidal closed category and $\perp \subseteq \mathbf{C}(I, c)$, the focussed orthogonal category denoted $\mathcal{O}_\perp(\mathbf{C})$ has

- objects: $(c, S, V)$, where $c \in \text{ob}(\mathbf{C})$, $S \subseteq \mathbf{C}(I, c)$, $V \subseteq \mathbf{C}(c, \perp)$ such that $S = V^\circ$ and $V = S^\circ$

- morphisms: $f: (c, S, V) \to (d, R, W)$, where $f: c \to d \in \text{mor}(\mathbf{C})$, such that $f \circ s \in R$ and $w \circ f \in V$

The condition that $S = V^\circ$ and $V = S^\circ$ is equivalent to $S = S^{\circ\circ}$, we can thus see the objects of $\mathcal{O}_\perp(\mathbf{C})$ as pairs $(c, S^{\circ\circ})$ instead of triples.

**Proposition 94 (stable).** A focussed orthogonality $\perp$ on $\mathbf{C}$ is stable, that is, for all $S \subseteq \mathbf{C}(I, c)$, $R \subseteq \mathbf{C}(I, d)$ and $W \subseteq \mathbf{C}(d, \perp)$

$$-(S^{\circ\circ} \otimes R^{\circ\circ})^\circ = (S \otimes R^{\circ\circ})^\circ = (S^{\circ\circ} \otimes R)^\circ$$

$$-(R^{\circ\circ} \to W^{\circ\circ})^\circ = (R \to W^{\circ\circ})^\circ = (R^{\circ\circ} \to W)^\circ$$

*Proof.* – For a focussed orthogonality we have the following equivalence

$$s \perp \langle r | f \rangle \iff r \perp \langle s | f \rangle \quad (18)$$

as the two composites below are equal

$$I \xrightarrow{s} S \otimes I \xrightarrow{S \otimes r} S \otimes R \xrightarrow{f} \perp$$

$$I \xrightarrow{r} R \otimes I \xrightarrow{s \otimes R} S \otimes R \xrightarrow{f} \perp$$

First equality: By Lemma 92 it is sufficient to prove that $(S \otimes R^{\circ\circ})^\circ \subseteq (S^{\circ\circ} \otimes R^{\circ\circ})^\circ$. Suppose $f \in (S \otimes R^{\circ\circ})^\circ$, by preciseness (Proposition 91) we have that $\langle r | f \rangle \in S^\circ$ for all $r \in R^{\circ\circ}$. Consequently we have $s \perp \langle r | f \rangle$ for all $s \in S^{\circ\circ}$ and $r \in R^{\circ\circ}$, hence by the equivalence above $f \in (S^{\circ\circ} \otimes R^{\circ\circ})^\circ$. The second equality can be proved in a similar way.
- For a focussed orthogonality

$$f \circ r \perp w \iff r \perp w \circ f$$

as $w \circ (f \circ u) = (w \circ f) \circ u \in \perp$.

First equality: By Lemma 92 it is sufficient to prove $(R \to W^{\circ\circ})^{\circ} \subseteq (R^{\circ\circ} \to W^{\circ\circ})^{\circ}$. Suppose $f \in (R \to W^{\circ\circ})^{\circ}$, then by preciseness $w \circ f \in R^{\circ}$ for all $w \in W^{\circ\circ}$. Consequently we have $r \perp w \circ f$ for all $r \in R^{\circ\circ}$ and $w \in W^{\circ\circ}$, hence by preciseness $f \in (R^{\circ\circ} \to W^{\circ\circ})^{\circ}$. The second equality can be proved in a similar way.

□

**Theorem 95 ([24, Prop. 62]).** Let $\mathbf{C}$ be a $*-\text{autonomous category and } \perp \subseteq \mathbf{C}(I, \perp)$, then the induced focussed orthogonal category $\mathcal{O}_{\perp}(\mathbf{C})$ is $*-\text{autonomous}$, with

- Tensor: $(c, S) \hat{\otimes}(d, R) \triangleq (A \hat{\otimes} B, (S \otimes U)^{\circ\circ})$ with unit: $(I, \{id_I\}^{\circ\circ})$, where $S \otimes U \triangleq \{s \otimes u \mid s \in S, u \in U\}$

