quantum state monad

[[!redirects quantum costate comonad]]

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### Context
#### Categorical algebra
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[[!include categorical algebra -- contents]]
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#### Quantum systems
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[[!include quantum systems -- contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The *[[state monads]]* and *[[costate comonads]]* ([[store comonads]]) are typically discussed on [[cartesian closed categories]] --- but, of course, the analogous construction exists on any [[closed monoidal category]].

An immediate non-cartesian example is a [[category of vector spaces]]. At least for a [[finite-dimensional vector space]] $\mathscr{H}$ (and generally on a [[compact closed category]]), the *linear $\mathscr{H}$-state/store (co)monads* merge into a single [[Frobenius monad]] (see [below](#FrobeniusStructure)) which reflects some core structure of [[open quantum systems]] ([[mixed states]], [[quantum channels]] and their [[quantum observables]]).

Due to this fact and the general broad identification of (multiplicative) [[linear logic]] with a form of [[quantum logic]], the linear (co)state (co)monad may deserve to be called the *quantum state monad* (cf. the similar situation for the *[[quantum reader monad]]*).

For example, [[unitary quantum channels]] turn out to be embodied by [[monad transformations]] between such quantum state monads, so that with this terminology the plausible and desireable statement "*quantum channels are quantum state transformations*" becomes an actual theorem (see [below](#QuantumChannelsAreQuantumStateTransformations)).

## Preliminaries

For definiteness (only), we consider the category $Mod_{\mathbb{C}}$ of [[complex vector spaces]] with, as usual:

* [[monoidal category|monoidal structure]] given by the [[tensor product of vector spaces]],

* [[tensor unit]]$\;$ $\mathbb{1} \,\equiv\, \mathbb{C}$ the [[complex numbers]] themselves,

* [[internal hom]]$\;$ $\mathscr{H} \multimap \mathscr{H}'$ the vector space of [[linear maps]]

  \[
    \label{InternalHomHomIsomorphism}
    \mathscr{H} \otimes (\text{-})
    \;\;\dashv\;\;
    \mathscr{H} \multimap (\text{-})
  \]

Throughout the following we fix

\[
  \label{TheFinDimHilbertSpace}
  \mathscr{H} \,\in\, FinDimMod_{\mathbb{C}} \hookrightarrow Mod_{\mathbb{C}}
\]

a [[finite-dimensional vector space|finite-dimensional]] [[complex vector space]].

But most or all of the following discussion works for $Mod_{\mathbb{C}}$ replaced by any [[symmetric monoidal closed category]] and $\mathscr{H}$ taken from a [[compact closed category|compact closed]] [[subcategory]].

In this vein, for notational purposes only (at this point) we choose on $\mathscr{H}$:

1. a [[Hermitian form|Hermitian]] [[inner product]] $\left\langle\cdot \vert \cdot\right\rangle$, hence consider it as a [[finite-dimensional Hilbert space]] and use [[bra-ket]] notation to denote the generic elements of $\mathscr{H}$ by

   $$
     \left\vert \psi \right\rangle
     \,\equiv\,
     \psi \,\in\, \mathscr{H}
   $$

   and those of its [[linear dual space]]

   $$
     \mathscr{H}^\ast
     \,\equiv\,
     \mathscr{H} \multimap \mathbb{1}
   $$

   by

   $$
     \left\langle \phi \right\vert
     \,\equiv\,
     \left\langle \psi \vert (\text{-}) \right\rangle
     \,\in\,
     \mathscr{H}^\ast
   $$

1. an [[orthonormal linear basis]] $\big(\left\vert w \right\rangle \,\in\, \mathscr{H} \big)_{w \colon W}$, hence such that

   \[
     \label{OrthonormalBasis}
     w,w' \,\colon\, W
     \;\;\;\;
       \vdash
     \;\;\;\;
     \left\langle w \vert w' \right\rangle
     \,=\,
     \delta_{w, w'}
   \]

   ([[Kronecker delta]]).

Furthermore, for $\psi \in \mathscr{H}$, $\phi \in \mathscr{K}$ we leave the [[tensor product]]-symbol implicit in

$$
  \left\vert \psi \right\rangle
  \left\langle \phi \right\vert
  \;\;
    \equiv
  \;\;
  \psi
    \otimes
  \left\langle \phi \vert \text{-} \right\rangle
  \;\;
  \in
  \;\;
  \mathscr{H} \otimes \mathscr{K}^\ast
  \,.
$$

In this [[bra-ket]]-notation, the
[[compact closed category|compact closure]] for [[finite-dimensional vector space|finite-dimensional]] $\mathscr{H}$ (see [there](finite-dimensional+vector+space#CompactClosure)) is witnessed by the following [[isomorphism]]

\[
  \label{CompactClosureViaBraKetNotation}
  \array{
    \Big(
      \mathscr{H}
      \multimap
      \mathscr{H}'
    \Big)
    &\overset{\;\; \sim \;\;}{\longrightarrow}&
    \mathscr{H}' \otimes \mathscr{H}^\ast
    \\
    \Big(
      \left\vert w \right\rangle
      \,\mapsto\,
      \underset{w'}{\sum}
      \left\vert w' \right\rangle
      \cdot
      A_{w', w}
    \Big)
    &\mapsto&
      \underset{w,w'}{\sum}
      \left\vert w' \right\rangle
      A_{w', w}
      \left\langle w \right\vert
    \mathrlap{\,.}
  }
\]

## Definition

For $\mathscr{H}$ a [[finite-dimensional Hilbert space]] --- or generally an object in a [[compact closed category]] $QuTypes$ --- its [[internal hom]]-[[adjoint functor|adjunction]] is [[ambidextrous adjunction|ambidextrous]] and associated to a pair of [[Frobenius monads]], in linear version of the classical [[state monads]]/[[store comonad]]-construction:

\begin{imagefromfile}
    "file_name": "QuantumStateFrobeniusMonads-230919.jpg",
    "width": 680,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

Moreover, these (co)monads may equivalently be understood as the ([[cowriter comonad|co]]-)[[writer monads]] induced by the [[endomorphism algebra]] $End(\mathscr{H}) \,\simeq\, \mathscr{H}^\ast \otimes \mathscr{H}$, which is a [[Frobenius algebra]] (by general arguments as recalled in [Lauda 2006](Frobenius+algebra#Lauda2006), made explicit for instance in [Vicary 2011 Lem 3.17](endomorphism+ring#Vicary11)) and as such makes its ([[cowriter comonad|co]]-)[[writer monad]] into a [[Frobenius monad]].

\linebreak

We next spell this out in more detail

> (Beware that there is now a switch in conventions. Will streamline later.)

### The defining adjunction
 {#TheDefiningAdjunction}

The [[internal hom]]-[[adjoint functor|adjunction]] for $\mathscr{H}$ (eq:TheFinDimHilbertSpace)

\[
  \label{TheInternalHomAdjunction}
  Mod_{\mathbb{C}}
  \underoverset
    {
       \underset{
         \mathscr{H} \multimap (\text{-})
       }{\longrightarrow}}
    {
      \overset{
        \mathscr{H} \otimes (\text{-})
      }{\longleftarrow}
    }
    {\;\; \bot \;\;}
  Mod_{\mathbb{C}}
\]

the $\big(\mathscr{H} \otimes (\text{-}) \,\dashv\, \mathscr{H} \multimap (\text{-})\big)$-[[unit of an adjunction|unit]] is:

\[
  \label{TheAdjunctionUnit}
  \array{
    (\text{-})
    &\overset{
      ret
       ^{ \mathscr{H}State }
       _{\mathbb{1},\mathbb{1}}
    }{\longrightarrow}&
    \mathscr{H}
      \multimap
    \big(
     \mathscr{H}
       \otimes
     (\text{-})
    \big)
    \\
    \left\vert \psi \right\rangle
    &\mapsto&
    \big(
      \left\vert \psi' \right\rangle
      \mapsto
      \left\vert \psi' \right\rangle
      \otimes
      \left\vert \psi \right\rangle
    \big)
    \mathrlap{\,.}
  }
\]

Using the [[symmetric monoidal category|symmetric braiding]], an orthonormal basis (eq:OrthonormalBasis) and the compact closure (eq:CompactClosureViaBraKetNotation)
we may equivalently write this as the "insertion of an identity" in the form known from [[quantum mechanics]] textbooks:

$$
  \array{
    (\text{-})
    &
    \overset{
      ret
       ^{ \mathscr{H}State }
       _{\mathbb{1},\mathbb{1}}
    }{\longrightarrow}&
    \mathscr{H}^\ast
      \otimes
    \mathscr{H}
      \otimes
    (\text{-})
    \\
    \vert \psi \rangle
    &\mapsto&
    \sum_w
    \left\vert w \right\rangle
    \left\langle w \right\vert
    \otimes
    \left\vert \psi \right\rangle
    \mathrlap{\,.}
  }
$$

The [[adjunction counit]] is the [[evaluation map]]:

\[
  \label{TheAdjunctionCounit}
  \array{
    \mathscr{H}
    \otimes
    \big(
      \mathscr{H}
      \multimap
      (\text{-})
    \big)
    &\overset{
      obt^{ \mathscr{H}Store }_{}
    }{\longrightarrow}&
    (\text{-})
    \\
    \left\vert \psi \right\rangle
    \left\langle \phi \right\vert
    \otimes
    \left\vert \text{-} \right\rangle
    &\mapsto&
    \left\langle \phi \vert \psi \right\rangle
    \,
    \left\vert \text{-} \right\rangle
    \mathrlap{\,.}
  }
\]

which --- again using the [[symmetric monoidal category|symmetric braiding]], an orthonormal basis (eq:OrthonormalBasis) and the compact closure (eq:CompactClosureViaBraKetNotation) --- is equivalently the [[partial trace]] over $\mathscr{H}$:

\[
  \label{TheAdjunctionCounit}
  \array{
    \mathscr{H}
    \otimes
    \mathscr{H}^\ast
    \otimes
    (\text{-})
    &\overset{
      obt^{ \mathscr{H}Store }_{}
    }{\longrightarrow}&
    (\text{-})
    \\
    \left\vert \psi \right\rangle
    \left\langle \phi \right\vert
    \otimes
    \left\vert \text{-} \right\rangle
    &\mapsto&
    \underset{w}{\sum}
    \left\langle w \vert \psi \right\rangle
    \left\langle \phi \vert w \right\rangle
    \otimes
    \left\vert \text{-} \right\rangle
    \mathrlap{\,.}
  }
\]

### The linear state monad

The adjunction (eq:TheInternalHomAdjunction)
[induces](monad#RelationBetweenAdjunctionsAndMonads) a linear version of the [[state monad]]:

The [[underlying]] [[endofunctor]] is

\[
 \label{UnderlyingEndofunctorOfQuantumStateMonad}
  \array{
    \mathllap{
      \mathscr{H}
      State
      \;\colon\;
    }
    Mod_{\mathbb{C}}
    &\longrightarrow&
    Mod_{\mathbb{C}}
    \\
    \mathscr{V}
    &\mapsto&
    \Big(
    \mathscr{H}
    \multimap
    \big(
      \mathscr{H}
      \otimes
      \mathscr{V}
    \big)
    \Big)
    \mathrlap{\,.}
  }
\]

The join operation is induced from the adjunction counit (eq:TheAdjunctionCounit) as:

$$
  \array{
    \mathscr{H}^\ast
    \otimes
    \mathscr{H}
    \otimes
    \mathscr{H}^\ast
    \otimes
    \mathscr{H}
    &
    \overset{
      join^{\mathscr{H}State}_{(\text{-})}
    }{\longrightarrow}
    &
    \mathscr{H}^\ast
    \otimes
    \mathscr{H}
    \\
    \left\langle \text{-} \right\vert
    \otimes
    \left\vert \psi \right\rangle
    \left\langle \phi \right\vert
    \otimes
    \left\vert \text{-} \right\rangle
    &\mapsto&
    \left\langle \phi \vert \psi \right\rangle
    \,
    \left\langle \text{-} \right\vert
    \otimes
    \left\vert \text{-} \right\rangle
    \mathrlap{\,.}
  }
$$

