数理解析研究所講究録 1227 巻 2001 年 61-82

61

QUANTUM SEX AND MUTUAL INFORMATION
VIACHESLAV P BELAVKIN

ABSTRACT. The operational structure of quantu pairings, couplings entanencodings is studied and classified for general von Neum ann
algebras. We show that the classical-quantum correspondences such as encodas diagonal CP semi-classical (c-) couplings, and tlllt. ellings an [)by
-CP truly quan ruu ((q-) couplings.
$\mathrm{m}$$\mathrm{k}^{1}1\mathrm{t}^{\backslash }111\mathrm{e}^{\iota}\mathrm{n}\mathrm{t}\mathrm{s}\dot{\mathrm{c}}111\mathrm{t}1$$)$$\mathrm{c}$(.

$\mathrm{t}1\mathrm{t}^{\backslash }\dot{\mathrm{c}}\iota\uparrow(.(1$$\mathrm{t}:\iota \mathrm{u}\mathrm{h}’1\mathrm{t}^{\backslash }111\mathrm{t}^{\backslash }11\mathrm{t}|\mathrm{s}\dot{\mathrm{c}}11\mathrm{t}^{1}\mathrm{t}.11_{\dot{\mathrm{f}}}\iota \mathrm{r}_{\dot{\mathrm{e}}}\iota \mathrm{t}.\mathrm{t}\mathrm{t}^{1}1\mathrm{i}^{r}/_{\lrcorner}\mathrm{t}^{\backslash }\mathrm{t}1$$\mathrm{t}.\mathrm{r}\dot{\mathrm{c}}1.11\mathrm{H}1$$\mathrm{o}\mathrm{s}\mathrm{e}$entangled states lead toentropy of the diagonal compound
two differ ent types of entropies for given quantum state 011 athe voll Neu)
entropy, which is achieved as theof tbcthe dimensional entropy, which is achieved at theall (fiue quant um entanglement, coinciding withof pure marginal states. The -capacity ofentangle ment only in the (.
,quantu 111 noiseless channel, defined as the supremum over all
is given [ the logarithm of the (lil1lensionality of the input algebra. It lnay
. supremum over all c-couplings,, achieved as
doubleclassical
mensionality of a
, which is bounded by the logarithm of the$\dot{‘}\iota 11\mathrm{t}1$$\mathrm{T}11\mathrm{t}^{\backslash }1(.1‘.\iota \mathrm{t}\mathrm{i}\mathrm{v}\mathrm{t}^{\backslash }$$\dot{(}\iota$$11\downarrow‘.\mathrm{t}1111$$.:.\iota\iota 1$$\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{f}\mathrm{t}’ 111\mathrm{l}\mathrm{i}\iota \mathrm{t}\mathrm{i}()11$$1\mathrm{t}^{\backslash }11111111$$‘)\mathrm{t}^{\backslash }1$$\iota_{-\mathrm{t}^{\backslash }11\uparrow j\iota 11}\mathrm{h}^{\prime 1\mathrm{t}^{\iota}111(^{\backslash }11\mathrm{t}\iota}\backslash \cdot,\dot{\prime}\iota 11(1$$‘ \mathrm{b}\uparrow \mathrm{i}\mathrm{l}\mathrm{l}\mathrm{l}(\mathrm{l}\mathrm{j}|\mathrm{l}\mathrm{t}\mathrm{l}$$(^{\backslash }11\mathrm{t}_{i1.11\mathrm{k}},1\mathrm{t}^{\backslash }111(^{\backslash }11\mathrm{t},$$\mathrm{t}11(^{\backslash }$$i\iota$$(1-$$j\iota$$\dot{‘}1\aleph 1^{\backslash }$$\mathrm{q}$$(^{\mathrm{t}}.\mathrm{n}\mathrm{t}j\iota \mathrm{I}\mathrm{l}\mathrm{g}1\mathrm{t}^{\backslash }.1\mathrm{u}\mathrm{t}^{\backslash }J\mathrm{I}1\mathrm{t}..\forall$$).\mathrm{V}$$\mathrm{t}1_{1}\mathrm{e}$$\mathrm{f}11\mathrm{t}^{\backslash }$$(.\mathrm{a}\mathrm{l})\mathrm{a}\mathrm{t}\cdot \mathrm{i}\mathrm{t}\mathrm{y}$$\mathrm{d}\mathrm{i}$$()1(^{\backslash }11(\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{s}$$111\dot{\mathrm{c}}\iota \mathrm{x}\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{l}\dot{\mathrm{t}}\iota 1$

Abelian subalgebra.

1. INTRODUCTION
In this paper we develop the operational approach to quantum entanglement [1],extending the notion of quantum conditional entropy and mutual information to thegeneral von Neumann algebras with normal semifinite faithful weights. By quantum
sex we call pairings, such as quantum couplings, entanglements and encoodings, of
two systems $(A, \mu)$ and $(B, \nu)$ , reffered in quantum communications as Allice andBob, with respect to the given weights , on the von Neumann algebras $A$ and8respectively.
Tlte entanglements as specifically quantum correlations, are used to study quantum information processes, in particular, quantum computations, quantum teleportation, quantum cryptography [2, 3, 4]. There have been mathematical studies ofthe entanglements in [5, 6, 7], in which the entangled state is defined by astate)
not written as aform of aconvex combination
with any states. However it is obvious that there exist several types of the correlated, andstates written as ‘separable’ forms above. Such correlated, or classically entangledstates have been also discussed in several contexts in quantum probability such as
quantum measurement and filtering $[8, 9]$ , quantum compound state $[10, 11]$ andlifting [12].
$\mu$

$\nu$

$\sum_{n}\rho_{n}\triangleright$$\sigma_{n}p(n)$$\sigma_{n}$$\llcorner 0,$

Daft$\mathrm{M}\dot{\mathrm{c}}\mathrm{t}1\mathrm{t}.112\mathrm{t}$$K\mathrm{c}$( $\tau natic\cdot \mathrm{s}$$l/(’()\prime\prime l\backslash 07\prime \mathrm{d}$$\mathrm{E}_{11}\mathrm{t}_{1\mathrm{t})}1)\mathrm{Y}$$\mathrm{T}11\mathrm{t}^{\backslash }$}, 2001.

Subject Classification. Quantum Probability and Information.
phrases. Quantum Pairing, Quan rum Coulpling, Compound States, Quantumand Mutual Information.
author acknowledges the Royal Society scheme for UK-Japan research collaboration

1991 Math62
VIACHESLAV P BELAVKIN

In this paper, we study the mathematical structure of classical-quantum andquantum-quantum couplings to provide afiner classification of quantum separable
and entangled states, and we discuss the informational degree of entanglement andentangled quantum mutual entropy.
The term entanglement was introduced by SchrOdinger in 1935 out of the needto describe correlations of quantum states not captured by mere classical statisticalcorrelations as the convex combinations of noncorrelated states. In this spirit theby now standard definition of entanglement is the state of acompound quantum
system ‘which cannot be prepared by two separated devices with only correlatedclassical data as their inputs’ (see for example Werner, 1989. We show that theentangled states can be achieved by quantum (q-) encodings, the nonseparable
couplings of states, in the same way as the separable states can be achieved byclassical (c-) encodings.
Tlze compound states, called -coupled, are defined by orthogonal decompositions of their marginal states. This is aparticular case of so called diagonal state ofacompound system, the convex combination of the special product states which wecall compound Tbe -compound states are most informative among c-compoundstates in the sense that the maximum of mutual entropy over all couplings to thequantum system is achieved on the extreme -coupled(even -coupled)states as theof agiven normal state on asimple algebra $A$ . Thusvon Neumann entropy
the maximum of mutual entropy over all classical couplings of (classical) probe sys, the logarithm of the rank
tems $A$ to aquantum system , is bounded by$\dim
H$ of the Hilbert spaceof the algebra which is defined as the dimensionality
for the simple ,$?t$ for irreducible representation of Q.
Due to $\dim B$
I only in theit is achieved on the normal tracial density operator
case of finite dimensional .
More general than -coupled states, the -entangled states, are defined as centangled states by orthogonal decomposition of only one marginal state on theprobe algebra $A$ . In general they can give larger mutual entropy for aquantum
noisy channel than the -coupled state (which gains the same information as dcoupled extreme states in the case of adeterministic channel).
We prove that the truly entangled pure states are most informative in the sensethat the maximum of mutual entropy over all entanglements to the quantum systemis achieved on the -compound state, given by an extreme (standard) entanglewith coinciding marginals, called standard forment of the probe systemagiven . The gained information for such extreme -compound state defines anwhich is bigger than the von Neumannother type of entropy, the entropy
in the case of mixed . The maximum of mutual entropy over all
entropyquantum couplings, including the true quantum entanglements of probe systems$A$ to the system, is bounded by lndimB, the logarithm of the dimensionality
in theof the von Neumann algebra , which is achieved on anormal tracial,becalled
the
can
which
case of finite dimensional S. Thus the entropy
dimensional entropy, is the true quantum entropy, in contrast to the von Neumann, the -entropy which is semi-classical entropy achieved as asupremum over all
couplings with the classical probe systems $A$ . In the case of finite-dimensional
the -capacity $\mathrm{C}_{q}=\ln\dim B$ is achieved as the supremum of mutual entropy overall -encodings, the. quantum-quantum correspondences, described by entangleof the identit
ments. It is strictly larger then the classical capacity$0$$\mathrm{d}$$\mathrm{d}$$\mathrm{c}$$0$$\mathrm{d}$$\mathrm{S}(\sigma)$$\sigma$