- Negation: $(c, S)^* \triangleq (c^*, S^\circ)$

**Proposition 96 (focussed morphism, [24, Sec. 5.2]).** Let $\mathbf{C}$ be a $*-\text{autonomous category with an orthogonality } \perp \subseteq \mathbf{C}(I, \perp)$ and $S \subseteq \mathbf{C}(I, c)$, $V \subseteq \mathbf{C}(d, \perp)$, then any $f: c \to d$ is focussed (w.r.t $S$ and $V$), meaning that

$$f \circ s \perp v \iff s \perp v \circ f \quad \forall s \in S \quad \forall v \in V$$

**Proposition 97 (co)products.** Let $\mathbf{C}$ be a $*-\text{autonomous category with an orthogonality } \perp \subseteq \mathbf{C}(I, \perp)$

- Suppose $\mathbf{C}$ is cartesian, then $\mathcal{O}_{\perp}(\mathbf{C})$ is cartesian. The binary products are explicitly defined as

$$(c, S) \times (d, R) \triangleq (c \times d, S \times R)$$

where $S \times R \triangleq \{\langle s, r \rangle : I \to c \times d \mid s \in S, r \in R\}$ and the terminal object is $(1, \mathbf{C}(I, 1))$.

- Suppose $\mathbf{C}$ is cocartesian, then $\mathcal{O}_{\perp}(\mathbf{C})$ is cocartesian. The binary coproducts are explicitly defined as

$$(c, S) + (d, R) \triangleq (c + d, (S^{\circ} + R^{\circ})^{\circ})$$

where $S^{\circ} + R^{\circ} \triangleq \{[f, g] : c + d \to \perp \mid f \in S^{\circ}, g \in R^{\circ}\}$ and the initial object is $(0, \mathbf{C}(0, \perp)^{\circ})$.

*Proof.* By [24, Prop. 63] it is sufficient to show that projections (injections) are focussed. We know by Proposition 96 that all morphisms in a focussed orthogonal category are focussed, hence projections (injections) are too. □

Now we have that $\mathbf{Q}$ is the focussed orthogonal category of $G(\mathbf{FdOS})$ with focus $\{1\} \subseteq \mathbf{FdOS}(\mathbb{C}, \mathbb{C}) = \text{Ball}(\mathbb{C})$.
**Theorem 48.** The category **Q** is *-autonomous and has finite (co)products: duals are given by $(X, S)^* \triangleq (X^*, S^\circ)$; the initial object by $(0, \emptyset)$; the terminal object by $(0, \{0\})$; binary products by $(X, S) \underset{\sim}{\oplus} (Y, R) \triangleq (X \underset{\sim}{\oplus} Y, S \times R)$; binary coproducts by $(X, S) \underset{\oplus}{\oplus} (Y, R) \triangleq (X \underset{\oplus}{\oplus} Y, (S^\circ + R^\circ)^\circ)$, where

$$S^\circ + R^\circ \triangleq \{[s', r'] : X \underset{\oplus}{\oplus} Y \to \mathbb{C} \mid s' \in S^\circ, r' \in R^\circ\} \subseteq \text{Ball}((X \underset{\oplus}{\oplus} Y)^*);$$

multiplicative conjunction is given by $(X, S) \widehat{\otimes}(Y, R) \triangleq (X \widehat{\otimes} Y, (S \otimes R)^{\circ\circ})$, where $S \otimes R \triangleq \{s \otimes r \mid s \in S, r \in R\}$; multiplicative disjunction by $(X, S) \widetilde{\otimes}(Y, R) \triangleq (X \widetilde{\otimes} Y, (S^\circ \otimes R^\circ)^\circ)$; tensor unit for both tensors is given by $(\mathbb{C}, \{1\})$.

*Proof.* Any focussed orthogonality is *precise* by Proposition 91 and *stable* by Proposition 94. Since **FdOS** is *-autonomous*, it follows that **Q** is *-autonomous* as well with the given structure by Theorem 95. We also know that **FdOS** has finite products, thus **Q** has finite products (and consequently also coproducts) as given by Proposition 97. □
## E Bipolar subsets in $\mathbb{Q}$

Our notion of polarity is a strong condition which implies that non-trivial polar sets must be subsets of the unit sphere.

**Proposition 98.** Given an operator space $X$ and a non-empty subset $S \subseteq \text{Ball}(X)$, if $S^\circ \neq \emptyset$, then $\forall s \in S$. $\|s\| = 1$ and $\forall s' \in S^\circ$. $\|s'\| = 1$.

*Proof.* Let $s \in S$ and $s' \in S^\circ$. By definition, $s'(s) = 1$ and therefore $1 = \|s'(s)\| \le \|s'\|\|s\| \le 1$, because $s$ and $s'$ are in the unit balls of $X$ and $X^*$. This obviously implies that $\|s\| = \|s'\| = 1$.