### The linear store comonad

The adjunction (eq:TheInternalHomAdjunction)
[induces](monad#RelationBetweenAdjunctionsAndMonads) a linear version of the [[costate comonad]]:

The [[underlying]] [[endofunctor]] is

\[
 \label{UnderlyingEndofunctorOfQuantumCostateComonad}
  \array{
    \mathllap{
      \mathscr{H}
      Store
      \;\colon\;
    }
    Mod_{\mathbb{C}}
    &\longrightarrow&
    Mod_{\mathbb{C}}
    \\
    \mathscr{V}
    &\mapsto&
    \mathscr{H}
    \otimes
    \big(
      \mathscr{H}
      \multimap
      \mathscr{V}
    \big)
    \mathrlap{\,.}
  }
\]

The duplication (cojoin) operation in the $\mathscr{H}$-CoState comonad (eq:UnderlyingEndofunctor) is induced from the adjunction unit (eq:TheAdjunctionUnit)
as follows:

\[
  \label{DuplicationOperation}
  \array{
    \mathscr{H}
      \otimes
    \big(
      \mathscr{H}
      \multimap
      (\text{-})
    \big)
    &
    \overset{
      dupl
        ^{ \mathscr{H}Store }
        _{ (\text{-}) }
    }{\longrightarrow}
    &
    \mathscr{H}
    \otimes
    \bigg(
      \mathscr{H}
      \multimap
      \Big(
        \mathscr{H}
          \otimes
        \big(
          \mathscr{H}
          \multimap
          (\text{-})
        \big)
      \Big)
    \bigg)
    \\
    \left\vert \psi \right \rangle
    \left\langle \phi \right\vert
    \otimes
    \left\vert \text{-} \right \rangle
    &\mapsto&
    \sum_w
    \left\vert \psi \right\rangle
    \left\langle w \right\vert
    \otimes
    \left\vert w \right\rangle
    \left\langle \phi \right\vert
    \otimes
    \left\vert \text{-} \right \rangle
    \mathrlap{\,.}
  }
\]

Finally, the obtain-operation (the [[counit of a comonad|comonad counit]]) is the adjunction counit (eq:TheAdjunctionCounit)

### Kleisli structure
 {#KleisliStructure}

We assume an ambient [[compact closed category]] such as that of [[finite-dimensional vector spaces]].

For the formula in Prop. \ref{KleisliCompositionIsSuperoperatorComposition} to come out naturally, we take -- without loss of generality -- the generic object on which to apply the linear store comonad to be a [[dual object]] $\mathscr{K}^\ast \coloneqq (\mathscr{K} \multimap \mathbb{1})$.

\begin{remark}
\label{TheCoKleisliMorphisms}
**(the coKleisli morphisms)**
\linebreak
The [[coKleisli morphisms]] of the quantum store comonad (eq:UnderlyingEndofunctor)

\[
  \label{KleisliMorphism}
  \mathcal{O}_A
  \;\colon\;
  \mathscr{H}
    \otimes
  \big(
  \mathscr{H}
    \multimap
  \mathscr{K}^\ast
  \big)
  \longrightarrow
  \mathscr{K}^\ast
\]

are [[isomorphism|isomorphically]] of the form

\[
  \label{KleisliMorphismUnderCompactClosure}
  \mathcal{O}_A
  \;\colon\;
  \mathscr{H}
    \otimes
  \mathscr{H}^\ast
    \otimes
  \mathscr{K}^\ast
    \longrightarrow
  \mathscr{K}^\ast
\]

and as such are equivalently ([[adjunct]] to) morphisms of the form

\[
  \label{SuperoperatorOnMatrices}
  A
  \;\colon\;
  \mathscr{H}
    \otimes
  \mathscr{K}^\ast
    \longrightarrow
  \mathscr{H}
    \otimes
  \mathscr{K}^\ast
\]

which in turn are isomorphically "*[[superoperators]]*" of the form

\[
  \label{Superoperator}
  \big(
    \mathscr{K} \multimap \mathscr{H}
  \big)
  \longrightarrow
  \big(
    \mathscr{K} \multimap \mathscr{H}
  \big)
  \,.
\]

\end{remark}

\begin{proposition}
\label{KleisliCompositionIsSuperoperatorComposition}
  The [[Kleisli composition]] on [[Kleisli morphisms]] (eq:KleisliMorphism) of the quantum store comonad is given by the plain composition of the linear duals of the corresponding [[superoperators]] (eq:SuperoperatorOnMatrices):
\[
  \label{KleisliComposition}
  \big(
    extend^{\mathscr{H}Store}
      \mathcal{O}_{A'}
  \big)
  \circ
  \big(
    extend^{\mathscr{H}Store}
      \mathcal{O}_{A}
  \big)
  \;\;=\;\;
  \Big(
    \left\vert \psi \right\rangle
    \left\langle \phi, \kappa \right\vert
    \,\mapsto\,
    \left\vert \psi \right\rangle
    \left\langle \phi, \kappa \right\vert
    A \circ A'
  \Big)
  \,.
\]
\end{proposition}
Here for $\psi \in \mathscr{H}$ and $\kappa \in \mathscr{K}$ we denote their tensor product as
$$
  \left\vert \psi, \kappa \right\rangle
  \,\in\,
  \mathscr{H} \otimes \mathscr{K}
  \,.
$$
\begin{proof}
The relation between a Kleisli morphism $\mathcal{O}_A$ (eq:KleisliMorphismUnderCompactClosure) and the corresponding super-operator $A$ (eq:SuperoperatorOnMatrices) on matrices is

$$
  \array{
    \mathllap{
      \mathcal{O}_A
      \;\colon\;
    }
    \mathscr{H}
      \otimes
    \mathscr{H}^\ast
      \otimes
    \mathscr{K}^\ast
    &\longrightarrow&
    \mathscr{K}^\ast
    \\
    \left\vert \psi \right\rangle
    \left\langle \phi \right\vert
    \otimes
    \left\langle \kappa \right\vert
    &\mapsto&
    \left\langle
      \phi, \kappa
    \right\vert
     A
    \left\vert
      \psi, -
    \right\rangle
    \mathrlap{\,.}
  }
$$

With this and the cojoin operation (eq:DuplicationOperation) it follows that the extend-operation turns $\mathcal{O}_A$ into

$$
  \array{
    \mathscr{H}
      \otimes
    \big(
      \mathscr{H}
      \multimap
      \mathscr{K}^\ast
    \big)
    &
      \overset{
        dupl^{ \mathscr{H}Store }_{\mathscr{K}^\ast,\mathscr{K}^\ast}
      }{\longrightarrow}
    &
    \mathscr{H}
    \otimes
    \bigg(
      \mathscr{H}
      \multimap
      \Big(
        \mathscr{H}
        \otimes
        \big(
          \mathscr{H}
          \multimap
          \mathscr{K}^\ast
        \big)
      \Big)
    \bigg)
    &\overset{
      \mathscr{H}
      \otimes
      \big(
        \mathscr{H}
          \multimap
        \mathcal{O}_A
      \big)
    }{\longrightarrow}&
    \mathscr{H}
    \otimes
    \big(
      \mathscr{H}
      \multimap
      (\mathscr{K}')^\ast
    \big)
    \\
    \left\vert \psi \right\rangle
    \left\langle \phi \right\vert
    \otimes
    \left\langle \kappa \right\vert
    &\mapsto&
    \sum_w
    \left\vert \psi \right\rangle
    \left\langle w \right\vert
    \otimes
    \left\vert w \right\rangle
    \left\langle \phi \right\vert
    \otimes
    \left\langle \kappa \right\vert
    &\mapsto&
    \;\;
    \underset{
      =
        \left\vert \psi \right\rangle
        \left\langle \phi, \kappa \right\vert
         A
    }{
    \underbrace{
    \sum_{w, k'}
        \left\vert \psi \right\rangle
        \left\langle \phi, \kappa \right\vert
         A
        \left\vert w, k' \right\rangle
        \left\langle w, k' \right\vert
    }
    }
  }
$$

hence
\[
  \label{StoreExtension}
  \array{
    \mathllap{
      exend^{\mathscr{H} Store}
      \mathcal{O}_A
      \;\colon\;
    }
    \mathscr{H}
      \otimes
    \mathscr{H}^\ast
      \otimes
    \mathscr{K}^\ast
    &\longrightarrow&
    \mathscr{H}
      \otimes
    \mathscr{H}^\ast
      \otimes
    \mathscr{K}^\ast
    \\
    \left\vert \psi \right\rangle
    \left\langle \phi, \kappa \right\vert
    &\mapsto&
    \left\vert \psi \right\rangle
    \left\langle \phi, \kappa \right\vert
    A
  }
\]
This implies the claim (eq:KleisliComposition).
\end{proof}

The same argument applies with the second copy of $\mathscr{K}$ replaced by some $\mathscr{K}'$: The Kleisli category is the [[opposite category|opposite]] of that of superoperators of the form

$$
  \mathscr{H}\otimes\mathscr{K}
  \longrightarrow
  \mathscr{H}\otimes\mathscr{K}'
  \longrightarrow
  \mathscr{H}\otimes\mathscr{K}''
  \,.
$$

## Examples

### Quantum observables are the Quantum state contextful scalars

> (The following now starts out a little repetitive, recalling a special case of the general statement above. Will streamline...)

For the classical [[costate comonad]] on a [[cartesian closed category]] its value and its operations on the [[tensor unit]] (the [[terminal object]] in this case) are vacuous. Quite in contrast, the linear CoState comonad on the tensor unit encodes core structure of [[quantum physics]]:

\begin{example}
\label{QuantumObservablesAsQuantumStateContextfulScalars}
**(quantum observables as quantum state contextful scalars)**
\linebreak

First notice that the quantum store on the tensor unit

$$
  \mathscr{H}Store(\mathbb{1})
  \;\equiv\;
  \mathscr{H}
    \otimes
  \big(
    \mathscr{H}
    \multimap
    \mathbb{1}
  \big)
  \;=\;
  \mathscr{H} \otimes \mathscr{H}^\ast
$$

is the home of [[density matrices]]/[[mixed quantum states]] generalizing the [[pure states]] in $\mathscr{H}$.