$\ln$

$B$

$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}/3$

$B$

$B$

$=(\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}B)^{2}$$\sigma=(\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}B)^{-1}$$B$$\mathrm{d}$

$0$

$0$

$B$

$\mathrm{q}$$A=I\mathit{3}$$\mathrm{q}$$\sigma$$\mathrm{H}(\sigma)$$\mathrm{q}$$\mathrm{S}(\sigma)$$\sigma$$B$$B$$\sigma$$\mathrm{H}(\sigma)$$\mathrm{q}$$\mathrm{S}(\sigma)$$\mathrm{c}$$B$$\mathrm{q}$$\mathrm{q}$$\mathrm{C}_{c}=\ln \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}B$$\mathrm{y}$63
QUANTUM SEX AND MUTUAL$1\mathrm{N}\mathrm{F}()\Gamma \mathrm{t}$MATI$()\mathrm{N}$channel, which is achieved as the supremum over usual encodings, described by theclassical-quantum correspondences $A^{0}arrow B$ .

In this short paper we consider the case of decomposable probe algebras $A$ but
simple algebra $B=\mathcal{L}(H)$ for which the proofs are rather straightforward. Moregeneral decomposable algebra including the classical discrete systems as aparticular Abelian case is considered in [13], and even more general case of von Neumann
algebras will be also published elsewhere.
$B$

2. PAIRINGS, COUPLINGS AND ENTANGLEMENTS
be theLet 7{ denote the Hilbert space of aquantum system, andalgebra of all linear bounded operators on 7#. It consists of all operators
is called astate on if
having the adjoints A\dagger on $H$ . Alinear functional$B=A^{\uparrow}A$) and normalizedifor any positive operator
it is positive$\sigma(I)=1$ for the identity operator I in $A$ . Anormal state can be expressed as$B$$=\mathcal{L}(H)$$A:\mathcal{H}arrow \mathcal{H}$$\sigma:B$$B$$arrow \mathrm{C}$$(\mathrm{i}.\mathrm{e}., \sigma(B)\geq 0$(1)$\mathrm{n}$$\sigma(B)=\mathrm{T}\mathrm{r}x^{\dagger}Bx\equiv\langle B, \sigma\rangle$,$B\in I\mathit{3}$$B$,where is alinear Hilbert-Schmidt operator from $H$ to (another) Hilbert space (;,in ,is the adjoint operator from to $H$ . Here Tr stands for the usualasiscalled. Thistheand in the case of ambiguity it will also be denotedas the square root
amplitude operator which can always be considered on(it is called simply amplitude if is one dimensional space ,of tlte operator$\chi\chi\dagger=||\eta||^{2}=1$ , in which caseis the functionalfrom $H$ to
with$\chi$$x^{1}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$\mathcal{G}$$\mathcal{G}$$\chi$$\mathrm{n}_{\mathcal{G}}$$\mathcal{G}=It$$J\mathrm{C}JC^{1}$$\mathbb{C}$$\mathcal{G}$$x^{\mathfrak{j}}$$\chi$$\eta^{\uparrow}$$\in \mathcal{H}$$=\mathrm{t})$.

$\mathbb{C})$)(and will equip all auxiliary Hilbert spaces, e.g.
We can always equip$J=J\dagger$$J^{2}=I$
complexproperties
ofhaving,thewith an isometric involution
conj ugation$\mathcal{G}$$\mathcal{H}$$J \sum\lambda_{j}\eta_{j}=\sum\overline{\lambda}_{j}J\eta_{j}$,$\forall\lambda_{j}\in \mathbb{C}$,$\eta_{j}\in 7\{$,class operatorsof fl with thethe tilda-pairingand denote by
the probability. We shall callasuch thatdensity of the state (1) with respect to this pairing, and assume that the supportof ais the minimal projector $E=E\dagger\in B$ for which $\sigma(E)=1$ , i.e. that
. The latter can also be expressed as the symmetricity property.
onwith respect to the tilda operation (transposition)
One can always assume that is the standard complex conjugation in an eigencoincides with as the real element
representation of asuch thatof all diagonal (and thusof the invariant maximal Abelian subalgebra
mmetric) operators in this basis.
The auxiliary Hilbert space and the amplitude operator in (1) are not unique,in , and (;however is defined uniquely up to aunitary transfor$??,${ for , thecan be always taken minimal, identified with the support
such that $\sigma E=\sigma$ ). In
is the minimal orthoprojector inclosure of all (must not be less than
general, (; is not one dimensional, the dimensionality
coinciding with the, the dimensionality of the range.
support for, one can define not only the state:
Given the amplitude operator
but also the normal state$\mathrm{b}B\tilde{\sigma}$$\langle B, \sigma\rangle$$\in \mathcal{T}(\mathrm{H})$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$\sigma=JxxJ\dagger=\tilde{J}<^{\mathfrak{s}}\tilde{\chi}$$\tilde{\sigma}=J\sigma\dagger J$$E_{\sigma}$$\underline{\overline{E_{\sigma}}}:=JEaJ=E\mathrm{a}$$\tilde{B}=JB\dagger J$$E_{\sigma}=E_{\sigma}$$\mathcal{L}(H)$$J$$\overline{\sigma}=XJ\gamma^{\uparrow}=\tilde{\sigma}$$\sigma$$A\subset \mathcal{L}(H)$$\mathrm{s}\mathrm{y}$$\mathcal{G}$$\mathrm{m}$$\chi$$\kappa^{1}\mapsto Ux\dagger$$=E_{\sigma}7$$\mathcal{G}$$\sigma$$B$$E_{\sigma}$$\dim \mathcal{G}$$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}x^{1}=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\sigma$$\mathcal{G}_{\rho}=\mathrm{r}\mathrm{a}\mathrm{n}x^{\uparrow}$$\rho=ux\simeq\dagger\tilde{\sigma}$$\chi$(2)$\mathcal{G}arrow Tt$$\rho(A)=\mathrm{b}\tilde{x}^{\uparrow}A\tilde{x}\equiv\langle A, \rho\rangle$$\sigma$,$A\in A$

64
VIACHESLAV P BELAVKIN

on

as the marginal of the pure compound state$A=\mathcal{L}(\mathrm{Q})$$\omega$on the algebra A$\mathcal{G}\ltimes)\mathcal{H}$) $B$$\alpha$.