We recall some facts from functional analysis. For a matrix $a \in \mathbb{M}_n$, we write $\|a\|$ for its operator norm, i.e. its norm as an element $a \in \mathbb{M}_n$. We write $\|a\|_{\text{tr}}$ for its trace norm, which is simply its norm as an element $a \in \mathbb{T}_n$.

**Lemma 99.** Let $a, b \in \mathbb{M}_n$ be two matrices and $u \in \mathbb{M}_n$ a unitary matrix. Then:

1. $\|u\| = 1$.

2. $\|u\|_{\text{tr}} = n$.

3. $|\text{tr}(ab)| \le \|a\|_{\text{tr}}\|b\|$.

*Proof.* The first two statements follow easily from the definitions. The last statement is a special case of [12, p. 267].

**Lemma 100.** Let $X$ be a strictly convex normed space and let $x_1, \dots, x_n \in X$ with $\|x_i\| = 1$ be $n$ points on the unit sphere. Assume further that $\|\sum_{k=1}^n x_k\| = n$. Then $x_1 = x_2 = \dots = x_n$.

*Proof.* A strictly convex normed space is characterised by the property [23, p. 263]:

$$ \text{if } \|x + y\| = \|x\| + \|y\| \text{ and } x \neq 0, \text{ then } y = cx \text{ for some } c \ge 0. $$

The lemma clearly holds for $n = 0$ or $n = 1$. The rest follows via a simple inductive argument. If the lemma is true for $n = k \ge 1$, then to prove it for $n = k + 1$, we reason as follows. Let $y = \sum_{i=1}^k x_i$. Then

$$ k + 1 = \|y + x_{k+1}\| \le \|y\| + \|x_{k+1}\| \le k + 1. $$

Therefore $\|y + x_{k+1}\| = \|y\| + \|x_{k+1}\|$ and also $\|y\| = k$. From the latter and the induction hypothesis, it follows $x_1 = \dots = x_k$. From the former and strict convexity, it follows $x_{k+1} = cy$ for some $c \ge 0$. It follows $c = \frac{\|x_{k+1}\|}{\|y\|} = \frac{1}{k}$ and therefore $x_1 = \dots = x_{k+1}$. $\square$

**Lemma 101.** Let $i: X \to Y$ be an isomorphism in $\mathbf{FdOS}$. If $(X, U)$ is an object of $\mathbb{Q}$, then $(Y, i[U])$ is also an object of $\mathbb{Q}$, where $i[U] \triangleq \{i(u) | u \in U\}$ is the image of $U$ under $i$.

*Proof.* This can be proven by straightforward verification.
## E.1 Bipolar sets relevant for von Neumann algebras

We now proceed to identify important bipolar sets that we can associate to f.d. vN-algebras in $\mathbf{Q}$.

**Lemma 102.** Let $X$ be an operator space whose underlying vector space is $M_n$ and such that for every $a \in M_n$ we have $\|a\|_{M_n} \le \|a\|_X$. Assume that $(X, S) \in \text{Ob}(\mathbf{Q})$, i.e. $S$ is bipolar. If $S^\circ \subseteq \text{Ball}(X^*)$ contains an element of the form $\frac{1}{n}\text{tr}(u^* \cdot)$, where $u \in M_n$ is a unitary matrix, then $S \subseteq \{u\}$.

*Proof.* Let $g \in S = S^{\circ\circ}$ be an arbitrary element. By definition of $S^{\circ\circ}$ we have that $\text{tr}(\frac{1}{n}u^*g) = 1$ and therefore $\text{tr}(u^*g) = n$. Let $\{e_k\}_{k=1}^n$ be an orthonormal basis of $\mathbb{C}^n$. Then

$$ 
\begin{aligned}
n &= \text{tr}(u^*g) \\
  &= \sum_{k=1}^{n} \langle u^* g e_k, e_k \rangle \quad (\text{definition of tr}) \\
  &= \sum_{k=1}^{n} \langle g e_k, u e_k \rangle \quad (\text{adjoint})
\end{aligned}
$$

and therefore $n = |\sum_{k=1}^n \langle g e_k, u e_k \rangle|$. The Cauchy-Schwarz inequality gives for every $1 \le k \le n$:

$$ |\langle g e_k, u e_k \rangle| \le \|g e_k\| \|u e_k\| = \|g e_k\| \le \|g\|_{M_n} \|e_k\| \le \|g\|_X \le 1 \quad (19) $$