Moreover, an $\mathscr{H}$-CoState [[coKleisli morphism]] on the tensor unit is equivalently a [[linear operator]]
$$
  A \,\colon\, \mathscr{H} \to \mathscr{H}
  \,,
$$
hence a [[quantum observables]] -- incarnated via its [[expectation values]] on quantum states

\[
  \label{CoKleisliMorphism}
  \array{
    \mathscr{H}
    \otimes
    \mathscr{H}^\ast
    &\overset{\mathcal{O}_A}{\longrightarrow}&
    \mathbb{1}
    \\
    \left\vert \psi \right\rangle
    \left\langle \phi \right\vert
    &\mapsto&
    \left\langle \phi \right\vert
    A
    \left\vert \psi \right\rangle
    \mathrlap{\,.}
  }
\]

\end{example}

\begin{proposition}
  Composition of quantum observables $\mathcal{O}_A$, $\mathcal{O}_{A'}$ (eq:CoKleisliMorphism) in the [[coKleisli category]] is given by ordinary composition $A \circ A'$ of the corresponding [[linear operators]]:
$$
  \mathcal{O}_{A'}
  \circ
  \big(
    ext^{ \mathscr{H}Store }_{\mathbb{1},\mathbb{1}}
    \mathcal{O}_A
  \big)
  \;\;
    =
  \;\;
  \mathcal{O}_{ A' \circ A }
  \;\colon\;
    \left\vert \psi \right\rangle
    \left\langle \phi \right\vert
    \,\mapsto\,
    \left\langle \phi \right\vert
    A A'
    \left\vert \psi \right\rangle
  \,.
$$
\end{proposition}
\begin{proof}
  From (eq:DuplicationOperation) we find that the extension operation is given as follows:
$$
  \array{
    \mathscr{H}
      \otimes
    \mathscr{H}^\ast
    &
      \overset{
        dupl^{ \mathscr{H}Store }_{\mathbb{1},\mathbb{1}}
      }{\longrightarrow}
    &
    \mathscr{H}
    \otimes
    \big(
      \mathscr{H}
      \multimap
      \mathscr{H}
      \otimes
      \mathscr{H}^\ast
    \big)
    &\overset{
      \mathscr{H}
      \otimes
      \big(
        \mathscr{H}
          \multimap
        \mathcal{O}_A
      \big)
    }{\longrightarrow}&
    \mathscr{H}
    \otimes
    \mathscr{H}^\ast
    \\
    \left\vert \psi \right\rangle
    \left\langle \phi \right\vert
    &\mapsto&
    \sum_w
    \left\vert \psi \right\rangle
    \left\langle w \right\vert
    \otimes
    \left\vert w \right\rangle
    \left\langle \phi \right\vert
    &\mapsto&
    \;\;
    \mathclap{
    \underset{
    {
      =\,
      \left\vert \psi \right\rangle
      \left\langle \phi \right\vert
      A
    }
    }
    {
      \underbrace{
        \sum_w
        \left\vert \psi \right\rangle
        \left\langle w \right\vert
        \otimes
        \left\langle \phi \right\vert
         A
        \left\vert w \right\rangle
      }
    }
    }
  }
$$

hence

$$
  ext^{ \mathscr{H}Store }_{\mathbb{1},\mathbb{1}}
  \,\colon\,
  \Big(
    \left\vert \psi \right\rangle
    \left\langle \phi \right\vert
    \,\mapsto\,
    \left\langle \phi \right\vert
    A
    \left\vert \psi \right\rangle
  \Big)
  \,\mapsto\,
  \Big(
    \left\vert \psi \right\rangle
    \left\langle \phi \right\vert
    \,\mapsto\,
    \left\vert \psi \right\rangle
    \left\langle \phi \right\vert
    A
  \Big)
$$

\end{proof}

\begin{remark}\label{Traces}
Understanding $\mathscr{H}Store$ as a [[Frobenius monad]] as [above](#FrobeniusStructure) makes it natural to combine the [[unit of a monad|monad unit]] (eq:TheAdjunctionUnit) with the Kleisli morphisms of Rem. \ref{TheCoKleisliMorphisms}.

This gives the *[[trace]]* of linear operators, as seen via (eq:StoreExtension):

$$
  \array{
    \mathbb{1}
    &\overset{ret^{\mathscr{H}^\ast State}_{\mathbb{1}}}{\longrightarrow}&
    \mathscr{H}
    \otimes
    \mathscr{H}^\ast
    &\overset{
      ext^{\mathscr{H}Store}_{\mathbb{1}, \mathbb{1}} \mathcal{O}_{A}
    }{\longrightarrow}&
    \mathscr{H}
    \otimes
    \mathscr{H}^\ast
    &\overset{obt^{\mathscr{H}Store}_{\mathbb{1}}}{\longrightarrow}&
    \mathbb{1}
    \\
    1
    &\mapsto&
    \underset{w}{\sum}
    \left\vert w \right\rangle
    \left\langle w \right\vert
    &\mapsto&
    \underset{w}{\sum}
    \left\vert w \right\rangle
    \left\langle w \right\vert
    A
    &\mapsto&
    tr\big( A \big)
  }
$$
and generally the [[partial trace]]:
$$
  \array{
    \mathscr{K}^\ast
    &\overset{ret^{\mathscr{H}^\ast State}_{\mathscr{K}^\ast}}{\longrightarrow}&
    \mathscr{H}
    \otimes
    \mathscr{H}^\ast
    \otimes
    \mathscr{K}^\ast
    &\overset{
      ext^{\mathscr{H}Store}_{\mathscr{K}^\ast, \mathscr{K}^\ast} \mathcal{O}_{A}
    }{\longrightarrow}&
    \mathscr{H}
    \otimes
    \mathscr{H}^\ast
    \otimes
    \mathscr{K}^\ast
    &\overset{obt^{\mathscr{H}Store}_{\mathscr{K}^\ast}}{\longrightarrow}&
    \mathscr{K}^\ast
    \\
    \left\langle \kappa \right\vert
    &\mapsto&
    &\mapsto&
    &\mapsto&
    \left\langle \kappa \right\vert
    tr_{\mathscr{H}}\big(A \big)
  }
$$
\end{remark}

### Quantum channels that are Quantum state transformations
 {#QuantumChannelsAreQuantumStateTransformations}

\begin{example}
\label{UnitaryQuantumChannelsAsQUantumStateTransformations}
**(unitary quantum channels are quantum state transformations)**
\linebreak
  If $U \,\colon\, \mathscr{H}_1 \to \mathscr{H}_2$ is an [[invertible map]] with [[inverse]] $U^\dagger \,\colon\, \mathscr{H}_2 \to \mathscr{H}_1$, then the "[[unitary quantum channel]]"
$$
  \array{
    \mathscr{H}_1
      \otimes
    \mathscr{H}_1^\ast
      \otimes
    (\text{-})
    &
    \overset{
      U \otimes U^{\dagger \ast}
    }{\longrightarrow}
    &
    \mathscr{H}_2
      \otimes
    \mathscr{H}_2^\ast
      \otimes
    (\text{-})
  }
$$
is a [transformation of](monad#TransformationOfMonadsOnFixedCategory) quantum state monads.
\end{example}
\begin{proof}
By direct unwinding of the definitions, we find that the counit is respected

$$
  \array{
    \mathscr{H}_1
      \otimes
    \mathscr{H}_1^\ast
      \otimes
    (\text{-})
    &
    \overset{
      U
        \otimes
      U^{\dagger \ast}
      \,
        \otimes
      (\text{-})
    }{\longrightarrow}
    &
    \mathscr{H}_2
      \otimes
    \mathscr{H}_2^\ast
      \otimes
    (\text{-})
    &
    \overset{
      obt^{\mathscr{H}_2Store}_{\mathscr{K}_2}
    }{\longrightarrow}
    &
    (\text{-})
    \\
    \left\vert \psi \right\rangle
    \left\langle \phi \right\vert
    \otimes
    \left\langle \text{-} \right\vert
    &\mapsto&
    U
    \left\vert \psi \right\rangle
    \left\langle \phi \right\vert
    U^\dagger
    \otimes
    \left\langle \text{-} \right\vert
    &\mapsto&
    \left\langle \phi \right\vert
      U^\dagger
      U
    \left\vert \psi \right\rangle
    \otimes
    \left\langle \text{-} \right\vert
    \\
    && &=&
    \left\langle \phi \vert \psi \right\rangle
    \otimes
    \left\langle \text{-} \right\vert
    \\
    && &=&
    obt^{\mathscr{H}_1 Store}_{\mathscr{K}^\ast}
    \big(
      \left\vert \psi \right\rangle
      \left\langle \phi \right\vert
      \otimes
      \left\langle \text{-} \right\vert
    \big)
  }
$$

by the property that $U^\dagger$ is a [[left inverse]] to $U$,

and the unit is respected

$$
  \array{
    (\text{-})
    &
    \overset{
      ret^{ \mathscr{H}_1State }_{(\text{-})}
    }{\longrightarrow}
    &
    \mathscr{H}_1
    \otimes
    \mathscr{H}_1^\ast
    \otimes
    (\text{-})
    &
    \overset{
      U \otimes U^{\dagger\ast}
      \otimes
      (\text{-})
    }{\longrightarrow}
    &
    \mathscr{H}_2
    \otimes
    \mathscr{H}_2^\ast
    \otimes
    (\text{-})
    \\
    \underset{w}{\sum}
    \left\vert w \right\rangle
    \left\langle w \right\vert
    \otimes
    \left\vert \text{-} \right\rangle
    &\mapsto&
    \underset{w_1}{\sum}
    U
    \left\vert w_1 \right\rangle
    \left\langle w_1 \right\vert
    U^{\dagger}
    \otimes
    \left\vert \text{-} \right\rangle
    \\
    && &=&
    \underset{w_2}{\sum}
    \left\vert w_2 \right\rangle
    \left\langle w_2 \right\vert
    \otimes
    \left\vert \text{-} \right\rangle
    \mathrlap{\,,}
  }
$$

because $U^\dagger$ is also a [[right inverse]] to $U$, using here the [[compact closed category|compact closure]] identification

\[
  \label{SeeingThatUnitaryQuantumChannelPreservesSatteReturn}
  \array{
    \mathscr{H}_2
    \otimes
    \mathscr{H}_2^\ast
    &\overset{\sim}{\longrightarrow}&
    \mathscr{H}_2 \multimap \mathscr{H}_2
    &\overset{\sim}{\longrightarrow}&
    \mathscr{H}_2
    \otimes
    \mathscr{H}_2^\ast
    \\
    \underset{w_1}{\sum}
    U
    \left\vert w_1 \right\rangle
    \left\langle w_1 \right\vert
    U^{\dagger}
    &\mapsto&
    \underset{
      \mathrm{Id}
    }{
    \underbrace{
      U
      \circ
      Id
      \circ
      U^{\dagger}
    }
    }
    &\mapsto&
    \underset{w_2}{\sum}
    \left\vert w_2 \right\rangle
    \left\langle w_2 \right\vert
    \mathrlap{\,.}
  }
\]

Therefore also the join and cojoin are preserved, eg.:
$$
  \array{
    \mathscr{H}_1
      \otimes
    \mathscr{H}_1^\ast
      \otimes
    (\text{-})
    &
    \overset{
      dupl^{\mathscr{H}_1 Store}_{\mathscr{K}^\ast}
    }{\longrightarrow}
    &
    \mathscr{H}_1
      \otimes
    \mathscr{H}_1^\ast
      \otimes
    \mathscr{H}_1
      \otimes
    \mathscr{H}_1^\ast
      \otimes
    (\text{-})
    &
    \overset{
      U \otimes U^{\dagger\ast}
      \otimes
      U \otimes U^{\dagger\ast}
      \,
      \otimes \mathrm{id}
    }{\longrightarrow}
    &
    \mathscr{H}_2
      \otimes
    \mathscr{H}_2^\ast
      \otimes
    \mathscr{H}_2
      \otimes
    \mathscr{H}_2^\ast
      \otimes
    (\text{-})
    \\
    \left\vert \psi \right\rangle
    \left\langle \phi \right\vert
    \otimes
    \left\langle \text{-} \right\vert
    &\mapsto&
    \mathclap{
    \underset{w_1}{\sum}
    \,
    \left\vert \psi \right\rangle
    \left\langle w_1 \right\vert
    \otimes
    \left\vert w_1 \right\rangle
    \left\langle \phi \right\vert U^\dagger
    \otimes
    \left\langle \text{-} \right\vert
    }
    &\mapsto&
    \mathclap{
    U \left\vert \psi \right\rangle
    \Big(
    \underset{
      =
    \underset{w_2}{\sum}
      \left\langle w_2 \right\vert
      \otimes
      \left\vert w_2 \right\rangle
    }
    {
    \underbrace{
      \underset{w_1}{\sum}
      \left\langle w_1 \right\vert
      U^\dagger
      \otimes
      U
      \left\vert w_1 \right\rangle
    }
    }
    \Big)
    \left\langle \phi \right\vert U^\dagger
    \otimes
    \left\langle \text{-} \right\vert
    }
    \\
    &&
    &=&
    \mathclap{
    dupl^{\mathscr{H}_2 Store}_{\mathscr{K}^\ast}
    \circ
    \big(
      U \otimes U^{\dagger \ast}
      \otimes id
    \big)
    \big(
    \left\vert \psi \right\rangle
    \left\langle \phi \right\vert
    \otimes
    \left\langle \text{-} \right\vert
    \big)
    }
  }
$$
\end{proof}

\begin{example}
\label{PartialTraceAsComonadicQuantumStateTransformation}
  A [[partial trace quantum channel]]
$$
  \array{
    (\text{-})
    \otimes
    \mathscr{H}
    \otimes
    \mathscr{B}
    \otimes
    \mathscr{B}^\ast
    \otimes
    \mathscr{H}^\ast
    &
    \xrightarrow{
      id
      \otimes
      trace^{\mathscr{B}}
    }
    &
    \mathscr{H} \otimes \mathscr{H}^\ast
    \\
    \left\vert \psi, \beta \right\rangle
    \left\langle \beta', \psi' \right\vert
    &\mapsto&
    \left\vert \psi \right\rangle
    \left\langle \beta' \vert \beta \right\rangle
    \left\langle \psi' \right\vert
    \mathrlap{\,.}
  }
$$
is a comonadic quantum state transformation
$$
  trace^{\mathscr{B}}
  \,\colon\,
  (\mathscr{H} \otimes \mathscr{B})State
  \underoverset{ comon\;transf }{}{\longrightarrow}
  \mathscr{H}State
  \,.
$$
\end{example}
\begin{proof}
Since the structure maps of the $(\mathscr{H} \otimes \mathscr{B})\mathrm{State}$-comonad are [[tensor products]] of structure maps of $\mathscr{H}\mathrm{State}$ and $\mathscr{B}\mathrm{State}$,
it is sufficient to show this for $\mathscr{H} = \mathbb{1}$ the [[tensor unit]], hence for the case that $\mathscr{H}\mathrm{State} = \mathrm{Id}$.
But in this case $\mathrm{trace}^{\mathscr{B}} \,=\, obt^{ \mathscr{H}^\ast\mathrm{Store} }{}$, which is a comonadic transformation (in fact the [[terminal object|terminal]] one) by [this example](monad#IdentityMonadIsInitial)).