(A$\alpha$) $B)=\mathrm{T}\mathrm{r}\tilde{A}x^{\uparrow}Bx=\mathrm{R}\tilde{x}^{\uparrow}A\tilde{x}\overline{B}$.

of all bounded operators on the Hilbert tensor product space

Indeed, thus defined bilinear form withis uniquely extended to suchastate, given on ( tt $H$ ) by the amplitude $=l$ , where.
for all
This pure compound stateis so called entangled state, unless its marginalstate (and p) is pure, corresponding to arank one operator, ill which), given by the amplitude $v=(;$case. The amplitude operator
corresponding to mixed states on $A$ and will be called the entangling operator ofto.
As follows from the next theorem, any pure entangled state$\tilde{A}=JA^{\uparrow}J$$\mathcal{L}$$(\in \mathcal{G},$$(\zeta \mathrm{W} \eta)^{\uparrow}!=\eta xJ\dagger\zeta$$\psi$$\mathcal{G}$$\eta\in \mathcal{H}$$\omega$$2t^{\uparrow}=\zeta\eta^{\uparrow}$$\sigma$$\omega$$=\rho\triangleright$$\omega$$\sigma$$\chi$$\eta$$B$$\sigma=\tilde{x}^{\uparrow}\tilde{x}$$\rho=x^{\uparrow_{y\zeta}}$$\omega$(A$\mathrm{o}0B$)

A$=\psi^{\mathrm{t}}(A(\kappa B)\psi,$$B\in \mathcal{L}(\mathcal{G}007\{)$$\omega$given by an amplitude, can be achieved as described by auniqueentanglement to the algebraof the marginal state on $B=\mathcal{L}(H)$ .
Before to formulate this theorem in the generality which we need for further
consideration, let us introduce the following notations.
Let $A$ be
on with anormal faithful semiflnite weight ,de$A$$\{A’\in\mathcal{L}(\mathcal{G}):[A’,A]=0,\forall
A\in
A\}$
note the commutant
of , anddenote$\psi$$\in \mathrm{C}\mathcal{G}$$\mathrm{M}H$$A=\mathcal{L}(\mathcal{G})$$\chi$$\sigma$$A^{J}$$\mathcal{G}$$\mathrm{a}*\mathrm{a}\mathrm{l}\mathrm{g}\mathrm{e}\mathrm{b}\mathrm{r}\mathrm{a}$$\mu$$(\tilde{A},\tilde{\mu})$the transposed algebra of the operatorswithwhich may
not coincide with $(A, \mu)$ (and with $A’$ ). We denote by
the space of all op, where ,erators $A\in A$ in the form
, with,and bythe standard representation :given by the left multiplication $(A)x=Ax$ on , with the standard isometric involution
defining normal faithful representationof the trans$\tilde{A}=JA^{\uparrow}J$$\tilde{\mu}(A)=\mu(\tilde{A})$$A_{\mu}\subseteq A$$x\dagger z$$(\mathrm{Q})\iota,$$x$$\mathfrak{a}_{\mu}=\{?\cdot\in A:l\iota (x^{\uparrow}.r)<\lambda \}$$z\in a_{\mu}$$J_{/\iota})$$\iota$$Aarrow \mathcal{L}(\mathcal{G}_{\mu})$$J_{\mu}:.?\cdot\mapsto.r\dagger$$\iota$$\alpha_{\mu}$$\tilde{\iota}(\tilde{A})=J_{\mu}\iota(A^{\uparrow})J_{\mu}\equiv\overline{\iota(A)}$posed algebraon the completionof the left modulewith respect to theinner product. We recall that the von Neumann algebra $A$ defined[$(A)’$ , and that
$A’=A$ is anti-isomorphic to $(A)’=J_{\mu}\iota(A)J_{\mu}$ and thus$\tilde{A}$$\mathcal{G}_{\mu}$$a_{\mu}$$(x|z)_{l^{l}}=\mu(x^{\uparrow}z)$$\tilde{A}\simeq\iota$$\iota$$)\mathrm{y}$as the space of all continuous functionals
the pairing$\tilde{A}=A_{\mu}^{*}$$\langle x^{\dagger}z,\overline{A}\rangle_{\mu}:=(x|\overline{\iota(A)}z)_{\mu}\equiv\langle A,\overline{x\dagger z}\rangle_{\mu}$The completion of$A_{\mu}$with respect to-norm$\mathrm{t}\mathrm{h}\mathrm{e}*$$\tilde{A}:\phi\mapsto\langle\phi,\tilde{A}\rangle_{\mu}$,$x^{\uparrow}z\in A_{\mu}$,.

with respect to$\tilde{A}\in\tilde{A}$.

$||x^{\uparrow}z||_{*}= \sup\{|\langle A,\overline{x\dagger z}\rangle_{\mu}|$: $||A||<1\}$

and the is indentified with the predual Banach space denoted as(if $\mu=$is the usual
on $A$ , thencoincides withas the classofoperators, where$A_{*}$$\tau|A$$\{x\in \mathcal{L}(\mathcal{G}) :$\mathcal{T}(\mathcal{G})=\{x^{\uparrow}z:x, z\in S(\mathcal{G})\}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$\mathrm{H}_{\mathcal{G}}x^{\mathrm{t}}x<\infty\})$$A_{*}$$A_{\mu}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\tau=\mathrm{n}_{\mathcal{G}}$$A_{\tau}=A\cap \mathcal{T}(\mathcal{G})$$S(\mathcal{G})=$.

Note that$=H_{\mu}$ , $J=J_{\mu}$ ,in the standard representation
unles $A$ is Abelian as only in this case $A’=A$ . If $A$ is not the algebra of
all operators, the density operator for anormal state (2) is not unique even$\tilde{A}\neq A,\tilde{\mu}\neq\mu$$7\{$$\overline{A}=A’$$\mathcal{L}(\mathcal{G})$$\rho$

65
QUANTUM SEX AND MUTUAL INFORMATI$()\mathrm{N}$. However it is uniquely defined as the bounded probability
with respect todensitywith respect to the restriction $\mu=\tau|A$ (i.e. as thedensity operator with respect to ) describing this state asby. Note that each probability density
the additional condition $x$
on $A\ni A$ is positive and normalized
describing the normal statemay, but the predual spaceas the -completion ofasto (however eachconsist of not only the bounded densities with).
can always be approximated by the bounded
In the following formulation can also be more general von Neumann algebra
defining the:, with anormal faithful semifinite weightthan, wherecoincides withinpairing
whenis the space ofthe case of the standard).
Hilbert-Schmidt operators $y\in B$ and$\tau=\mathrm{T}_{\mathrm{i}\mathrm{g}}$$\rho=Jx\dagger xJ=\overline{x}^{\uparrow}\overline{x}$$\langle A, \rho\rangle_{\mu}=\mu(xAx^{\uparrow})$$\mu$$=\overline{x}\in\overline{A_{\mu}}$$\rho\in\overline{A_{\mu}}$$\rho(A)=\langle A, \rho\rangle_{\mu}$$\tilde{A}_{*}=A_{*}$$\langle I, \rho\rangle_{\mu}=1$$\overline{A_{\mu}}$$*$$\rho\in\overline{A}_{*}$$\mu$$\mathrm{r}\mathrm{e}\mathrm{s}\underline{\mathrm{p}\mathrm{e}\mathrm{c}}\mathrm{t}$$\rho_{n}\in A_{\mu}$$B$$\mathcal{L}(\mathcal{H})$$B_{\nu}\mapsto \mathbb{C}$$\nu$$v\in\overline{\mathrm{b}_{\nu}}(B_{\nu}=\mathrm{b}_{\nu}^{1}\mathrm{b}_{\nu}$$\langle B, v^{1}v\rangle_{\nu}=(\overline{v}|\overline{\iota(B)}\overline{v})_{\nu}$$B_{*}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\nu(B\tilde{\sigma})=\mathrm{T}\mathrm{r}B\tilde{\sigma}=\langle B, \sigma\rangle_{\nu}$$\mathrm{b}_{\nu}$$\mathit{1}\mathit{3}=I\mathit{3}$Theorem 2.1. Let(3)$\omega$$\omega$:$A\ltimes$) $B$$arrow \mathbb{C}$

be a normal compound state$(A\ltimes\}B)=(\overline{v}|\iota(\overline{A\propto)}B)\overline{v})\equiv\langle A\omega B, v^{\uparrow}v\rangle$

described by an amplitude operator : (; C4 $H$
Hilbert spaces and , satis fying the condition$arrow \mathcal{E}\infty$$v$$v^{\dagger}v\in\overline{A}\omega\overline{B}$$\aleph\nu$$(A \omega B)_{*}$:$\chi$$\mathcal{G}\omega$,$(\mu \mathrm{t}\triangleleft \nu)(\overline{v}^{\uparrow}\overline{v})=1$

is the product weight the pair$r^{*}ing$, and $\overline{v}=JvJ$ .

$\mathcal{F}arrow \mathcal{E}\omega$(4)on the tensor product of$\mathcal{F}$$\mathcal{F}$$\mathcal{E}$

Here 74 (,$\mathcal{H}$.

of A ci 13 in (3) with$(A \mathrm{c}\triangleleft B)_{\mathrm{T}}=$

Then this state is achieved by an entangling operatoras$\langle A, \nu(x^{\dagger} (I\infty B)x) \rangle_{\mu}=\omega$(A$\omega B$)$=\langle B, \mu(\tilde{r}\tau^{\uparrow}(A\infty I)\tilde{x})\rangle_{\nu}$for all $A\in A$ and $B\in B$ such that$\nu$

The operator(5)$lll)$$\chi$.