Writing $c_k \triangleq \langle g e_k, u e_k \rangle \in \mathbb{C}$, we therefore have that $0 \le |c_k| \le 1$ and

$$ n = \left| \sum_{k=1}^{n} c_k \right| \le \sum_{k=1}^{n} |c_k| \le n $$

and therefore $|c_k| = 1$ for every $1 \le k \le n$. It follows that $c_1 = c_2 = \dots = c_n = 1$ by Lemma 100. Returning to (19), we have that

$$ 1 = |\langle g e_k, u e_k \rangle| = \|g e_k\| \|u e_k\| $$

and by applying the Cauchy-Schwarz argument again (for the equality) we get that

$$ g e_k = \frac{\langle g e_k, u e_k \rangle}{\|u e_k\|^2} u e_k = u e_k $$

for every $1 \le k \le n$. Therefore $g = u$. $\square$

**Lemma 103.** Let $u \in M_n$ be a unitary element with $n \in \mathbb{N}$. Then the pair $(M_n, \{u\})$ is an object of $\mathbf{Q}$.
Proof. If $n = 0$, then $M_0 \cong 0$ and the unique (zero) element $0 \in M_0$ is unitary, because it is also the identity element. The pair $(M_0, \{0\})$ is the terminal object of $\mathbf{Q}$ by Theorem 48.

If $n > 0$, then let $U \triangleq \{u\}$. Consider the pair $(M_n^*, U^o)$. By definition, we have that

$$U^o = \{f: M_n \to \mathbb{C} \mid f(u) = 1\} \subseteq \mathbf{FdOS}(M_n, \mathbb{C}).$$

It follows that $\mathrm{tr}(\frac{1}{n}u^*) \in U^o$, where $u^*$ is the conjugate transpose (and inverse) of $u$. Indeed, we have that $\mathrm{tr}(\frac{1}{n}u^*u) = \frac{1}{n}\mathrm{tr}(1_n) = 1$ and to see that $\mathrm{tr}(\frac{1}{n}u^*)$ is a (necessarily complete) contraction, we verify that if $b \in M_n$, then

$$ \begin{aligned} \left| \mathrm{tr} \left( \frac{1}{n} u^* b \right) \right| &= \frac{1}{n} \left| \mathrm{tr} (u^* b) \right| \\ &\le \frac{1}{n} \|u^*\|_{\mathrm{tr}} \|b\| \qquad (\text{Lemma 99}) \\ &\le \|b\| \qquad (\text{Lemma 99}) \end{aligned} $$

We always have that $U \subseteq U^{\circ\circ}$ and Lemma 102 gives the other direction. $\square$

**Proposition 104.** Let $M$ be a f.d. $vN$-algebra and $u \in M$ a unitary element, i.e. $uu^* = 1_M = u^*u$. Then $(M, \{u\}) \in \mathrm{Ob}(\mathbf{Q})$.

Proof. From Example 28, we have a $vN$-algebra isomorphism $M \cong \bigoplus_{1 \le i \le n}^{\infty} M_{k_i}$ for some $n \in \mathbb{N}$ and $k_i \ge 1$. This isomorphism preserves unitaries and therefore by Lemma 101, it suffices to show that

$$ \left( \bigoplus_{1 \le i \le n}^{\infty} M_{k_i}, \{u\} \right) $$

is an object of $\mathbf{Q}$ for any unitary $u \in \bigoplus_{1 \le i \le n}^{\infty} M_{k_i}$. The unitary $u$ is of the form

$$u = (u_1, \dots, u_n)$$

with $u_i \in M_{k_i}$ and such that each $u_i$ is a unitary in $M_{k_i}$. By Lemma 103 each pair $(M_{k_i}, \{u_i\})$ is an object of $\mathbf{Q}$ and by Theorem 48 we have that

$$ \left( \bigoplus_{1 \le i \le n}^{\infty} M_{k_i}, \{(u_1, \dots, u_n)\} \right) \in \mathrm{Ob}(\mathbf{Q}) $$

is the categorical product of the objects $(M_{k_i}, \{u_i\})$. $\square$

It is now easy to prove that we have a fully faithful inclusion $H: \mathbf{H} \xrightarrow{\text{full}} \mathbf{Q}$.