Alternatively, onee readily checks the required conditions explicitly:

\begin{imagefromfile}
    "file_name": "PartialTraceAsQuantumStateComonTransf-230924b.jpg",
    "width": 800,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

\end{proof}

### Heisenberg evolution is action of Quantum state transformations
 {#HeisenbergEvolution}

namely on the quantum-state contextful scalars:

\begin{example}
  The quantum state transformation induced by a [[unitary quantum channel]] $U$, according to Exp. \ref{UnitaryQuantumChannelsAsQUantumStateTransformations}, is invertible (its inverse given by the inverse unitary quantum channel) and hence it induces a functor between the [[Kleisli categories]] (by the discussion [here](monad#ExtensionOfFreeModalesAlsoIsomorphicMonadTransformation)) which on the quantum-state contextful scalars $\mathcal{O}_A$ (Exp. \ref{QuantumObservablesAsQuantumStateContextfulScalars}) is given by [[conjugation action|conjugation]] with $U$:
$$
  \array{
    (\text{-})
    \otimes
    \mathscr{H}_1
    \otimes
    \mathscr{H}_1^\ast
    &
    \overset{
      (\text{-})
      \otimes
      U \otimes U^{\dagger\ast}
    }{\longleftarrow}
    &
    (\text{-})
    \otimes
    \mathscr{H}_2
    \otimes
    \mathscr{H}_2^\ast
    \\
    Kl(\mathscr{H}Store)
    &\longrightarrow&
    Kl(\mathscr{H}Store)
    \\
    \left[
    \array{
      \mathscr{H}_1
      \otimes
      \mathscr{H}_1^\ast
      \\
      \Big\downarrow\mathrlap{ \mathcal{O}_A }
      \\
      \mathbb{1}
    }
    \right]
    &\mapsto&
    \left[
    \array{
      \mathscr{H}_2
      \otimes
      \mathscr{H}_2^\ast
      \\
      \Big\downarrow\mathrlap{
        \mathcal{O}_{ U \cdot A \cdot U^\dagger }
      }
      \\
      \mathbb{1}
    }
    \;\;\;\;\;\;
    \right]
  }
$$

This transformation

$$
  A \;\mapsto\; U \cdot A \cdot U^\dagger
$$

is of course the [[Heisenberg picture]]-evolution of [[quantum observables]] $A$ under unitary transformations $U$ of the [[quantum states]].

\end{example}

## Properties

### Interaction with QuantumEnvironment monad

\begin{proposition}
\label{QuantumStoreDistributesOverQuantumReader}
For

* $\mathscr{H}$ a [[dualizable object|dualizable]] linear type

* $W$ a [[finite set|finite]] classical type,

we have:

1. the $\mathscr{H}$QuantumState comonad $\mathscr{H}^\ast Store$ [distributes over](distributive+law#ComonadDistributingOverMonad) the $W$[[quantum reader monad|QuantumEnvironment]] monad $\bigcirc_W$,

1. the $W$-[[quantum reader monad|QuantumEnvironment]] comonad $\bigstar_W$ [distributes over](distributive+law#ComonadDistributingOverMonad) over the $\mathscr{H}$QuantumState monad $\mathscr{H}^\ast Store$.

\begin{imagefromfile}
    "file_name": "DistrStrucQuantumStoreOverQuantumReader-231002.jpg",
    "width": 700,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 0
    }
\end{imagefromfile}

\end{proposition}
\begin{proof}
  The relevant conditions (from  [Brookes & Van Stone 1993 Def. 3](distributivity+law#BrookesVanStone93), see [there](distributive+law#MonadDistributingOverComonad)) all follows immediately from the distributivity of the [[tensor product]] (being a [[left adjoint]]) over the [[direct sum]] (being a [[coproduct]]).

For definiteness, we spell it out. In the first direction we have:

\begin{imagefromfile}
    "file_name": "DistrCondQuantumStoreOverQuantumEnvrn-231002.jpg",
    "width": 800,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 0
    }
\end{imagefromfile}

and in the converse direction we have:

\begin{imagefromfile}
    "file_name": "DistrCondQuantumEnvrnOverQuantumStore-231002.jpg",
    "width": 800,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 0
    }
\end{imagefromfile}

\end{proof}

\begin{example}
\label{QuantumStateReaderKleisliCategory}
  The two-sided [[Kleisli category]] induced by Prop. \ref{QuantumStoreDistributesOverQuantumReader} (via [Brookes & Van Stone 1993 Thm. 2](distributivity+law#BrookesVanStone93)) equivalently has (by arguing $W$-wise just as in Prop. \ref{KleisliCompositionIsSuperoperatorComposition}):

* as morphisms $W$-indexed sets of linear operators

\begin{imagefromfile}
    "file_name": "QuantumStateReaderKleisliMap-230929b.jpg",
    "width": 700,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 0
    }
\end{imagefromfile}

* with composition given by their $W$-wise operator products

\begin{imagefromfile}
    "file_name": "QuantumStateReaderKleisliComposition-230929b.jpg",
    "width": 700,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 0
    }
\end{imagefromfile}

\end{example}

## Related concepts

* [[quantum reader monad]]

[[!redirects quantum costate comonads]]

[[!redirects quantum store comonad]]
[[!redirects quantum store comonad]]

[[!redirects linear costate comonad]]
[[!redirects linear costate comonads]]

[[!redirects linear store comonad]]
[[!redirects linear store comonad]]


quantum state preparation

+-- {: .rightHandSide}
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum physics]] and especially in [[quantum information theory]], by *state preparation* one refers to a process (either in actual [[experiments]] or theoretically as a kind of [[quantum gate]]-operation) which, possibly conditioned on a classical parameter $b$, "produces" a predefined [[quantum state]] $\vert b \rangle \,\in\, \mathscr{H}$ in a given [[space of quantum states]] $\mathscr{H}$.

For example, for a [[qbit]] [[data type]] there are, by definition of qbits, two canonical basis states $\vert 0 \rangle, \vert 1 \rangle \,\in\, \mathbb{C}^2$ to be prepared, depending on a classical parameter which may be understood as of [[boolean]] [[type]] (where we set $Bool \,\coloneqq\, \{0, 1\}$).

The various fundamental quantum physics phenomena at play here may be identified with four [[modality|modal]]-[[unit of a monad|units]] of [quantum modal logic](necessity+and+possibility#ModalQuantumLogic) (as discussed at *[[quantum circuits via dependent linear types]]*):

\begin{imagefromfile}
    "file_name": "QCwthLHT-QuantumModalUnits-221110.jpg",
    "width": "800",
    "unit": "px",
    "margin": {
        "top": -10,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

|  |  |
|--|--|
| [[quantum measurement]] | quantum state preparation |
| [[quantum superposition]] | [[quantum parallelism]] |

Explicitly, in the language[^1] of [modal quantum logic](necessity+and+possibility#ModalQuantumLogic) on [[dependent linear types]] (see at *[[quantum circuits via dependent linear types]]*), the conditional preparation of a [[qbit]]-state reads as follows:

[^1]: The above text follows *[[schreiber:Quantum Certification via Linear Homotopy Types]]*.

\begin{imagefromfile}
    "file_name": "QCwthLHT-QBitStatePreparationCondt-221021.jpg",
    "width": "500",
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

and the unconditional state preparation is of this form:

\begin{imagefromfile}
    "file_name": "QCwthLHT-QBitStatePreparation-221021.jpg",
    "width": "500",
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

Other states may be prepared by first preparing a [[qbit]]-state and then sending it through some [[quantum gate]].

For instance, the [[Bell state]] on two [[qbits]] may be prepared by first preparing a [[tensor product]] of two $\vert 0 \rangle$-states and then sending one through a [[Hadamard gate]] and then the resulting two qbits through a [[CNOT gate]]:

\begin{imagefromfile}
    "file_name": "QCwthLHT-BellStatePreparation-221021.jpg",
    "width": "800",
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

This is used in many common [[quantum circuits]], such as for instance in the [[quantum teleportation]]-protocol.

## References

Standard textbook accounts:

* [[Michael A. Nielsen]], [[Isaac L. Chuang]], §7.2 in: *Quantum computation and quantum information*, Cambridge University Press (2000) [[doi:10.1017/CBO9780511976667](https://doi.org/10.1017/CBO9780511976667), [pdf](http://csis.pace.edu/~ctappert/cs837-19spring/QC-textbook.pdf), [[NielsenChuangQuantumComputation.pdf:file]]]

[[!redirects quantum state preparations]]

[[!redirects state preparation]]
[[!redirects state preparations]]


quantum state tomography

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#Contents#
* table of contents
{:toc}

## Idea

In [[quantum information theory]], by *quantum state tomography* one means [[quantum measurement]]-protocols which, when applied successively to copies of one and the same [[quantum state]] ([[quantum state collapse|collapsing]] one copy at a time), allow to determine that quantum state, to prescribed accuracy.

## References

* Joseph B. Altepeter, Daniel F.V. James & Paul G. Kwiat, *Qubit Quantum State Tomography*, in *Quantum State Estimation*, Lecture Notes in Physics **649**, Springer (2004) [[doi:10.1007/978-3-540-44481-7_4](https://doi.org/10.1007/978-3-540-44481-7_4), [pdf](http://research.physics.illinois.edu/QI/Photonics/tomography-files/tomo_chapter_2004.pdf)]

See also:

* Wikipedia, *[Quantum tomography](https://en.wikipedia.org/wiki/Quantum_tomography)*

* [[Matthias Christandl]], [[Renato Renner]], *Reliable Quantum State Tomography*, Physical Review Letters **109** 120403 (2012) [[arXiv:1108.5329](https://arxiv.org/abs/1108.5329), [doi:10.1103/PhysRevLett.109.120403](https://doi.org/10.1103/PhysRevLett.109.120403)]


quantum statistical mechanics

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## Idea

The [[quantum physics]]-version of [[statistical mechanics]].

For the moment see at _[[quantum probability]]_.