,$\mu(\tilde{\chi}^{\uparrow}(A \omega I)\tilde{x})\subseteq\overline{B}$$(\chi^{\uparrow}(I \mathrm{t}\triangleleft B) x)\subseteq\overline{A}$together with$\tilde{x}=Jx^{\uparrow}J$

is uniquely defined by $v=Ul$ , where$(\xi(\aleph^{)}\eta’)^{\dagger}x’(\zeta\omega J\eta)=(\xi \mathrm{c}\triangleleft \eta)^{\dagger}x (\zeta\triangleright)J\eta’)$to a unitary $transf\dot{o}rmation$$U$,$\xi\in \mathcal{E}$, $\eta’\in F$ ,$\zeta\in \mathcal{G}$of the minimal subspace space ranv

, $\eta\in H$ ,$\subseteq \mathcal{E}\mathrm{C}\triangleleft F$, $F$
Proo . Without loss of generality we can assume thatand(A Ea) as the support)
for’is contained in
$f$

$l’$

$\mathcal{E}=\mathcal{G}_{\rho}$

By virtue$v^{1^{1}}v\in(\overline{A’}\mathrm{i}\cross J\overline{B}’)’$(A$\omega B$)$(\overline{A}\alpha$$J$$\mathcal{F}=\mathcal{G}_{\rho}\omega$check for such$\mathcal{H}_{\sigma}$

$v$

$v$

$\mathcal{G}_{\rho}\triangleright$$H_{\sigma}$.

is invariant under the action) $v=v(AE_{\rho}\omega BE_{\sigma})$ ,$\forall A\in\overline{A’}$,$B\in\overline{B}’$. Let us equip and with the involutions
, and
and
,
{, withwith the induced involution. It easy to(5) that for any $A\in A’$ andand $ff$ $=v’$ defined byof the commutant
leaving invariant$\mathcal{E}\omega$

the range of$v^{\uparrow}=$$=7\{_{\sigma}$$v^{\uparrow}\tau$$(\mathcal{G}\ltimes)H)_{\tau’ v}\mathrm{f}=\mathrm{r}\mathrm{a}\mathrm{n}v^{\uparrow}$$(E_{\rho}$.

$B-)’=\overline{A’}\mathrm{c}\triangleleft\overline{B}’$$\mathcal{G}_{\rho}=E_{\rho}\mathcal{G}$$\mathcal{G}$$H_{\sigma}=E_{\sigma}7$$\mathcal{H}$$J_{\rho}=\mathrm{E}\mathrm{a}\mathrm{J}$$J$$(\zeta \mathrm{C}\triangleleft \eta)=J_{\rho}\zeta\infty$$v=p\swarrow \mathrm{i}\mathrm{n}$$J_{\sigma}=E_{\sigma}J$$J_{\sigma}\eta$66
VIACHESLAV P BELAVKIN$B\in\overline{B}’$$(\overline{A}\xi\propto)\eta)^{\uparrow}x$$=$$((\propto)\overline{B}J\eta’)$

$=$

where

$\overline{A}=JAJ\in\overline{A’}$

(A

where

$\mathrm{t}\triangleleft$

$B’$

)

$A\in\overline{A_{\rho}’}:=\overline{A’}E_{\rho}$(I

Moreover, due to$(\overline{A}\zeta\triangleright)J\eta’)$)$x$ $=x^{\uparrow}(A\infty B’B)x$ $=x^{\uparrow}(I*)B)x$$\chi$$\in(\overline{A_{\rho}’}\omega$$B)x$$\tilde{x}^{\uparrow}(A \alpha)$$\omega$

$B’$

),

,

. Thus for all $A\in A$ and $B\in B$,$B_{\sigma}’)’$$A_{\rho}^{JJ}=E_{\rho}AE_{\rho}\equiv A_{\rho}$$x^{\uparrow}(I \alpha)$(A$I)\tilde{x}=x$ $(A’A\infty B)x^{\uparrow}=\tilde{x}^{\uparrow}(A\infty I)\tilde{x}(A’\mathrm{M} B)$$B\in\tilde{B}’$$\mathrm{t}\triangleleft B$$(\xi \mathrm{t}\triangleleft\tilde{B}\eta)^{\uparrow}x$, $B’\in B_{\sigma}’:=B’E_{\sigma}$ , and for any $A\in A$$(A’\mathrm{c}\triangleleft B)\tilde{x}^{\uparrow}(A \alpha)$$x^{1}$$B\eta’)^{\uparrow}v(\xi N J\eta)=(\xi \mathrm{M} \eta’)^{\mathfrak{j}}v(\overline{A}\xi\triangleright\iota J\overline{B}\eta)$, $\overline{B}=JBJ\in B’$ . Hence for any $B\in B$$x^{\uparrow}(I \mathrm{C}\triangleleft B)$

where $A’\in A’$ and$(\tilde{A}\xi \mathrm{c}\triangleleft$$\tilde{x}^{\uparrow}(A\infty I)\tilde{x}\in(A_{\rho}’\mathrm{C}\mathrm{A}\overline{B_{\sigma}’})’$

and$B_{\sigma}’=E_{\sigma}BE_{\sigma}\equiv B_{\sigma}$$\subseteq J_{\rho}A_{\mu}J_{\rho}N$ $E_{\sigma}B_{\nu}E_{\sigma}:=(\overline{A_{\rho}}00$$I)x\sim\subseteq E_{\rho}A_{\mu}E_{\rho}\infty$$B_{\sigma})_{\overline{\mu}\otimes\nu}$,$J_{\sigma}B_{\nu}J_{\sigma}:=(A_{\rho}\omega\overline{B_{\sigma}})_{\mu\otimes\overline{\nu}}$

respectively. The partial weights
and byas bounded byon these reduced algebras are defined as$||A||\tilde{x}^{\uparrow}\grave{x}$$||B||x^{\uparrow}r\mathrm{c}$(6)$|/(x^{1}(I(\cross)B)x)$

according to,$=\langle B, v^{\dagger}v\rangle_{|J}$

$\mu$

(

$\tilde{x}^{\mathrm{t}}$(A$\mathrm{o}0$$\langle A, \langle B,v^{\uparrow}v\rangle_{J},\rangle_{\mu}=\langle A\infty B, v^{\uparrow}v\rangle=\langle B$,$\nu(x^{\uparrow}x)=\tilde{\nu}(vv)\dagger=\rho$

Any other choice of
. I

$v$

with the minimal,$I)\tilde{y}\zeta$)$=\langle A, v^{\dagger}v\rangle_{\mu}$

$\nu$

and

$\mu$

,

. In particular$\langle A, v^{\uparrow}v\rangle_{\mu}\rangle_{\nu}$.

$\mu(\tilde{\chi}^{\uparrow}\tilde{\chi})=\tilde{\mu}(v^{\uparrow}v)=\sigma$$\mathcal{E}\omega F$$\simeq \mathcal{G}_{\rho}(\triangleleft$$\mathcal{H}_{\sigma}$is unitary equivalent to$\sqrt$Note that the entangled state (3) is written in (4) as$\langle B, \varpi (A)\rangle_{\nu}=\omega$ $(A\infty B)=\langle A, \varpi^{\mathrm{T}}(B)\rangle_{\mu}$

in terms of the mutually adjoint maps
given in (6) as(7)$\varpi$$(A)=\langle A, v^{\uparrow}v\rangle_{\mu}=\overline{\pi^{*}(A})$

where the linear map
partial weights$\pi$$\varpi$,: $Aarrow B_{*}$ and,: $Barrow A_{*}$ . They are,$\varpi^{\mathrm{T}}(B)=\langle B,v^{\uparrow}v\rangle_{\nu}=\overline{\pi(B)}$: $Barrow A_{\mu}$ and the adjoint$\pi(B^{\uparrow})=J\langle B, v^{\uparrow}v\rangle_{\nu}J$$\varpi^{\mathrm{T}}$$\pi^{*}$: $Aarrow B_{\nu}$ are defined as.