**Theorem 49.** If $A \in \mathrm{vNAlg}$ is a $vN$-algebra, then $(A, \{1_A\}) \in \mathrm{Ob}(\mathbf{Q})$. Moreover, the functor $H: \mathbf{H} \to \mathbf{Q}$ defined by $H(A) \triangleq (A, \{1_A\})$ and $H(f) \triangleq f$, is fully faithful, strict monoidal w.r.t $\otimes$, and it strictly preserves finite products.
**Proof.** By the previous proposition, we see that the functor $H$ is well-defined on objects. Using Proposition 34, it follows that $H$ is well-defined on morphisms. Faithfulness is obvious and fullness follows by Proposition 34 (again) because any morphism $f: (A, \{1_A\}) \to (B, \{1_B\})$ in $\mathbf{Q}$ is clearly a unital complete contraction and therefore completely positive.

Theorem 48 shows immediately that $H$ strictly preserves products. It remains to show that $H$ strictly preserves $\tilde{\otimes}$. We show this first for vN-algebras of the form $M_n$. This amounts to proving that $S \triangleq (\{1_n\}^\circ \otimes \{1_m\}^\circ)^\circ = \{1_n \otimes 1_m\}$ when viewed as subsets of $M_n \tilde{\otimes} M_m$. To see that $1_n \otimes 1_m \in S$, let

$$ \varphi \otimes \psi \in (\{1_n\}^\circ \otimes \{1_m\}^\circ). $$

Then, by definition $\varphi(1_n) = 1 = \psi(1_m)$ and so $(\varphi \otimes \psi)(1_n \otimes 1_m) = 1$, as required. To see that $S \subseteq \{1_n \otimes 1_m\}$, observe that we have linear functionals $\frac{1}{n} \text{tr} \in \{1_n\}^\circ$ and $\frac{1}{m} \text{tr} \in \{1_m\}^\circ$, which we know from the proof of Lemma 103. The functional $\frac{1}{n} \text{tr} \otimes \frac{1}{m} \text{tr}$ may now be identified with $\frac{1}{nm} \text{tr}$ via the vN-algebra isomorphism $M_n \tilde{\otimes} M_m \cong M_{nm}$ and the proof of Lemma 103 shows that the only element $g \in \text{Ball}(M_{nm})$ that satisfies $\text{tr}(g) = nm$ is $g = 1_{nm}$ which corresponds to $1_n \otimes 1_m$ via the isomorphism $M_{nm} \cong M_n \tilde{\otimes} M_m$, as required.

Therefore $H(M_n) \tilde{\otimes} H(M_m) = H(M_n \tilde{\otimes} M_m)$. If $A = M_n \tilde{\oplus} M_k$, then

$$ \begin{align*} H(A \tilde{\otimes} M_m) & \xrightarrow{H(\cong)} H((M_n \tilde{\otimes} M_m) \tilde{\oplus} (M_k \tilde{\otimes} M_m)) \\ &= H(M_n \tilde{\otimes} M_m) \tilde{\oplus} H(M_k \tilde{\otimes} M_m) \\ &= (H(M_n) \tilde{\otimes} H(M_m)) \tilde{\oplus} (H(M_k) \tilde{\otimes} H(M_m)) \\ &\xrightarrow{\cong^{-1}} (H(M_n) \tilde{\oplus} H(M_k)) \tilde{\otimes} H(M_m) \\ &= H(A) \tilde{\otimes} H(M_m) \end{align*} $$

and since the action of $H$ on morphisms is trivial and the MALL structure of $\mathbf{Q}$ coincides with that of $\mathbf{FdOS}$ (under the forgetful functor), it follows that the above composite is the identity map. This argument can be obviously generalised and symmetrised and it follows $H(A) \tilde{\otimes} H(B) = H(A \tilde{\otimes} B)$ for all concrete vN-algebras. The result for all (abstract) f.d. vN-algebras follows by using Example 28 and similar arguments. $\square$

## E.2 Bipolar sets relevant for von Neumann coalgebras

We now turn our attention to the inclusion of f.d. vN-coalgebras in $\mathbf{Q}$.