## Related concepts

* [[Bayesian interpretation of quantum mechanics]]

## References

* {#BratteliRobinson79} [[Ola Bratteli]], [[Derek W. Robinson]], *Operator Algebras and Quantum Statistical Mechanics* -- vol 1: *$C^\ast$- and $W^\ast$-Algebras. Symmetry Groups. Decomposition of States.*, Springer (1979, 1987, 2002) [[doi:10.1007/978-3-662-02520-8](https://doi.org/10.1007/978-3-662-02520-8)]

See also

* Wikipedia, _[Quantum statistical mechanics](https://en.wikipedia.org/wiki/Quantum_statistical_mechanics)_


quantum superposition

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* table of contents
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## Idea

In [[quantum theory]] the [[state]] of a [[physical system]] is given by a [[vector]] in a [[complex vector space]] (usually a [[Hilbert space]]). This means that given any two states $\psi_1$ and $\psi_2$, their sum, the vector $\psi_1 + \psi_2$, also represents a state. This is called the _quantum superposition_ of the two states.

If these quantum states are thought of as [[wavefunctions]] solving a [[linear differential equation]] such as the [[Schroedinger equation]], [[Dirac equation]], [[Klein-Gordon equation]] or [[Tomonaga-Schwinger equation]], their quantum superposition is an example of the general principle of *[[superposition]]* of solutions to [[linear differential equations]].

The concept of quantum superposition is at the heart of what makes [[quantum physics]] peculiar, notably via the phenomenon of [[entanglement]].

The various fundamental quantum physics phenomena at play here may be identified with four [[modality|modal]]-[[unit of a monad|units]] of [quantum modal logic](necessity+and+possibility#ModalQuantumLogic) (as discussed at *[[quantum circuits via dependent linear types]]*):

\begin{imagefromfile}
    "file_name": "QCwthLHT-QuantumModalUnits-221110.jpg",
    "width": "800",
    "unit": "px",
    "margin": {
        "top": -10,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

|  |  |
|--|--|
| [[quantum measurement]] | [[quantum state preparation]] |
| quantum superposition | [[quantum parallelism]] |

## Related concepts

* [[quantum parallelism]]

* [[superposition]]

[[!include states and observables -- content]]

## References

See also

* Wikipedia, _[Quantum superposition](https://en.wikipedia.org/wiki/Quantum_superposition)_

[[!redirects quantum superpositions]]

[[!redirects superposition state]]
[[!redirects superposition states]]


quantum supremacy

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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion that for certain tasks the power of [[quantum computation]] potentially drastically surpasses that of [[classical physics|classical]] [[computation]] (e.g. via [[Shor's algorithm]]) has been called *quantum supremacy* ([Preskill 2013](#Preskill13), there related to the computational resource of [[quantum entanglement]]) or *quantum advantage*.

## Related concepts

* [[quantum complexity theory]]

## References
 {#References}

### General

The term "quantum supremacy" is due to:

* {#Preskill13} [[John Preskill]]: *Quantum computing and the entanglement frontier*: pp. 63-80 in: *The Theory of the Quantum World -- Proceedings of the 25th Solvay Conference on Physics*, World Scientific (2013) [[arXiv:1203.5813](https://arxiv.org/abs/1203.5813), [doi:10.1142/8674](https://doi.org/10.1142/8674), slides: [pdf](https://simons.berkeley.edu/sites/default/files/docs/394/preskilljohn.pdf)]

See also:

* Rønnow et al. *Defining and detecting quantum speedup*, Science **345** 6195 (2014) 420-424 [[doi:10.1126/science.1252319](https://doi.org/10.1126/science.1252319)]

* Wikipedia, *[Quantum supremacy](https://en.wikipedia.org/wiki/Quantum_supremacy)*

Review:

* [[Scott Aaronson]], *How Much Structure Is Needed for Huge Quantum Speedups?*, talk at: *28th Solvay Physics Conference* in Brussels on May 21, 2022 [[arXiv:2209.06930](https://arxiv.org/abs/2209.06930), slides: [ppt](https://www.scottaaronson.com/talks/aar-solvay.ppt)]

Emphasis of the need of [[topological quantum computation|topological stabilization]]:

* {#DasSarma22} [[Sankar Das Sarma]], *Quantum computing has a hype problem*, [MIT Technology Review (March 2022)](https://www.technologyreview.com/2022/03/28/1048355/quantum-computing-has-a-hype-problem/)

  > The [[qubit]] systems we have today are a tremendous scientific achievement, but they take us no closer to having a quantum computer that can solve a problem that anybody cares about. $[\cdots]$ What is missing is the breakthrough $[\cdots]$ bypassing [[quantum error correction]] by using far-more-stable qubits, in an approach called [[topological quantum computing]].

See also:

* Takashi Yamakawa, Mark Zhandry, *Verifiable Quantum Advantage without Structure* [[arXiv:2204.02063](https://arxiv.org/abs/2204.02063)]

### Experimental realizations

Claimed experimental demonstration of "quantum supremacy" ("quantum advantage") in (very) special purpose applications:

* F. Arute et al. *Quantum supremacy using a programmable superconducting processor*, Nature **574** (2019) 505–510 ([doi:10.1038/s41586-019-1666-5](https://doi.org/10.1038/s41586-019-1666-5))

* Han-Sen Zhong et al., *Quantum computational advantage using photons*, Science  **370** 6523 (2020) 1460-1463 ([doi:10.1126/science.abe8770](https://science.sciencemag.org/content/370/6523/1460))

Critical analysis:

* [[Gil Kalai]], Yosef Rinott, Tomer Shoham, *Google's 2019 "Quantum Supremacy" Claims: Data, Documentation, and Discussion* [[arXiv:2210.12753](https://arxiv.org/abs/2210.12753)]

{#ExperimentalReview} Review and status reports:

* Adrian Cho, *Google claims quantum computing milestone*, Science **365** 6460 (2019) 1364 ([doi:10.1126/science.365.6460.1364](https://science.sciencemag.org/content/365/6460/1364))

* [[Philip Ball]], *Physicists in China challenge Google’s "quantum advantage"*, Nature **588** 380 (2020) [[doi:10.1038/d41586-020-03434-7](https://doi.org/10.1038/d41586-020-03434-7)]

* Barry C. Sanders, *[Quantum Leap for Quantum Primacy](https://physics.aps.org/articles/v14/147)*, Physics **14** 147 (2021)

Quantum advantage for [[Grover's algorithm]]:

* Bibek Pokharel, Daniel Lidar, *Better-than-classical Grover search via quantum error detection and suppression* [[arXiv:2211.04543](https://arxiv.org/abs/2211.04543)]

See also:

* Réouven Assouly, Rémy Dassonneville, Théau Peronnin, Audrey Bienfait, Benjamin Huard, *Demonstration of Quantum Advantage in Microwave Quantum Radar* [[arXiv:2211.05684](https://arxiv.org/abs/2211.05684)]

### Potential applications

Discussion of [[algorithms]] for which quantum computers are expected to outperform classical computers:

The case of [[Shor's algorithm]], relevant for [[quantum cryptography]]:

* [[Peter W. Shor]], *Algorithms for quantum computation: discrete logarithms and factoring*,  Proceedings 35th Annual Symposium on Foundations of Computer Science, IEEE Comput. Soc. Press: 124-134 (1994) [[doi:10.1109/SFCS.1994.365700](https://doi.org/10.1109/SFCS.1994.365700)]

* [[Daniel R. Simon]], *On the power of quantum computation*, SIAM Journal on Computing **26** 5 (1997) [[doi:10.1137/S0097539796298637](https://doi.org/10.1137/S0097539796298637), [pdf](https://courses.cs.washington.edu/courses/cse599/01wi/papers/simon_qc.pdf)]

The case of [[Grover's algorithm]], for database searches:

* [[Lov K. Grover]], *A fast quantum mechanical algorithm for database search*, STOC '96: Proceedings of the twenty-eighth annual ACM symposium on Theory of Computing (1996) 212-219 [[arXiv:quant-ph/9605043](https://arxiv.org/abs/quant-ph/9605043), [doi:10.1145/237814.237866](https://doi.org/10.1145/237814.237866)]

The case of the [[Deutsch-Jozsa algorithm]]:

* [[David Deutsch]], [[Richard Jozsa]], *Rapid solution of problems by quantum computation*. Proc. R. Soc. Lond. A **439** 1907 (1992) 553-558 [[pdf](https://www.isical.ac.in/~rcbose/internship/lectures2016/rt08deutschjozsa.pdf), [doi:10.1098/rspa.1992.0167](https://doi.org/10.1098/rspa.1992.0167)]

For [[integral transforms]]:

* Doğa Veske et al., *Quantum Advantage for Integral Transforms* [[arXiv:2204.04159](https://arxiv.org/abs/2204.04159)]

Survey:

* Andrew M. Childs, Wim van Dam, *Quantum algorithms for algebraic problems*, Rev. Mod. Phys. **82** 1 (2010) [[arXiv:0812.0380](https://arxiv.org/abs/0812.0380), [doi:10.1103/RevModPhys.82.1](https://doi.org/10.1103/RevModPhys.82.1)]

Application of quantum computing to [[quantum chemistry]]:

* [[Seth Lloyd]], *Universal Quantum Simulators*, Science New Series **273** 5278 (1996) 1073-1078 [[jstor:2899535](https://www.jstor.org/stable/2899535), [doi:10.1126/science.273.5278.1073](https://doi.org/10.1126/science.273.5278.1073)]

* *Quantum Chemistry in the Age of Quantum Computing*, Chem. Rev. (2019) **119** 19 (2019) 10856–10915 [[doi:10.1021/acs.chemrev.8b00803](https://doi.org/10.1021/acs.chemrev.8b00803)]

Argument that quantum supremacy in quantum chemistry is elusive:

* [[Garnet Kin-Lic Chan]] et al., *Is there evidence for exponential quantum advantage in quantum chemistry?* [[arXiv:2208.02199](https://arxiv.org/abs/2208.02199), [talk video](https://youtu.be/DZPH7ENcRLU)]

Application to [[solid state physics]]:

* Nobuyuki Yoshioka, Tsuyoshi Okubo, Yasunari Suzuki, Yuki Koizumi, Wataru Mizukami, *Hunting for quantum-classical crossover in condensed matter problems* [[arXiv:2210.14109](https://arxiv.org/abs/2210.14109)]

### Quantum complexity theory

Discussion of [[complexity theory|complexity-theoretic]] aspects:

* [[Scott Aaronson]], [[Lijie Chen]], *Complexity-Theoretic Foundations of Quantum Supremacy Experiments*, CCC '17: Proceedings of the 32nd Computational Complexity Conference (2017) **22** 1–67 [[arXiv:1612.05903](https://arxiv.org/abs/1612.05903), [doi;10.5555/3135595.3135617](https://dl.acm.org/doi/10.5555/3135595.3135617)]

[[!redirects quantum advantage]]


quantum system

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## Idea

A system in

* [[quantum mechanics]]

* [[quantum field theory]]

* [[quantum probability]]

* [[quantum topology]]

* [[quantum information theory]]

* [[quantum computing]]

## Related concepts

* [[classical mechanical system]]

* [[subsystem]], [[composite system]]

* [[open quantum system]]

## References

See also the references at *[[quantum mechanics]]*.