$\pi^{*}(A^{\uparrow})=J\langle A,v^{\uparrow}v\rangle_{\mu}J$

The linear normal map in (6) is written in the Kraus-Steinspring form [17] andthus is completely positive (CP) but not unital, normalized to the density operators$(I)$ with respect to tlte weight.
Alinear map : $Barrow A_{*}$ is called tilda-positive if $\pi^{-}(B):=J\pi(B)^{\dagger}J$ is positive
for any positive (and thus Hermitian) operator $B\geq 0$ in the sense of non-negative
definiteness of $B$ . It is called tilda-completely positive (TCP) if the operator-matrix,
is positive for every positive operator-matrix$\varpi$$\sigma=\omega$

$\nu$

$\pi$

$\pi-(\mathrm{B})=J\pi(\mathrm{B})^{\uparrow}J$$\mathrm{B}=[B_{ik}]=\mathrm{B}^{*}$

67
QUANTUM SEX AND MUTUAL INFORMATI$()\mathrm{N}$,for $\mathrm{A}=[A_{ik}]\geq 0$ , and(and thus). Obviously every tilda-positive and tilda-completely positiveforis positive as positive isfor every positive , but it is not necessarily$\tilde{A}=A$
for all $A\in A$ , in which case $A$ is Abelian (or thecompletely positive unless
Abelian is ).
, is obviously
defined in (8) as aTCP \dagger -map,
The mapwher$\mathrm{e}$$\mathrm{A}\dagger=[A_{ik}^{\dagger}]$$\mathrm{B}^{4}=\mathrm{B}$$\mathrm{A}^{\uparrow}=[A_{ki}]$$\mathrm{B}^{*}=[B_{ki}^{\dagger}]$$\mathrm{B}\geq 0$$\tilde{A}=JA^{\mathfrak{j}}J$

$A$

$\pi$

$B$

$\pi(B^{\uparrow})=\pi(B)^{\dagger}$$\pi$

for anytranspose-CP in the sense of positivity ofbeevery
repretranspose-CP map can. Because, but it is in general not
for whichissented as tilda-CP, there might be apositive-definite matrix
TCP,
maps
as
as
map
is
also
well
the
not positive. Note that the adjoint, obtained from (6) as partial tracings, whereand$\pi(\mathrm{B})^{\dagger}=[\pi(B_{ki})]=\pi(\mathrm{B}^{\mathrm{t}})$$\mathrm{C}\mathrm{P}$$\mathrm{B}\geq 0$$\pi(\mathrm{B})$$\mathrm{B}$$\pi^{*}=\overline{\pi}^{\mathrm{T}}$$\overline{\pi}(B)=J\pi(\overline{B})J$$\pi^{\mathrm{T}}=\overline{\pi}^{*}$$\tilde{\pi}=\overline{\pi}$(8),$\overline{\pi}(B)=\nu(x^{1}(I\infty\tilde{B})x)$$\pi^{\mathrm{T}}(A)=\mu(\tilde{x}^{1}(\tilde{A}\infty I)\tilde{x})$.

In these terms of the compound state (4) is written as$\langle A|\pi(B)\rangle_{\mu}=\omega$(A$\infty B$)$=\langle\pi^{*}(A)|B\rangle_{\nu}$,defines an inner product which coincides in the case of traces
wherewith the GNS product $(x|y)$ .
(as well as)
In the following definition the predual spacewith the space of generalized density
is identified by the pairing
operators 4which are thus uniquely defined as selfadjoint operators (could be unbounded) in . Note that.
andif$\langle x|y\rangle=\langle y,\overline{x}\rangle$$A_{\mathrm{T}}=\tilde{A}_{*}$$B_{\mathrm{T}}=\overline{B}_{*}$$\langle B, \sigma\rangle_{\nu}=\sigma(B)$$/\mathcal{H}$$\mathit{1}\mathit{3}=I\mathit{3}$$B_{\mathrm{T}}=B_{\nu}$$\nu=\mathrm{T}\mathrm{r}_{\mathcal{H}}=\tilde{\nu}$(or $Barrow A_{\mu}\subseteq A_{*}$ ) nor malized as:
Definition 2.1. A $TCP$ map
is called normal{ $(\pi(/))=1$ and having an adjoint with
on $A$ .
on to the state
coupling (bounded coupling) of the state) normalized to the probability density(or:
The $CP$ map$\sigma=\varpi(I)$will be called normal entanglementof with(bounded entanglement) of the system $(A, \rho)$ with the probability density(entanglement ) is called truly quantum if it is not. The couplingto$CP$ (not $TCP$). The self-adjoint entanglement(oron) is called standard for the system
intosymmetric coupling
if it is given by
$\pi$

$B$

$arrow A_{*}$$\pi^{*}(A)\subseteq B_{*}(\pi^{*}(A)\underline{\subseteq}B_{\nu})$$l$$\sigma$$\varpi$$Aarrow B_{\mathrm{T}}$$B$$=\mu\circ\pi$$\llcorner 0$$=\nu\circ\pi$’$Aarrow B_{/},\subseteq B_{\underline{\mathrm{T}}}$$\varpi^{\mathrm{T}}(I)\in B_{*}(\varpi^{\mathrm{T}}(I)\in A_{\mu})$$\sigma$$\rho=\varpi^{\mathrm{T}}(I)$$(B, \sigma)$$\varpi$$\pi$$(A, \rho)=(\tilde{B},\tilde{\sigma})$$\varpi_{q}=\varpi_{q}^{*}$(9)$(B, \sigma)$$A_{*}=B_{\mathrm{T}}$$\pi_{q}=\pi_{q}^{\mathrm{T}}$$\varpi_{q}(A)=\sigma^{1/2}A\sigma^{1/2}$,$\pi_{q}(B)=\sigma^{1/2}\overline{B}\sigma^{1/2}$.

Note that the standard entanglement is true as soon as the reduced algebraof the state is not Abelian, i.e. is not oneon the support, corresponding to apure normal on $B=\mathcal{L}(H)$ .
dimensional in the caseIndeed,is the composition of the nondegenerated multiplication
restricted to) and the transpositionon(which is(which is TCP but not CP if dim7{\sigma $>1$ ).
The standard entanglement in the purely quantum case $B=B(7\{)=\tilde{B}$, $\nu=$
corresponds to the pure standard compound stateTr$7\{_{\sigma}=E_{\sigma}H$$B_{\sigma}=E_{\sigma}BE_{\sigma}$$B$$\pi^{q}$$\sigma$$=\mathcal{L}(H)$$\sigma$$B_{\sigma}$$B_{\sigma}\ni B\mapsto\tilde{\sigma}^{1/2}B\tilde{\sigma}^{1/2}$$\overline{B}=JB\dagger J$$\mathrm{C}\mathrm{P}$$B_{\sigma}$$=\tilde{\nu}$(10)$\mathrm{T}\mathrm{r}A\sigma^{1/2}\overline{B}\sigma^{1/2}=\omega_{q}$(A C4 $B$ )$=\mathrm{b}B\tilde{\sigma}^{1/2}\overline{A}\tilde{\sigma}^{1/2}$

on the algebra (A . It is given by the amplitude$l$fordefined in (5) as !
$B$$\equiv(\sigma^{1/2}|$$v’\simeq|\sigma^{1/2}$$B$$(\zeta \mathrm{t}\triangleleft J\eta)=\eta^{\uparrow}x\zeta$$\chi$$=\sigma^{1/2}$).