**Proposition 105.** Let $C$ be a finite-dimensional von Neumann coalgebra with counit $\varepsilon: C \to \mathbb{C}$ and let $S_C \triangleq \{c \in C \mid \|c\| = 1 = \varepsilon(c)\}$. Then $(C, S_C) \in \text{Ob}(\mathbf{Q})$. Moreover, the double dual isomorphism $d: C \cong C^{**} : \text{Id} \to \text{vNCoalg}$ lifts to an isomorphism of $\mathbf{Q}$ in the following way

$$ d: (C, S_C) \cong (C^{**}, \{\varepsilon\}^\circ) = (C^*, \{\varepsilon\})^*, $$

with $(C^*, \{\varepsilon\}) = H(C^*)$ the canonical inclusion of the vN-algebra $C^*$.
*Proof.* By Theorem 44 we know that $C^*$ is a vN-algebra with unit $\varepsilon$. Proposition 104 shows that $(C^*, \{\varepsilon\}) \in \text{Ob}(\mathbf{Q})$ and Theorem 48 shows that $(C^{**}, \{\varepsilon\}^\circ)$ is also an object of $\mathbf{Q}$. By definition

$$ \{\varepsilon\}^{\circ} = \{f \in \text{Ball}(C^{**}) \mid f(\varepsilon) = 1\}. $$

But for $f \in \{\varepsilon\}^\circ$, we have $1 = \|f(\varepsilon)\| \le \|f\|\|\varepsilon\| \le \|f\| \le 1$ and therefore $f(\varepsilon) = 1 = \|f\|_{C^{**}}$. Let

$$ V \triangleq \{f \in C^{**} \mid f(\varepsilon) = 1 = \|f\|_{C^{**}}\} = \{\varepsilon\}^{\circ}. $$

We have a completely isometric isomorphism $d: C \cong C^{**} :: c \mapsto (\varphi \mapsto \varphi(c))$ and by Lemma 101, it follows that $(C, d^{-1}[V])$ is an object of $\mathbf{Q}$. It is easy to see that

$$ d^{-1}[V] = \{c \in C \mid \varepsilon(c) = 1 = \|c\|\} = S_C $$

which completes the proof. $\square$

*Remark 106.* Note that when $\dim(C) = 0$, the above proposition shows that $(0, \emptyset) \in \text{Ob}(\mathbf{Q})$. In fact, this is the initial object of $\mathbf{Q}$ (Theorem 48).

**Corollary 107.** If $C$ is a f.d. vN-coalgebra, then $(C, P_C) \in \text{Ob}(\mathbf{Q})$.

*Proof.* Combine Proposition 105 and Corollary 86. $\square$

We can also show that the inclusion functors $H$ and $S$ behave well with respect to the duality.

**Proposition 108.** Let $A$ be a f.d. vN-algebra and $C$ a f.d. vN-coalgebra. Then $H(A)^* = S(A^*)$ and $S(C)^* = H(C^*)$.

*Proof.* We verify

$$
\begin{align*}
H(A)^* &= (A, \{1_A\})^* \\
&= (A^*, \{1_A\}^\circ) \\
&= (A^*, \{f \in \text{Ball}(A^*) \mid f(1_A) = 1\}) \\
&= (A^*, \{f \in A \mid f(1_A) = 1 = \|f\|\}) \\
&= (A^*, P_{A^*}) \tag*{(Corollary 86)} \\
&= S(A^*)
\end{align*}
$$

and for the other statement we have

$$
\begin{align*}
S(C)^* &= (C, P_C)^* \\
&= (C^*, P_C^\circ) \\
&= (C^*, \{\varepsilon\}) \tag*{(Proposition 105)} \\
&= H(C^*)
\end{align*}
$$

which completes the proof. $\square$
In order to prove that we have a fully faithful inclusion $S: \mathbf{S} \xrightarrow{\text{full}} \mathbf{Q}$, we prove an extra proposition first.

**Proposition 109.** Let $f: (C, P_C) \to (D, P_D)$ be a morphism in $\mathbf{Q}$, where $C = \bigoplus_{1 \le i \le n} T_{k_i}$ is a concrete $vN$-coalgebra and $D$ a f.d. $vN$-coalgebra. Then $f$ is counital, i.e. trace-preserving.

*Proof.* Let $c \in C$ be a positive element with $c \neq 0$. From the results in Appendix C, we know that $c = (c_1, \dots, c_n)$ where each $c_i \in T_{k_i}$ is a positive matrix. It is easy to see that

$$ \|c\| = \sum_{i} \|c_i\|_{\text{tr}} = \sum_{i} \text{tr}(c_i) = \varepsilon_C(c) $$