### General

* Kurt Engesser, Dov M. Gabbay, Daniel Lehmann (eds.) *[[Handbook of Quantum Logic and Quantum Structures]]* Elsevier (2009) [[ISBN:9780080931661](https://www.sciencedirect.com/book/9780444528698)]

In a context of [[topological phases of matter]]:

* [[Bei Zeng]], [[Xie Chen]], [[Duan-Lu Zhou]], [[Xiao-Gang Wen]]:

  Sec. 2 of: *[[Quantum Information Meets Quantum Matter]] -- From Quantum Entanglement to Topological Phases of Many-Body Systems*, Quantum Science and Technology (QST), Springer (2019) $[$[arXiv:1508.02595](https://arxiv.org/abs/1508.02595), [doi:10.1007/978-1-4939-9084-9](https://doi.org/10.1007/978-1-4939-9084-9)$]$

### Further resources

* [[Center for Quantum and Topological Systems]]

[[!redirects quantum systems]]


quantum systems -- contents

**[[quantum logic]]**

* [[linear logic]], [[dependent linear type theory|dependent]] [[linear type theory]]

  [[string diagrams]] in [[quantum information theory via dagger-compact categories|†-compact categories]]

  [[tensor networks]]

* [[Bohr topos]]

  [[order-theoretic structure in quantum mechanics|order-theoretic structure]]

* [[quantum probability]]

\linebreak

**[[quantum physics]]**

* [[quantum systems]]

  ([[parameterized quantum systems|parameterized]], [[open quantum system|open]])

  [[quantum measurement]]

  [[quantum state collapse]]

  [[quantum decoherence]]

  [[quantum adiabatic theorem]]

  [[Berry phases]]

  [[Dyson formula]]

* [[quantum many-body physics]]

* **[[quantum field theory]]**

  [[functorial quantum field theory]]

  [[algebraic quantum field theory]]

  ([[non-perturbative quantum field theory|non-]])[[perturbative quantum field theory]]

* **[[solid state physics]]**

  [[quantum material]]

  ([[topological phases of matter|topological]]) [[phases of matter]]

\linebreak

[[!include states and observables -- content]]

\linebreak

**[[quantum information]]**

* [[quantum information via dagger-compact categories]]

* [[quantum operation]], [[quantum channel]]

* [[quantum teleportation]]

* [[quantum entanglement]]

* [[entanglement entropy]]

  [[holographic entanglement entropy]]

  [[topological entanglement entropy]]

\linebreak

**[[quantum computation]]**

* [[adiabatic quantum computation]]

* [[measurement-based quantum computation]]

* [[topological quantum computation]]

* [[quantum gate]], [[quantum circuit]]

* [[quantum programming language]]

* [[quantum error correction]]

  * [[HaPPY code]]

  * [[Majorana dimer code]]

[[qbit]]

* [[spin resonance qbit]]

quantum algorithms:

* [[Grover's algorithm]]

* [[Shor's algorithm]]

\linebreak

**[[quantum sensing]]**

\linebreak

**[[quantum communication]]**


quantum Teichmüller theory

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## Idea

Quantum Teichnüller theory studies a [[deformation quantization]] of [[Teichmüller space]].

According to a conjecture in ([Verlinde 90](#Verlinde90)) the corresponding [[Hilbert space]] may be identified with the space of [[conformal blocks]] of [[Liouville theory]].

## References

The conjectured relation to [[Liouville theory]] is due to

* {#Verlinde90} [[Herman Verlinde]], _Conformal field theory, two-dimensional quantum gravity and quantization of Teichmüller space_. Nuclear Physics B 337.3 (1990): 652-680.

A review is in section 2 of

* {#Allegretti} [[Dylan Allegretti]], _Notes on Quantum Teichmüller theory_ ([pdf](http://users.math.yale.edu/~dga4/TeichmullerNotes.pdf))

[[!redirects quantum Teichmuller theory]]


quantum teleportation

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## Idea

What has been called quantum "teleportation" ([BBCJPW 1993](#BBCJPW93)) is a procedure (a "protocol") in [[quantum information theory]] by which the exact transmission of a [[quantum state]] (such as that of some [[qbits]]) is equivalently established by the transmission of [[classical physics|classical]] information, *provided* that sender and receiver "share a [[Bell state]]", hence that each have access to one half of a pair of [[subsystems]] which have been [[quantum state preparation|prepared]] in a maximally [[entanglement|entangled]] [[quantum state]] with each other. The protocol involves a [[quantum measurement]] and its ensuing [[wavefunction collapse]] destroys ("uses up") this entangled state in order to transmit the remaining quantum state -- which somewhat relativizes the imagery of "teleportation".

While this prose may appear somewhat exotic, the actual mathematics underlying the "quantum teleportation protocol" is an elementary and direct consequence of the most basic [[axioms]] of [[quantum mechanics]] (in fact it is all but a self-evident consequence when formulated in modern notation, see [below](#HistoryAndPerspective)). What makes quantum teleportation interesting is the ([[quantum information theory|quantum]]) [[information theory|information theoretic]] perspective on it.

## History and perspective
 {#HistoryAndPerspective}

Given that the basic axioms of quantum physics were established in the 1920s and widely appreciated by the 1930s, the observation of the possibility of "quantum teleportation" only [in 1993](#BBCJPW93), using tools and notation available to the founding fathers, may make one wonder (e.g [Coecke 2010, p. 1](#Coecke10)):

> why did it take us 60 years to discover the conceptually
intriguing and easily derivable physical phenomenon of 'quantum teleportation'?

Traditional reviews of the matter maintain that the phenomenon just happens to be intrinsically tricky to see through -- e.g. [Aaronson 2018](#Aaronson18), [p. 71](https://www.scottaaronson.com/qclec.pdf#page=71), apparently in reply to his student's puzzlement "How do people come up with this stuff?", asserts that:

> These sorts of protocols can be hard to find.

On the other hand, it was observed around [Coecke 2004](#Coecke04), [Abramsky & Coecke 2004](#AbramskyCoecke04) -- and pronouncedly brought out in [Coecke 2005](#Coecke05), [2010](#Coecke10) -- that the logical structure of [[quantum information theory]] in general and that of quantum teleportation in particular is transparently captured by [[string diagram]]-calculus for the ambient [[compact closed category|compact closed]] [[tensor category]] of [[finite dimensional vector space|finite dimensional]] [[Hilbert spaces]] (see *[[finite quantum mechanics in terms of dagger-compact categories]]*):

In this [[category theory|category-theoretic]] language, the quantum teleportation protocol is essentially just the [[zig-zag identity]] for [[dualizable objects]] (here: [[finite dimensional vector space|finite-dimensional]] [[Hilbert spaces]]) in any [[monoidal category]], see [below](#StatementInStringDiagramLanguage).

This suggests the following alternative answer, from [Coecke 2005, p. 1](#Coecke05):

> Why did discovering quantum teleportation take 60 year?

> We claim that this is due to a ‘bad quantum formalism’ (bad $\neq$ wrong) and this badness is in particular due to the fact that the formalism is ‘too low level'.

Here "bad quantum formalism" refers to the traditional notation as in the original [BBCJPW93](#BBCJPW93) and in traditional reviews (e.g. [Aaronson 2018](#Aaronson18), see also [Coecke 05](#Coecke05) [ftn. 5 on p. 8](https://arxiv.org/pdf/quant-ph/0510032.pdf#page=8)), while the suggested high-level improvement is [[string diagram]]-calculus for the ambient [[tensor category]] of finite-dimensional Hilbert spaces, further discussed at *[[finite quantum mechanics in terms of dagger-compact categories]]*.

## Statement
 {#Statement}

{#StatementInStringDiagramLanguage}The quantum teleportation protocol is neatly expressed in [[string diagram]]-calculus as follows (essentially following [Coecke 2005](#Coecke05)):

Here, by the general laws of [[string diagrams]]: a solid line pointing to the right represents a given [[finite dimensional vector space|finite dimensional]] [[Hilbert space]] $\mathscr{H} \,\simeq\, \mathbb{C}^D$ (of [[dimension of a vector space|dimension]] $D$), the same line pointing backwards denotes its [[dual vector space]], two lines running parallel reflects the [[tensor product of Hilbert spaces|tensor product]] of the corresponding Hilbert spaces, and the curve "$\supset$" denotes [[evaluation]] $\mathscr{H}^\ast \otimes \mathscr{H} \xrightarrow{ev} \mathbb{C}$.

The wiggly line denotes a classically [[controlled quantum gate]], controlled by the [[quantum measurement]]-outcome which is reflected by the projection operator shown at the top.

That --  for any fixed measurement outcome at the top with the correspondingly controlled correction gate operation at the bottom -- this [[string diagram]] (represents a [[linear map]] which) takes the [[quantum state]] $\Psi$ from "Agent A" ("Alice") to the same state $\Psi$ for "Agent B" ("Bob"), as shown is *immediate* from the [[zig-zag identity]], hence from the fact that the curved line may be "yanked straight" to an [[identity]] line, without changing the operational value of the diagram. This is the transparent [[proof]] of the quantum teleporation protocol.

Notice that this is a [[rigorous proof]], and yet this [[string diagram]] is close to informal flow-charts traditionally used to illustrate the idea of quantum teleportation (e.g [Bouwmeester, Ekert & Zeilinger 2020, Fig. 3.1](#BouwmeesterEkertZeilinger20)).

In more detail, here the top box shows the [[projector]] onto a [[Bell state]] with one variable transformed by a [[unitary operator]] $\sigma$. This reflects the (random/unpredictable!) result of a [[quantum measurement]] in a corresponding  [[linear basis]] for $\mathscr{H} \otimes \mathscr{H}^\ast$ (according to the [[wavefunction collapse|collapse postulate]]).

Concretely, if the [[Hilbert space]] represented by a single solid line is that of a single [[qbit]], i.e. if $\mathscr{H} = \mathbb{C}^2$ ($D = 2$), then  $\sigma$ may be taken to range over the [[unitary operator|unitary]] [[Pauli matrices]]:
$$
  \sigma_0
  \coloneqq
  \left(
    \begin{array}{cc}
      1 & 0
      \\
      0 & 1
    \end{array}
  \right)
   ,\;\;\;\;
  \sigma_1
  \coloneqq
  \left(
    \begin{array}{cc}
      0 & 1
      \\
      1 & 0
    \end{array}
  \right)
   ,\;\;\;\;
  \sigma_2
  \coloneqq
  \left(
    \begin{array}{cc}
      0 & -\mathrm{i}
      \\
      \mathrm{i} & 0
    \end{array}
  \right)
   ,\;\;\;\;
  \sigma_3
  \coloneqq
  \left(
    \begin{array}{cc}
      1 & 0
      \\
      0 & -1
    \end{array}
  \right)
  \,.
$$

{#TraditionalCircuit} In this case, the component-evaluation of the above diagram yields the quantum teleportation protocol in its traditional form (due to [BBCJPW93](#BBCJPW93)) which is the following [[quantum circuit]] (cf. [GLRSV13, p. 5](Quipper#GLRSV13)) on [[qbits]] (where "$H$" denotes the [[Hadamard gate]], the circles denote the [[quantum CNOT gates]] and the boxed pointers denote [[quantum measurement]] in the [[qbit]]-basis):

\linebreak

The concrete implementation of the above protocol on given quantum hardware will look more complicated.

{#AsABraidGateCircuit} For instance, the implementation of the quantum teleportation protocol as a circuit of [[braid representation|braid]] gates of [[Ising anyons]] ([[topological quantum computation]]) is given in [Xu & Zhou 2022](#XuZhou22):

See also [Valera (2023)](#Valera23).