$\equiv\psi$, with$|\sigma^{1/2})^{\uparrow}=$

68
VIACHESLAV P BELAVKIN

corresponding to apure compound,
Any entanglement on,
is not one. If the space is also minimal,state is true ifbeIndeed,.
cantostandardonetheequivalentunitaryisdecomposed as$A=\mathcal{L}(\mathcal{G})$$\mu=\mathrm{T}\mathrm{r}$$\mathcal{G}$$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\rho=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\sigma$$\mathcal{G}=\mathcal{G}_{\rho}$$\varpi$$\pi^{\mathrm{T}}$$\pi_{q}$$(A)=\tilde{x}^{\uparrow}A\tilde{x}$$\varpi(A)=\sigma^{1/2}U^{\uparrow}AU\sigma^{1/2}=\varpi_{q}(U^{\uparrow}AU),$ofonto the support
is aunitary operator from
=U\sigma U\dagger with nonabelian.
andtothe standard
corresponding
Note that the compound state (4) within the standard reprecan always be extended to avector state on$=B’$ , but it cannot be extended$(B)$ whenofsentation)in the case of nonatomic N. If is afactor, this state
to anorlnal state on
bv the unit vector
is pure, given in the standard representation.
, ; however it is not normal on 13
unless is type

where

$U$

:

$\sigma^{1/2}\eta\mapsto\tilde{x}\eta$$\mathcal{H}_{\sigma}$$/’$$\mathcal{G}_{\rho}$$B_{\sigma}=U^{\uparrow}A_{\rho}U=\mathcal{L}(H_{\sigma})$$A_{\rho}=\mathcal{L}(\mathcal{G}_{\rho})$$\tilde{x}=\sigma^{1/2}$$\tilde{B}\vee B$$\varpi$$=\varpi_{q}$$\tilde{B}=J_{t/}BJ,,$$\mathit{1}\mathit{3}\equiv\iota$$(\mathrm{W}/\mathrm{y}, \iota, J_{/},)$$\tilde{B}\triangleright$

$B$

$B$

$\tilde{B}\vee B=\mathcal{L}(H_{J},)$$?)=\grave{\sigma}^{1/2}\in H,$$\mathrm{c}\triangleleft$

3. C-, D- AND$\mathrm{O}$

$B$

$B$

$\mathrm{I}:B\simeq \mathcal{L}(H)$-COUPLINGS AND ENCODINGS

Tlle compound states play the role of joint input-0utput probability measuresin classical information channels, and can be pure in quantum case even if themarginal states are mixed. The pure compound states achieved by an entanglementof mixed input and output states exhibit new, non-classical type of correlations
which are responsible for the EPR type paradoxes in the interpretation of quantum
theory. However mixed, so called separable states on $A$ QO , the convex productombinations$B$$\mathrm{c}$$\omega_{c}$(A 00 $B$ ) $= \sum_{n}\rho_{n}(A)\sigma_{n}(B)p(n)$ ,which we refer as the -compound states, do not exhibit such paradoxical behavior.
, , ::Here $p(n)>0$ , $\sum p_{n}=1$ , is aprobability distribution, andofare usually normal states defined by the product densities:. Such compound states are achieved by -couplings, wheregiven by$c$$\rho_{n}$$Aarrow \mathbb{C}$$\rho_{n}\omega$$\omega,,$$=\rho_{r\iota}(n$$\sigma_{n}\in A_{\mathrm{T}}(n$$c$$\sigma_{n}$$Barrow \mathbb{C}$$\sigma_{1}$$B$$\pi_{c}$$B_{\mathrm{T}}$$arrow A_{*}$$\pi_{c}=\varpi_{c}^{\mathrm{T}}$$\varpi_{c}(A)=\sum_{n}\rho_{n}(A)\sigma_{n}p(n)$,,$\varpi_{c}^{\mathrm{T}}(B)=\sum_{n}\sigma_{n}(B)\rho_{n}p(n)$with respectandare the probability densities for
Here
a112
$A$
, being theto given weights and on
and . Note that the entanglement
, is
maps
primitive
CP-TCP
convex combinations of the
not truly quantum.
The separable states of the particular form$\sigma_{?1}\in B_{*}$$\rho_{n}\in A_{*}$$\rho_{n}$

$B$

$\nu$

$\mu$

$c$

$\sigma_{n}$$\varpi_{c}$$\varpi_{n}(A)=\rho_{n}(A)\sigma_{n}\in B_{\mathrm{T}}$(11)$\omega_{d}$(A$\omega B$)$= \sum_{n}\langle n|A|n\rangle\sigma(n, B)$,

given by an orth0-normal
are pure states on
where
, and
with $\sigma(n)=\sigma_{n}p(n)$ , are usually consystemsidered as the proper candidates for the input-0utput states in the communication
channels involving the classical-quantum (c-q) encodings. Such separable state wasintroduced by Ohya $[10, 22]$ using aSchatten decomposition $\rho=\sum|n\rangle$ $n|p(n)$ of
into the orthogonal one dimensional projectors
the input density operator. Here we note that such state is the mixture of the classical-quantum$A=\mathcal{L}(\mathcal{G})=\tilde{A}$$\rho_{n}(A)=\langle n|A|n\rangle$$\{|n\rangle\}\subset \mathcal{G}$$\sigma(n, B)=\langle B, \sigma(n)\rangle_{\nu}$$\langle$$\rho\in \mathcal{T}(\mathcal{G})$$\rho_{n}=|n\rangle\langle n|$

69
QUANTUM SEX AND MUTUAL lNFt)R M ATI$()\mathrm{N}$correspondences
which can be described as the composition ofquantum channelingand the errorless encodingsin the$n|\mapsto n$ as input decodsense that they can be inverted by the measurements
ings. We shall call such separable states -compound as they are achieved by the( -couplings)to the subalgebra
diagonal couplingsof the diagonal
operators $A= \sum a(n)|n\rangle\langle n|$ , where$n\mapsto|n\rangle\langle$ $n|\omega$$|n\rangle$$\langle$$\sigma_{n}$$n|\mapsto\sigma_{n}$$n\mapsto|n\rangle$$\langle$$n|$$|n\rangle\langle$

$d$

$d$

$\pi_{d}=\varpi_{d}^{\mathrm{T}}$(12)$A_{d}\subseteq A$$\varpi_{d}(A)=\sum_{n}\langle n|A|n\rangle\sigma(n)$,.

$\varpi_{d}^{\mathrm{T}}(B)=\sum_{n}\sigma(n, B)|n\rangle\langle n|$

for the respect to the standard transpositionin the eigenbasis$\langle n|\tilde{A}|m\rangle=\langle m|A|n\rangle$of .
Actually Ohya obtained the compound states$\rho$$\omega_{d}(A(\triangleleft B)=\omega_{o}$(A tt$\omega_{d}$$\Lambda(B)$of quantum channels as normal unital CP maps
compouncl states(13)

as the result of composition$\Lambda$:)$arrow A$$B$

and the special, 0-$\omega_{o}(A\ltimes)B)=\sum_{n}\langle n|A|n\rangle p(n)\langle n|B|n\rangle$

corresponding to the orthogonal decompositions(14)$\varpi_{o}(A)=\sum_{n}\langle n|A|n\rangle p(n)|n\rangle\langle n|=\varpi_{o}^{\mathrm{T}}(A)$,, wheresuch that.
, we can extend this construction to any discretelyAssuming thatwith invariant comdecolnposable algebra $A=A$ on the Hilbert sum
ponentsin the eigen-ba is of theunder the standard complex conjugation
density operator $\tilde{\rho}=J\rho J=\rho$ .
particular, the von Neumann algebra $A$ might
be Abelian, as it is in the case $A=A$ for all $A\in A$ ,. whenis thediagonal algebra of pointwise multiplications $Ag=ag=Ag$ by the bounded functions $n\mapsto a(n)\in \mathbb{C}$ on the functional Hilbert space $(;=\ell^{2}\ni g$ with the standard. In this case the densities
complex conjugation
are given by thesum mable functions, andwith respect to the standard
any compound state has the separable form with $\rho_{n}(A)=a(n)$ corresponding to. The normal states on thethe Kronecker densities
are described
the probability densities $p(n)\geq 0$ , $\sum p(n)=1$ with respect to the standard$\sigma_{n}(B)=\langle n|\Lambda(B)|n\rangle$$\sigma_{n}=\Lambda^{\mathrm{T}}(|n\rangle\langle n|)$$\langle B, \Lambda^{\mathrm{T}}(\rho)\rangle_{\nu}=\mathrm{T}\mathrm{r}_{\mathcal{G}}\Lambda(B)\tilde{\rho}$$\langle A, \rho\rangle=\mathrm{T}\mathrm{r}\sigma_{-}A\tilde{\rho}$$\mathcal{G}=(|)\mathcal{G}_{i}$$J$$\mathcal{G}_{i}$$\mathrm{I}\underline{\mathrm{n}}$$A=\overline{A}$$\mathrm{e}_{-}.\mathrm{g}$$Jg=\overline{g}$$\rho\in A_{*}$$p\in\ell^{1}$$\delta$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\mu(\rho)=\sum p(n)$$A\simeq\ell^{\infty}$$\rho_{n}\simeq\delta_{n}$$1\supset \mathrm{y}$$1)\mathrm{a}\mathrm{i}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}$.