using the fact that the trace norm coincides with the trace on positive matrices. If we define $c' \triangleq \frac{c}{\|c\|}$, then we see that $c' \in P_C$ and therefore $f(c') \in P_D$. Therefore $\varepsilon_D(f(c')) = 1 = \varepsilon_C(c')$ and by multiplying by $\|c\|$ we obtain $\varepsilon_D(f(c)) = \varepsilon_C(c)$. Therefore $f$ is trace-preserving on non-zero positive elements. But it is obviously trace-preserving for $c = 0$, so $f$ is trace-preserving on all positive elements of $C$. For the general case, recall that every element in (the underlying vector space of) $C$ is a finite linear combination of positive elements of $C$ (special case of [27, Corollary 4.2.4]), so if $c = \sum_j \lambda_j p_j$, where each $p_j \in C$ is positive, then

$$ \varepsilon_D(f(c)) = \sum_j \lambda_j \varepsilon_D(f(p_j)) = \sum_j \lambda_j \varepsilon_C(p_j) = \varepsilon_C(c), $$

and therefore $f$ is indeed counital, i.e. trace-preserving. $\square$

**Corollary 110.** Let $f: (C, P_C) \to (D, P_D)$ be a morphism in $\mathbf{Q}$, where $C$ and $D$ are f.d. $vN$-coalgebras. Then $f$ is counital, i.e. trace-preserving.

*Proof.* Combine the previous proposition with Example 40 which states that $C$ is isomorphic as a $vN$-coalgebra to a concrete one.

**Theorem 50.** If $C \in \text{vNCoalg}$ is a $vN$-coalgebra, then $(C, P_C) \in \text{Ob}(\mathbf{Q})$. Moreover, the functor $S: \mathbf{S} \to \mathbf{Q}$ defined by $S(C) \triangleq (C, P_C)$ and $S(f) \triangleq f$, is fully faithful, strict monoidal w.r.t $\otimes$, and it strictly preserves finite coproducts.

*Proof.* Corollary 107 shows that the functor $S$ is well-defined on objects. From Proposition 41 it follows that the morphisms in $\mathbf{S}$ may be equivalently described as the completely contractive trace-preserving maps. To prove that $S$ is well-defined on morphisms, it remains to check that for a completely contractive trace-preserving map $f: C \to D$, we have $f[C] \subseteq D$. If $c \in P_c$, then $\varepsilon_D(f(c)) = \varepsilon_C(c) = 1$, using trace-preservation and therefore

$$ 1 = \|\varepsilon_D(f(c))\| \le \|f(c)\| \le \|c\| = 1 $$

so that $\|f(c)\| = 1 = \varepsilon_D(f(c))$, as required. Faithfulness is obvious and fullness follows by Corollary 110.
We can now prove that $S$ strictly preserves $\hat{\otimes}$ by verifying

$$
\begin{align*}
S(C) \hat{\otimes} S(D) &\xrightarrow{d} (S(C) \hat{\otimes} S(D))^{**} \\
&\stackrel{\cong}{\longrightarrow} (S(C)^* \tilde{\otimes} S(D)^*)^* \\
&= (H(C^*) \tilde{\otimes} H(D^*))^* && \text{(Proposition 108)} \\
&= H(C^* \tilde{\otimes} D^*)^* && \text{(Theorem 49)} \\
&= S((C^* \tilde{\otimes} D^*)^*) && \text{(Proposition 108)} \\
&\xrightarrow{S(\cong^{-1})} S((C \hat{\otimes} D)^{**}) \\
&\xrightarrow{S(d^{-1})} S(C \hat{\otimes} D)
\end{align*}
$$

and recognising that the above composite is the identity. The proof for $\oplus^1$ is fully analogous. $\square$

### E.3 Bipolar sets relevant for pure state computation

Finally, we prove the claims that we made in the final paragraph of the paper where we discuss pure state computation. Note that in the following proposition we are not using the $\tilde{\otimes}$ tensor, so $M_n \hat{\otimes} M_n$ need not be a vN-algebra, nor a vN-coalgebra.

**Proposition 111.** Let $u, v \in M_n$ be two unitary matrices. Then we have the following object equalities in $\mathbb{Q}$

$$ (M_n, \{u\}) \hat{\otimes} (M_n, \{v\}) = (M_n \hat{\otimes} M_n, \{u \otimes v\}^{\circ\circ}) = (M_n \hat{\otimes} M_n, \{u \otimes v\}). $$