\linebreak

## Related concepts

* [[superdense coding]]

* [[quantum entanglement]]

* [[measurement-based quantum computation]]

* [[no-cloning theorem]]

* [[quantum information theory]]

## References

The original article:

* {#BBCJPW93} [[Charles H. Bennett]], [[Gilles Brassard]], [[Claude Crépeau]], [[Richard Jozsa]], [[Asher Peres]], [[William K. Wootters]]: *Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels*, Phys. Rev. Lett. **70** 1895 (1993) [[doi:10.1103/PhysRevLett.70.1895](https://doi.org/10.1103/PhysRevLett.70.1895)]

Traditional review:

* [[Michael A. Nielsen]], [[Isaac L. Chuang]], §1.3.7 in: *Quantum computation and quantum information*, Cambridge University Press (2000) [[doi:10.1017/CBO9780511976667](https://doi.org/10.1017/CBO9780511976667), [pdf](http://csis.pace.edu/~ctappert/cs837-19spring/QC-textbook.pdf), [[NielsenChuangQuantumComputation.pdf:file]]]

* {#Aaronson18} [[Scott Aaronson]], §10.1 in *Introduction to Quantum Information Science* (2018) [[pdf](https://www.scottaaronson.com/qclec.pdf), [webpage](https://www.scottaaronson.com/cs378/)]

and phrased in the [[quantum programming language]] [[Quipper]]:

* [[Alexander Green]], [[Peter LeFanu Lumsdaine]], [[Neil Ross]], [[Peter Selinger]], [[Benoît Valiron]], p. 5 of:
_An Introduction to Quantum Programming in Quipper_, Lecture Notes in Computer Science 7948:110-124, Springer, 2013 ([arXiv:1304.5485](https://arxiv.org/abs/1304.5485))

and with focus on the [[experiment|experimental]] realization:

* {#BouwmeesterEkertZeilinger20} [[Dirk Bouwmeester]], [[Artur Ekert]], [[Anton Zeilinger]] (eds.), §3.3 of: *The Physics of Quantum Information -- Quantum Cryptography, Quantum Teleportation, Quantum Computation*, Springer (2020) [[doi:10.1007/978-3-662-04209-0](https://doi.org/10.1007/978-3-662-04209-0)]

See also:

* Wikipedia, *[Quantum teleportation](https://en.wikipedia.org/wiki/Quantum_teleportation)*

Discussion via [[string diagram]]-calculus ([[finite quantum mechanics in terms of dagger-compact categories]]):

* {#Coecke04} [[Bob Coecke]], §4 of: *The logic of entanglement*, in: *Horizons of the Mind. A Tribute to Prakash Panangaden*, Lecture Notes in Computer Science **8464** (2014)  [[arXiv:quant-ph/0402014](https://arxiv.org/abs/quant-ph/0402014), [doi:10.1007/978-3-319-06880-0_13](https://doi.org/10.1007/978-3-319-06880-0_13)]

* {#AbramskyCoecke04} [[Samson Abramsky]], [[Bob Coecke]], §2 in: *A categorical semantics of quantum protocols*, Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS'04). IEEE Computer Science Press (2004) [[arXiv:quant-ph/0402130](https://arxiv.org/abs/quant-ph/0402130), [doi:10.1109/LICS.2004.1319636](https://doi.org/10.1109/LICS.2004.1319636)]

* {#Coecke05} [[Bob Coecke]], §3c in: *Kindergarten Quantum Mechanics*, in *Quantum Theory: Reconsideration of Foundations* (QTRF 3) Vaxjo, Sweden, June 6-11, 2005, AIP Conf. Proc. **810** (2006) [[arXiv:quant-ph/0510032](https://arxiv.org/abs/quant-ph/0510032), [doi:10.1063/1.2158713](https://doi.org/10.1063/1.2158713)]

* {#Coecke10} [[Bob Coecke]], *Quantum Picturalism*, Contemporary Physics **51** (2010) 59-83 [[arXiv:0908.1787](https://arxiv.org/abs/0908.1787), [doi:10.1080/00107510903257624](https://doi.org/10.1080/00107510903257624)]

Realization in [[experiment]]:

* [[Dirk Bouwmeester]], Jian-Wei Pan, Klaus Mattle, Manfred Eibl, [[Harald Weinfurter]], [[Anton Zeilinger]], *Experimental quantum teleportation*, Nature **390** 575–579 (1997) [[doi:10.1038/37539](https://doi.org/10.1038/37539)]

* Syed Tahir Amin, [[Aeysha Khalique]], *Practical Quantum Teleportation of an Unknown Quantum State*, Can. J. Phys. **95** 5 (2017) 498 [[arXiv:1508.01141](https://arxiv.org/abs/1508.01141), [doi:10.1139/cjp-2016-0758](https://doi.org/10.1139/cjp-2016-0758)]

* Dario Lago-Rivera, Jelena V. Rakonjac, Samuele Grandi, Hugues de Riedmatten, *Long-distance multiplexed quantum teleportation from a telecom photon to a solid-state qubit* [[arXiv:2209.06249](https://arxiv.org/abs/2209.06249)]

Implementation of the quantum teleportation protocol in [[topological quantum computation]] via [[braiding]] of [[Ising anyons]]:

* {#XuZhou22} Cheng-Qian Xu, D. L. Zhou, *Quantum teleportation using Ising anyons*, Phys. Rev. A **106** 012413 (2022) [[arXiv:2201.11923](https://arxiv.org/abs/2201.11923), [doi:10.1103/PhysRevA.106.012413](https://doi.org/10.1103/PhysRevA.106.012413)]

and via fusion of anyons:

* {#Valera23} [[Sachin J. Valera]], *Topological Quantum Teleportation and Superdense Coding Without Braiding* [[arXiv:2303.17700](https://arxiv.org/abs/2303.17700)]

Implementation in [[quantum programming languages]]:

* *[Qiskit: Quantum Teleportation](https://qiskit.org/textbook/ch-algorithms/teleportation.html)*

Argument that the image of quantum teleportation under [[AdS-CFT duality]] is *[[wormhole]] traversal* (see also at *[[ER = EPR]]*):

* [[Erik Verlinde]], [[Herman Verlinde]], *A Conversation on ER = EPR* [[arXiv:2212.09389](https://arxiv.org/abs/2212.09389)]

* [[Ping Gao]], [[Daniel Louis Jafferis]], [[Aron C. Wall]], *Traversable Wormholes via a Double Trace Deformation*, Journal of High Energy Physics **2017** 151 (2017) [[arXiv:1608.05687](https://arxiv.org/abs/1608.05687), doi:10.1007/JHEP12(2017)151]

* [[Juan Maldacena]], [[Douglas Stanford]], [[Zhenbin Yang]],  *Diving into traversable wormholes*,Fortsch. Phys. **65** 5 (2017) 1700034 [[arXiv:1704.05333](https://arxiv.org/abs/1704.05333), [doi:10.1002/prop.201700034](https://doi.org/10.1002/prop.201700034)]

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quantum topology

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## Idea

In [[mathematics]], the term *quantum topology* is used for the [[low-dimensional topology]] (such as [[knot theory]]) and related [[algebra]] (such as [[quantum groups]]) which is informed by [[Chern-Simons theory|Chern-Simons]] [[topological quantum field theory|topological]] [[quantum field theory]], and its various facets and generalizations.

## References

* [[David Yetter]] (ed.), *Quantum Topology*, World Scientific (1994) [[doi:10.1142/2322](https://doi.org/10.1142/2322)]

* [[Louis Kauffman]], *Quantum Topology and Quantum Computing*, in: [[Samuel J. Lomonaco]] (ed.), *Quantum Computation: A Grand Mathematical Challenge for the Twenty-First Century and the Millennium*, Proceedings of Symposia in Applied Mathematics **58**, AMS (2002) [[pdf](https://redirect.cs.umbc.edu/~lomonaco/ams/lecturenotes/Kauffman.pdf), [doi:10.1090/psapm/058](https://doi.org/10.1090/psapm/058)]

  > (in relation to [[topological quantum computation]])

See also:

* Wikipedia, *[Quantum Topology](https://en.wikipedia.org/wiki/Quantum_topology)*


quantum tunneling

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## Related concepts

* [[quantum physics]]

* [[instanton]]

## References

### General

See also

* Wikipedia, _[Quantum tunneling](https://en.wikipedia.org/wiki/Quantum_tunnelling)_

### Quantum halos
 {#ReferencesQuantumHalos}

* A. S. Jensen, K. Riisager, D. V. Fedorov, and E. Garrido, *Structure and reactions of quantum halos*, Rev. Mod. Phys. **76** 215 (2004) [[doi:10.1103/RevModPhys.76.215](https://doi.org/10.1103/RevModPhys.76.215)]

  > In very general terms a *halo* is a diluted or less intense component surrounding a stronger or more massive central object. Quite a few quantum systems could fit this general description, but the way the term has been applied for the last 15 years on the quantum level has been more restrictive, namely, to denote systems in which a wave-function component has an unusually large spatial extension. If this were used as the sole criterion, many systems with widely differing properties could be counted as halos, e.g., Rydberg states in atoms, and the term would lose its scientific usefulness. The tradition has been to restrict the name to a more exclusive set of structures. We shall in Sec. II consider a more precise definition of the quantum halo; it suffices here to note that tunneling into a classically forbidden region should be a pronounced feature.

*  K Riisager, *Halos and related structures*,  Physica Scripta **2013** 014001 (2013)  [[doi:10.1088/0031-8949/2013/T152/014001](https://iopscience.iop.org/article/10.1088/0031-8949/2013/T152/014001)]

See also:

* Wikipedia, *[Halo nucleus](https://en.wikipedia.org/wiki/Halo_nucleus)*

[[!redirects tunneling]]

[[!redirects quantum halo]]
[[!redirects quantum halos]]


quantum Yang-Baxter equation

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#### Quantum systems
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#Contents#
* table of contents
{:toc}

## Idea

Given a [[monoidal category]] $C$ with [[tensor product]] $\otimes$, and an [[object]] $V$ in $C$, a *quantum Yang-Baxter operator* is a [[morphism]] of the form

$$
  R \,\colon\, V\otimes V\to V\otimes V
$$

which satisfies the following *quantum Yang-Baxter equation* in $V\otimes V\otimes V$

$$
  R_{12}
  \,
  R_{13}
  \,
  R_{23}
  \;=\;
  R_{23}
  \,
  R_{13}
  \,
  R_{12}
  \,,
$$

where the subscripts indicate which tensor factors are being utilized, for instance $R_{12} = R\otimes id_V\in V\otimes V\otimes V$.

This equation is in particular satisfied by the component $\mathcal{R}_{V V}$ at $V$ of any [[braided monoidal category|braiding]] $\mathcal{R}$ on $C$.

Typical categories where the equation is considered are

1. the [[category of vector spaces]] when the solutions are called *$R$-matrices* (or *quantum Yang-Baxter matrices*),

1. [[categories of representations]] of [[quantum groups]]; often (a completion of) a quantum group $G$ itself has a particular element $\mathcal{R}$, called a *universal $\mathcal{R}$-element*, satisfying axioms ([[quasitriangular bialgebra|quasitriangularity]]) ensuring that its image in every representation satisfies qYBE

1. the [[category of sets]] where one speaks about [[set theoretic Yang-Baxter equation|set theoretic solutions of Yang-Baxter equation]].

\begin{remark}\label{HistoricalMotivation}
**(Historical motivation)**
\linebreak

The _quantum Yang-Baxter equation_ has been proposed by Baxter in the context of a particular model of [[statistical mechanics]] (the 8-vertex model) and called *star-triangle relation* [[Baxter (1978)](#Baxter78), [Baxter (1982)](#Baxter82)].

Later it has been generalized and axiomatized to a number of contexts: it is most notably satisfied by the universal R-element in a [[quasitriangular Hopf algebra]]. In some context it is equivalent to a braid relation for certain transposed matrix. Some solutions to quantum Yang-Baxter equation have good limits in classical mechanics which are [[classical r-matrices]], and the latter satisfy the [[classical Yang-Baxter equation]].
\end{remark}

## Equation with spectral parameter

With multiplicative spectral parameter, the equation reads

$$
  R_{12} (u)
  \,
  R_{13} (u v)
  \,
  R_{23} (v)
  \;=\;
  R_{23}(v)
  \,
  R_{13}(u v)
  \,
  R_{12}(u)
$$

where the subscripts indicate which tensor factors are being utilized.

## Related concepts

[[!include Yang-Baxter equations -- contents]]

## References

Original texts:

* {#Baxter78} [[Rodney J. Baxter]], *Solvable eight-vertex model on an arbitrary planar lattice*, Philos. Trans. Royal Society A **289** 1359 (1978) [[doi:10.1098/rsta.1978.0062](https://doi.org/10.1098/rsta.1978.0062)]

* {#Baxter82} [[Rodney J. Baxter]], *Exactly Solved Models in Statistical Mechanics*, Academic Press (1982) [[webpage](https://physics.anu.edu.au/research/ftp/mpg/baxter_book.php), [pdf](https://physics.anu.edu.au/research/ftp/_files/Exactly.pdf)]

Introductions:

* [[Michio Jimbo]], *Introduction to the Yang-Baxter Equation*, Int. J. Modern Physics A **4** 15 (1989) 3759-3777 [[doi:10.1142/S0217751X89001503](https://doi.org/10.1142/S0217751X89001503)]

  reprinted in: *Braid Group, Knot Theory and Statistical Mechanics*, Advanced Series in Mathematical Physics **9**, World Scientific (1991) [[doi:10.1142/0796](https://doi.org/10.1142/0796)]

See also:

* Wikipedia, *[Yang-Baxter equation](https://en.wikipedia.org/wiki/Yang%E2%80%93Baxter_equation)*

* [[eom]], *[Yang-Baxter equation](https://encyclopediaofmath.org/index.php?title=Yang-Baxter_equation)*

Further discussion in the context of [[quantum groups]]:

* A. U. Klymik, K. Schmuedgen, _Quantum groups and their representations_, Springer 1997.