$\langle A, \rho\rangle_{\mu}=\sum a(n)p(n)$,$p\in\ell^{1}$,$a\in\ell^{\infty}$of $A_{l},=A$ with the commutative algebra $A$ . Every normal compound state$A\ltimes)B$ isdefined by$\omega_{c}$(A$\omega B$) $= \sum_{n}a(n)$$\langle B, \mathrm{a} (n)\rangle_{\nu}$$\omega$on,where a(n) $=\sigma_{?1}p(n)$ is the function with positive values a(n)normalized
to tlte probability density. Thus all normal compound states
on
:
with
given byare achieved by couplings
convex combinations of the primitive CP-TCP maps $\varpi_{n}(a)=a(n)\sigma_{n}\in B_{*}$ ,$\in B_{\mathrm{T}}$$p(n)=\langle I, \mathrm{a} (n)\rangle_{/},$$l^{)\infty}\omega B$$c$$\varpi_{c}(A)=\sum_{ll}a(n)\sigma(n)$$\pi_{c}=\varpi_{c}^{\mathrm{T}}$,$Barrow\ell^{1}$$\pi_{c}^{\mathrm{T}}--\varpi_{c}$$\varpi_{c}^{\mathrm{T}}(B)=\sum_{n}\sigma(n, B)\delta_{n}$,70
VIACHESLAV P BELAVKIN.
Note that any -coupling can be regarded as such quantum-classical c-coupling
which is achieved by the identification $a(n)=\langle n|A|n\rangle$ of the reduced diagonal. This simply follows from theand$n|p(n)$ for the induced statescommutativity of the density operators$\rho(A)=\omega_{d}$ (A) identified withdescribed by probability
and pure elementary states
In the case, we have density, whereamplitudesof rank one. The total compound amplitude isandoperators
the amplitude operators
obviously $v= \sum|n\rangle$ $v(n)$ , wheresatisfying the orthogonality relations$\mathrm{w}1_{1}\mathrm{e}\mathrm{r}\mathrm{e}$$\backslash ^{-}(7\mathfrak{l}, B)=\langle B, \sigma(??)\rangle_{\iota\prime}$$\mathrm{d}$$. \mathrm{d}1_{\xi \mathrm{i}}\mathrm{e}\mathrm{b}\mathrm{r}\mathrm{a}A^{0}=\{\sum|??\rangle a(n,)\langle_{71},| :$\ell\infty\ni a$A\in A\}$$\rho=\sum|n\rangle\langle$$p\in\ell^{1}$$\omega I$$\omega_{n}$$A=\mathcal{L}(\mathcal{G})$$\tilde{\chi}_{n}\equiv|\chi_{n}\rangle$$v_{?\mathrm{t}}=\lambda_{n}^{\prime\alpha)\psi_{?\mathit{1}}}$$\in \mathcal{G},\tilde{\psi}_{n}\equiv|\psi_{n}\rangle$$\in \mathcal{H}$$\sigma_{n}=\psi_{n}^{\uparrow}\psi_{n}$$\rho_{n}=\chi_{n}^{\uparrow}\chi_{?l}$$v(n)=\chi_{n}\infty\psi_{n}p(n)^{1/2}\mathrm{a}\mathrm{r}\mathrm{e}$$\mathcal{G}\ltimes)H$$arrow\ell^{2}$$v(n)^{\uparrow}v(m)=\rho_{n}00$ $\sigma_{n}p(n)\delta_{n}^{m}$. The “entangling” opcorresponding to the decomposition$(n)$ or as
erator for the separable state $x$ can be chosen as either as $x$, wherewith$= \sum\chi$or even as
corresponding to d. In particular, -entangling operator, corresponding to theond-encodings (12) is diagonal,. Thus, we have proved the Theorem 2below in the case oforthogonal
and . But before formulating this theorem in anatural generality
pure states
let us introduce the following notations.
are defined as integral convex combiTlle general -compound states onnations$v^{\uparrow}v= \sum\rho_{n}\mathrm{W}$$\sigma_{n}p(n)$$= \sum|n\rangle$ $\chi$$(n.)\langle$$\chi$$’\grave{\psi}_{1\uparrow},(7l)$$\chi$$7\mathrm{t}|$$= \sum|n\rangle$ $\chi$ $(n)\langle n|$$=\grave{\psi}_{J_{\gamma 1}}p(n)^{1/2}$$(n)=\chi_{n}\omega\tilde{\psi}(n)$$\chi$$\chi$$\mathrm{d}$$\chi$$= \sum|n\rangle$$\tilde{\psi}(n)\langle n|$$\mathcal{G}=\ell^{2}$$\grave{\chi}_{?1}=|?l\rangle$$\rho_{?1}$$\sigma_{t\mathrm{t}}$$A$ $\mathrm{o}0B$$\mathrm{c}$(A$\omega$$\lambda’B$)$= \int\rho_{x}(A)\sigma_{x}(B)p(\epsilon 1\downarrow\cdot)$

Such compound
given by aprobability distribution on the product-statesstates are achieved by convex combinations of the primitive CP-TCP maps
with $\varpi_{T}(A)=\rho_{x}$. (A) :$\rho_{x}.\omega\sigma_{x}.\cdot$$p$$\pi_{x}=\varpi_{x}^{\mathrm{T}}$$\sigma_{x}$(15)$\varpi_{c}(A)=J^{\cdot}\rho_{x}(A)\sigma_{x}p(\mathrm{d}x)$,$\varpi_{c}^{\mathrm{T}}(B)=\int\sigma_{x}(B)\rho_{x}.p(\mathrm{d}x)$.

This is always the case when the von Neumann algebra $A$ is Abelian, and thus canbe identified with the diagonal algebra of multiplications (Ag) $(x)=a(x)g(x)$ by
with respect to aon the functional Hilbert space (the functions
as theon(not necessarily finite) measure on $X$ . It defines
for the bounded multiplication densities $(\rho g)(x)=$integral,. The normal states on $A$ are given by the probability densities
pairingwith respect to the standard$;=L_{\mu}^{2}$$a\in L_{\mu}^{\infty}$$A_{\mu}\simeq L_{\mu}^{1}\cap L_{\mu}^{\infty}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\mu$$\mu$$/ \iota(\rho)=\int p(x)/\iota(\mathrm{d}x)$$p\in L_{\iota}^{1}$$p(\mathrm{r}\mathrm{c})g(x)$$\langle A, \rho\rangle_{\mu}=\int.a(x)p(x)\mu(\mathrm{d}x)$,$p\in L_{\mu}^{1}$,$a\in L_{\mu}^{\infty}$.

corresponding to the trivial transposition
andis the c-compound
it $=n$ . Any normal compound state on $A$$A$ bystate, defined on the diagonal algebra$()\mathrm{f}A_{*}=A_{\mathrm{T}}\simeq L_{l^{\iota}}^{1}$$A=\tilde{A}\simeq L_{\mu}^{\infty}$$\omega$(16)$\omega_{d}$(A$\omega B$$\mathrm{o}0$$B\simeq L_{\mu}^{\infty}(Xarrow B)$) $= \int a(x)\sigma(x, B)\mu(\mathrm{d}x)$ ,is absolutely integrable function with density operator
where$=\sigma_{x}p(x)$normalized to the probability density $p(x)=\langle I, \sigma(x)\rangle,/=$values a(x)$\sigma(x, B)=\langle B, \sigma(x)\rangle_{\iota/}$

71$\mathrm{Q}1^{\mathrm{Y}}$ANTUM SEX AND MUTUAL INFORMATION$\nwarrow(.\iota\cdot I))$

. It corresponds to

into$(x, A)=a(x)\sigma(x)$ :$\varpi$(17)-couplings

with$\pi_{d}=\varpi_{d}^{\mathrm{T}}=\pi_{\overline{d}}$,$\varpi_{d}(A)=./\cdot a(x)\sigma(x)\mu(\mathrm{d}x)$$\pi_{d}^{\mathrm{T}}=\varpi_{d}$$\delta_{x}$Theorem 3.1. Let(18)$\omega_{c}$: AChdecomposing$\varpi_{d}^{\mathrm{T}}(B)=\int\sigma(x, B)\delta_{x}\mu(\mathrm{d}x)$

is the (generalized) density operator of the Dirac stateon the diagonal algebra $A$ .