*Proof.* The first equality holds by definition, so we only have to show $\{u \otimes v\}^{\circ\circ} = \{u \otimes v\}$. The inclusion $\{u \otimes v\}^{\circ\circ} \supseteq \{u \otimes v\}$ is obvious, so we prove the other one. For simplicity, we identify $M_n \hat{\otimes} M_n$ with $M_{n^2}$ and write $X$ for the operator space with the induced matrix norm from $M_n \hat{\otimes} M_n$, so that we have a c.i.i. $X \cong M_n \hat{\otimes} M_n$ given by the aforementioned isomorphism. Note that for any $a \in M_n \hat{\otimes} M_n$ we have that $\|a\|_{M_n \hat{\otimes} M_n} \le \|a\|_{M_n \hat{\otimes} M_n}$ and therefore for any $a \in M_{n^2}$ we have $\|a\|_{M_{n^2}} \le \|a\|_X$, because $M_n \hat{\otimes} M_n \cong M_{n^2}$. Next, observe that the linear functional $\frac{1}{n^2} \text{tr}((u^* \otimes v^*) \cdot): X \to \mathbb{C}$ satisfies

$$ \frac{1}{n^2} \text{tr}((u^* \otimes v^*)(u \otimes v)) = 1 $$

and this is a (necessarily complete) contraction, because for any $x \in X$ we have

$$
\begin{align*}
\left| \frac{1}{n^2} \operatorname{tr}((u^* \otimes v^*)x) \right| &= \frac{1}{n^2} \left| \operatorname{tr}((u^* \otimes v^*)x) \right| \\
&\le \frac{1}{n^2} \|u^* \otimes v^*\|_{\operatorname{tr}} \|x\|_{M_{n^2}} \\
&= \|x\|_{M_{n^2}} \\
&\le \|x\|_X.
\end{align*}
$$
Therefore $\frac{1}{n^2} \text{tr}((u^* \otimes v^*) \cdot) \in \{u \otimes v\}^\circ$ and we complete the proof via Lemma 102 which shows that $\{u \otimes v\}^{\circ\circ} \subseteq \{u \otimes v\}$. $\square$

**Proposition 112.** Let $u, v \in M_n$ be two unitary matrices. Then

$$ (M_n \overset{h}{\otimes} M_n, \{u \otimes v\}) \in \text{Ob}(\mathbf{Q}). $$

*Proof.* The proof is analogous to the previous proposition, because for any $a \in M_n \otimes M_n$ we have that $\|a\|_{M_n \underset{M_n}{\otimes}} \le \|a\|_{M_n \overset{h}{\otimes} M_n}$. $\square$

**Proposition 113.** Let $u, v \in M_n$ be two unitary matrices. Then

$$ (M_{2n}, \{|0\rangle \langle 0| \otimes (uv) + |1\rangle \langle 1| \otimes (vu)\}) \in \text{Ob}(\mathbf{Q}). $$

*Proof.* Straightforward verification shows that $|0\rangle \langle 0| \otimes (uv) + |1\rangle \langle 1| \otimes (vu)$ is a unitary matrix in $M_{2n}$, so the proposition follows by Lemma 103. $\square$

**Proposition 114.** Let $u, v \in M_n$ be two unitary matrices. Then

$$ \text{qsw: } (M_n \hat{\otimes} M_n, \{u \otimes v\}) \to (M_{2n}, \{|0\rangle \langle 0| \otimes (uv) + |1\rangle \langle 1| \otimes (vu)\}) $$

is a morphism of $\mathbf{Q}$, but

$$ \text{qsw: } (M_n \overset{h}{\otimes} M_n, \{u \otimes v\}) \to (M_{2n}, \{|0\rangle \langle 0| \otimes (uv) + |1\rangle \langle 1| \otimes (vu)\}) $$

is not a morphism of $\mathbf{Q}$.

*Proof.* All the pairs are valid objects of $\mathbf{Q}$ and qsw: $M_n \hat{\otimes} M_n \to M_{2n}$ is a complete contraction [30, §4]. Obviously, qsw($u \otimes v$) = $|0\rangle \langle 0| \otimes (uv) + |1\rangle \langle 1| \otimes (vu)$, so

$$ \text{qsw: } (M_n \hat{\otimes} M_n, \{u \otimes v\}) \to (M_{2n}, \{|0\rangle \langle 0| \otimes (uv) + |1\rangle \langle 1| \otimes (vu)\}) $$

is a valid morphism of $\mathbf{Q}$. However, qsw: $M_n \overset{h}{\otimes} M_n \to M_{2n}$ is not a complete contraction [30, §4], so we do not have a morphism

$$ \text{qsw: } (M_n \overset{h}{\otimes} M_n, \{u \otimes v\}) \to (M_{2n}, \{|0\rangle \langle 0| \otimes (uv) + |1\rangle \langle 1| \otimes (vu)\}). \square $$