* V. Chari, A. Pressley, _A guide to quantum groups_, Cambridge Univ. Press 1994

* [[V. G. Drinfel'd]], _Quantum groups_, Proceedings of the International Congress of Mathematicians 1986, Vol. 1, 798–820, AMS 1987, [djvu:1.3M](http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0798.0820.ocr.djvu), [pdf:2.5M](https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1986.1/ICM1986.1.ocr.pdf)

* D. Gurevich, [[V. Rubtsov]], _Yang-Baxter equation and deformation of associative and Lie algebras_, in: Quantum Groups, Springer Lecture Notes in Math. __1510__ (1992) 47-55,[doi](https://doi.org/10.1007/BFb0101177)

* P. P. Kulish, [[Nicolai Reshetikhin]], E. K. Sklyanin, _Yang-Baxter equation and representation theory: I_, Lett. Math. Phys. __5__:5 (1981), 393-403, [doi](http://dx.doi.org/10.1007/BF02285311)

Discussion in the context of [[braid group representations]]:

* [[Camilo Arias Abad]], §5.1 *Introduction to representations of braid groups*, Rev. colomb. mat. vol.49 no.1 (2015) [[arXiv:1404.0724](https://arxiv.org/abs/1404.0724), [doi:10.15446/recolma.v49n1.54160](https://doi.org/10.15446/recolma.v49n1.54160)]

[[!redirects quantum R-matrix]]

[[!redirects quantum Yang-Baxter matrix]]

[[!redirects quantum Yang-Baxter equation]]


quark

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#### Fields and quanta
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#Contents#
* table of contents
{:toc}

## Idea

_Quarks_ ([Gell-Mann 64](#GellMann64), [Zweig 64](#Zweig64)) are one of the fundamental [[particles]]/[[matter]] [[field (physics)|fields]] in the [[standard model of particle physics]]. Quarks couple to the [[Yang-Mills theory]] given by [[QCD]].

Quarks come in three [[generations of fermions]]:

[[!include flavors of fundamental particles -- table]]

At room-[[temperature]] quarks always form [[bound states]] to [[hadrons]]. This phenomenon of _[[confinement]]_ is quantitatively well-reproduced by [[lattice QCD]] computations (see [Fodor-Hoelbling 12](#FodorHoelbling12)) and qualitatively well reproduced by conceptual arguments such as the [[AdS/QCD correspondence]], but a full analytic [[proof]] of confinement from a rigorous [[AQFT]]-like foundation of [[QCD]] remains open, see the _[[mass gap problem]]_.

However, at high [[temperature]] [[QCD]] goes through a _deconfinement_ [[phase transition]] and enters another [[phase of matter]] known as the [[quark-gluon plasma]]. As the name suggests, here quarks and [[gluons]] are free.

## Related concepts

* [[constituent quark]]

  * [[diquark]]

* [[colour charge]], [[flavour (particle physics)]]

* [[quantum chromodynamics]]

* [[confinement]]

  * [[quark bag model]], [[Cheshire cat principle]]

  * [[chiral perturbation theory]]

  * [[AdS/QCD]]

* [[quark-hadron duality]]

* [[quark-gluon plasma]]

* [[generation of fermions]]

* [[hadron]]

  * [[meson]], [[baryon]]

  * [[pentaquark]]

* [[B meson]]

* [[squark]]

## References

### General

* [[Jiri Chyla]], _Quarks, partons and Quantum Chromodynamics_, ([spire:674163/)](http://inspirehep.net/record/674163/), [[ChylaQuarksQCD.pdf:file]])

* Jean-Marc Richard, _An introduction to the quark model_ ([arXiv:1205.4326](https://arxiv.org/abs/1205.4326), [spire:1115489](http://inspirehep.net/record/1115489))

* [[Particle Data Group]], *Quark Model* ([pdf](https://pdg.lbl.gov/2020/reviews/rpp2020-rev-quark-model.pdf), [web](https://pdg.lbl.gov/2021/reviews/contents_sports.html)), Chapter 15 in: *[The Review of Particle Physics](https://pdg.lbl.gov/)*,  Prog. Theor. Exp. Phys. 2020, 083C01 (2020) ([doi:10.1093/ptep/ptaa104](https://doi.org/10.1093/ptep/ptaa104))

Textbooks:

* {#HalzenMartin84} [[Francis Halzen]], [[Alan Martin]], _Quarks and Leptons: An Introductory Course in Modern Particle Physics_, Wiley 1984 ([pdf](http://ajbell.web.cern.ch/ajbell/Documents/eBooks/Quarks%20&%20Leptons.pdf), [spire:205394](https://inspirehep.net/record/205394))

* [[Howard Georgi]], §15  in: *Lie Algebras In Particle Physics*, Westview Press (1999), CRC Press (2019) [[doi:10.1201/9780429499210](https://doi.org/10.1201/9780429499210)]

Quark [[masses]] in terms of [[chiral perturbation theory]]:

* [[Jürg Gasser]], [[Heinrich Leutwyler]], _Quark masses_, Physics Reports Volume 87, Issue 3, July 1982, Pages 77-169 (doi:10.1016/0370-1573(82)90035-7)

### History

The quark model was proposed independently in 1964 by

* {#GellMann64} [[Murray Gell-Mann]], _A Schematic Model of Baryons and Mesons_, Phys.Lett. 8 (1964) 214-215 ([spire:11880](http://inspirehep.net/record/11880), doi:10.1016/S0031-9163(64)92001-3)

* {#Zweig64} [[George Zweig]], _An SU(3) model for strong interaction symmetry and its breaking_,  version 1 is CERN preprint 8182/TH.401, Jan. 17, 1964, version 2 in _Developments in the Quark Theory of Hadrons_ Volume 1. Edited by D. Lichtenberg and S. Rosen. Nonantum, Mass., Hadronic Press, 1980. pp. 22-101 ([spire:4674](http://inspirehep.net/record/4674))

Review of this history:

* [[George Zweig]], _Origins of the quark model_, 1980 ([pdf](https://authors.library.caltech.edu/18969/1/Origins_of_the_Quark_Model_Final_Zweig%5B1%5D.pdf), [[ZweigQuarkModel1980.pdf:file]])

### Ab-initio lattice computation
 {#ReferencesAbInitioLatticeComputation}

Due to [[confinement]], before the [[quark-gluon plasma]] was seen in [[experiment]] it was a logical possibility that the quark-model of [[QCD]] is not actually correct. But more recend ab-initio computation in [[lattice QCD]] show that starting with the quark model, at least the light [[hadron]] [[bound states]] observes in [[experiment]] are reproduced by these ab-initio computations. This is discussed in the following references, see the good review [Fodor-Hoelbling 12](#FodorHoelbling12)

* S. Durr, Z. Fodor, J. Frison, C. Hoelbling, R. Hoffmann, S.D. Katz, S. Krieg, T. Kurth, L. Lellouch, T. Lippert, K.K. Szabo, G. Vulvert,

  _Ab-initio Determination of Light Hadron Masses_,

  Science 322:1224-1227,2008 ([arXiv:0906.3599](https://arxiv.org/abs/0906.3599))

* {#FodorHoelbling12} Zoltan Fodor, Christian Hoelbling, _Light Hadron Masses from Lattice QCD_, Rev. Mod. Phys. 84, 449, ([arXiv:1203.4789](https://arxiv.org/abs/1203.4789))

* S. Aoki et. al. _Review of lattice results concerning low-energy particle physics_ ([arXiv:1607.00299](https://arxiv.org/abs/1607.00299))

See also

* Wikipedia, _[Quark](https://en.wikipedia.org/wiki/Quark)_

[[!redirects quarks]]


quark bag model

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#Contents#
* table of contents
{:toc}

## Idea

In [[nuclear physics]] ([[quantum chromodynamics]]) the _bag model_ is an _ad hoc_ [[effective field theory|effective]] [[model (in theoretical physics)|model]] of [[quark]] [[confinement]] in [[hadrons]], assuming literally that the [[quarks]] are constrained within an impenetrable membrane, the "bag".

The _[[Cheshire cat principle]]_ is the observation that this "bag" is not actually [[observable]].

The Cheshire Cat Principle follows in [[holographic QCD]], where the baryons are holographically realized as [[Skyrmions]]:

> The [[Skyrmion]] is the ultimate [[soliton|topological]] [[quark bag model|bag model]] with zero size bag radius, lending further credence to the [[Cheshire cat principle]]. ([Nielsen-Zahed 09](#NielsenZahed09))

## Related concepts

* [[QCD]]

* [[confinement]]

* [[Cheshire cat principle]]

[[!include effective field theories of nuclear physics -- contents]]

## References

### General

Textbook account:

* [[Ulrich Mosel]], *Bag Models of Hadrons* ([doi:10.1007/978-3-662-03841-3_16](https://doi.org/10.1007/978-3-662-03841-3_16)), Chapter 16 in: *Fields, Symmetries, and Quarks*, Springer 1999 ([doi:10.1007/978-3-662-03841-3](https://link.springer.com/book/10.1007/978-3-662-03841-3))

See also:

* Atsushi Hosaka, Hiroshi Toki, _Chiral bag model for the nucleon_, Phys. Rept. 277 (1996) 65-188 ([spire:429251](http://inspirehep.net/record/429251), doi:10.1016/S0370-1573(96)00013-0)

* Julius Kuti, Section 3 of: _Quark confinement and the quark model_, CERN - JINR School of Physics, Nafplion, Greece, 22 May - 4 Jun 1977, pp.79-128 (CERN-1977-018) ([doi:10.5170/CERN-1977-018.79](http://cds.cern.ch/record/878468), [[KutiQuarkModel.pdf:file]])

* Luiz L. Lopes, Carline Biesdorf, Débora P. Menezes, _Modified MIT Bag Models: Thermodynamic consistency, stability windows and symmetry group_ ([arXiv:2005.13136](https://arxiv.org/abs/2005.13136))

* Luiz L. Lopes, Carline Biesdorf, K. D. Marquez, Debora P. Menezes, _Modified MIT Bag Models pt II: QCD phase diagram, hot quark stars and speed of sound_ ([arXiv:2009.13552](https://arxiv.org/abs/2009.13552))

### In relation to the Skyrme model

In relation to the [[Skyrmion model]]:

* {#BrownRho88} G. E. Brown, [[Mannque Rho]], _The Chiral Bag_, Comments Nucl. Part. Phys. 18 (1988) no.1, 1-29 ([spire:18025](http://inspirehep.net/record/18025))

Combination of the [[omega-meson]]-stanilized [[Skyrme model]] with the [[bag model for quark confinement|bag model]] for [[nucleons]]:

* Atsushi Hosaka, _Omega stabilized chiral bag model with a surface $\omega q q$ coupling_, Nuclear Physics A Volume 546, Issue 3, 31 (1992) Pages 493-508 (doi:10.1016/0375-9474(92)90544-T)

### Realized on D-branes

From the point of view of [[holographic QCD]]:

* {#NielsenZahed09} [[Holger Bech Nielsen]], [[Ismail Zahed]], _The Cheshire Cat Principle from Holography_ ([doi:10.1142/9789814280709_0017](https://doi.org/10.1142/9789814280709_0017), [arXiv:0904.0429](https://arxiv.org/abs/0904.0429)), Chapter 17 in: _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

[[!redirects bag model of quark confinement]]

[[!redirects bag model for quark confinement]]