$\backslash \tau’\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}$( $1(.\iota\cdot)$$\mathrm{d}$,$\rho_{x}(A)=\langle A, \delta_{J}\rangle_{\mu}=$be a normal -compound state given as$13arrow \mathbb{C}$$c$$\omega_{c}(A\ltimes)$ $B)= \int.\mu_{x}(\tilde{\chi}^{1}.A\tilde{\chi}.)\nu_{x}(\tilde{\psi}^{\dagger}.B\tilde{\psi}.)p(\mathrm{d}x)$

,

: $H$
are linear operators having bounded transpose, .
spaces
with
.
. J.
.
. J. on Hilbert. We also assume that
respect to pointwise involution .

where$\lambda_{J}’$$\tilde{\lambda}=J\chi^{\uparrow}$:$\mathcal{G}arrow \mathcal{E}_{x}$$\psi\sim$,$arrow F_{x}$$\psi_{x}$$=J\psi^{\dagger}$$\mathcal{E}$$J$$\chi_{x}^{\uparrow}\chi_{x}\in\tilde{A}$,$= \int^{\oplus}\mathcal{E}_{x}p(\mathrm{d}x)$$\mathcal{F}$$= \int^{\oplus}\mathcal{F}_{x}p(\mathrm{d}x)$$=J^{\dagger}.$,$\psi_{x}^{\uparrow}\psi_{x}\in\overline{B}$$\mu_{x}(\tilde{\chi}^{\uparrow}.\tilde{\chi}.)=1=\nu_{x}(\tilde{\psi}^{\dagger}.\tilde{\psi}.)$with respect to the weights(19)$\mu_{x}(\tilde{\chi}^{\dagger}.\tilde{\chi}.)=\langle I, \chi_{x}^{\dagger}\chi_{x}\rangle_{\mu}$,$\nu_{x}(\tilde{\psi}^{\uparrow}.\tilde{\psi}.)=\langle I$,$\psi_{x}^{\dagger}\psi_{x}\rangle_{\nu}$Then this siate is achieved by decomposable entangling operator
defining -entanglement(15) $with$$\mathrm{t}_{\Gamma}^{/}l)\sim,(\mathrm{d}\mathrm{x})$$\chi$$= \int^{\oplus}\chi_{x}\omega$

$c$

(20)

$\rho_{x}(A)=\mu_{x}(\tilde{\chi}^{\dagger}.A\tilde{\chi}.)$,$\sigma_{x}(B)=\nu_{x}(\tilde{\psi}^{\dagger}.A\tilde{\psi}.)$,,. In particular,
corresponding to the probability densities
every -compound state (16) corresponding to $p(\mathrm{d}x)=p(x)\mu(\mathrm{d}x)$ with the Abelian
algebra $A$ can be achieved by the orthogonal sum of entangling operators $x_{x}=$,defining -entanglement(17) with$\sigma_{x}=\psi_{x}^{\dagger}\psi_{x}$$\rho_{x}=\chi_{x}\chi_{x}\dagger$

$d$

$d$

$\delta_{l}\cross)\tilde{\psi}_{x}$$\sigma(x)=\psi_{x}^{1}\psi_{x}p(x)$,$\sigma(x, B)=\nu_{x}(\tilde{\psi}^{1}.A\tilde{\psi}.)p(x)$.

corresponding to -compound state
Tlte amplitude operator))(18) is defined on as the orthogonal sum ofon (; ($nH$ into,
. Without loss of generality we can assume that
and
because the support
for$Proo|f$.

$v= \int^{\otimes}v_{x}p(\mathrm{d}x)$$\mathrm{c}$$v_{x}=\chi_{x}\aleph$$\mathcal{F}_{\alpha}p(\mathrm{d}x)$$\mathcal{E}_{x}=\mathcal{G}_{\rho}$$\mathrm{t}_{T}^{)^{1}}=v_{x}\mathrm{J}(E_{\rho}(\aleph E_{\sigma})$$(\mathcal{G}N H)_{v_{J}v}\dagger,$$v_{x}^{\dagger}v_{x}=\chi_{x}^{\uparrow}\chi_{x}\infty$. Due tocommute with $A\in A’$ and$B\in B’$ for almost all. Thus,is inand$\mathcal{G}_{\rho}\omega$$\int^{\oplus}\mathcal{E}_{x}\triangleright$$\psi_{x}$$\psi_{x}^{\uparrow}\psi_{x}=\rho_{x}\mathrm{t}\triangleleft$$\sigma_{x}$for almost all , the operators
respectively, andcommutes with$\chi_{x}\chi_{x}\in\dagger\overline{A’}_{:}-’\psi_{x}^{\uparrow}\psi_{x}\in\overline{B}^{\prime’}$$H_{\sigma}$$\psi_{T}$$F_{x}=\gamma\{_{\sigma}$$=\mathrm{r}\mathrm{a}\mathrm{n}v_{x}\dagger$$B\in\tilde{B}’$$x$$\chi_{x}$$\tilde{\psi}_{x}$$x$$\tilde{\chi}_{x}^{1}A\tilde{\chi}_{x}\in A$,$\tilde{\psi}_{x}B\tilde{\psi}_{x}\in B$) $A$ and
which defines the weights (19) on
for almost all . The restof the proof is the repetition of the proof of the Theorem 1for each with theaddition thatis the productfor each . The total entangling. C4
acts componentwise as
operator $x$ :.
$L_{p}^{\infty}\aleph$$L_{p}^{\infty}\mathrm{C}\triangleleft$

$B$

$x$

$x$

$v_{x}’=\chi_{x}\mathrm{c}\triangleleft\tilde{\psi}_{x}$$\chi_{T}$$\mathcal{G}\mathrm{c}\triangleleft F$$arrow \mathcal{E}$$\mathcal{H}$$x$$x_{x}(\zeta\ltimes)\eta.)=\chi_{x}(\propto)\tilde{\psi}_{x}\eta_{x}$72
VIACHESLAV P BELAVKIN,
In the case of -compound state (16) one should take$\chi_{J}g=g(x)$ . Thus the entangling operator in this case is given as$\mathrm{d}$$\chi$$\mathcal{G}=L_{\mu}^{2}$$(g \mathrm{t}\triangleleft \eta.)=\int^{\otimes}.g(x)\tilde{\psi}_{x}\eta_{x}\mu(\mathrm{d}x)$,$\forall g\in L_{\mu}^{2}$, .

$\mathcal{E}_{x}=\mathbb{C}$, and$= \int^{\oplus}\eta_{x}\mu(\mathrm{d}x)\in \mathcal{F}\ldots$$\eta$1in (15) are both CP and TCP and thus are notNote that -entanglements:with and Abelian algebra $A$ in (17) istrue quantum. The map
normalized to the input-valued measure a(dx)described by a
. This gives the concise form for the
probability measure aswith thedescription of random classical-quantum state correspondences.
given probability measure , called encodings of a$\varpi_{c}$$\mathrm{c}$$\varpi_{c}$$Aarrow B_{\mathrm{T}}$$=\sigma(x)\mu(\mathrm{d}x)$$B_{\mathrm{T}}$$p(\mathrm{d}x)=\langle I, \mathrm{a} (\mathrm{d}x)\rangle_{\nu}$$x\mapsto\sigma_{x}$$= \int\sigma(\mathrm{d}x)$$p$Definition 3.1. Let both algebras $A$ and be non-Abelian. The map :is called -encoding of
if it is a convex combination of the primitive maps.
and nor$mal$ states:
given by the probability densities$A$
It is called -encoding if it has the diagonalizing form (12) on , and it is called
are mutually orthogonal:-ereading if all density operators
for all$CP$
map
as in (14) The entanglement which is described by non-separable
will be called q-encoding.
:$B$$\varpi$$Aarrow B_{\mathrm{T}}$$(B, \sigma)$$\mathrm{c}$$\sigma_{n}\in B_{\mathrm{T}}$$\sigma_{?l}\rho_{n}$$\rho_{n}$$Aarrow \mathbb{C}$$\mathrm{d}$$\sigma_{m}\sigma_{n}=0$$\sigma_{n}$$\mathrm{o}$$?\prime 1\neq n$$\varpi$$Aarrow B_{\mathrm{T}}$,Note that due to the commutativity of the operators A I with I $B$ on
one can treat the encodings as nondemolition measurements [9] in $A$ with respect to
N. The corresponding compound state is the state prepared for such measurementson the input . It coincides with the mixture of the states, corresponding to thoseafter the measurement without reading the message sent. The set of all d-encodings
for aSchatten decomposition of the input state on $A$ is obviously convex withon , corresponding to theextreme points given by the pure output states
not necessarily orthogonal (not Schatten) decompositions $\sigma=\sum\sigma(n)$ into the
one-dimensional density operators $\sigma(n)=p(n)\sigma_{n}$ .
The Schatten decompositions $\sigma=\sum_{n}q(n)\sigma_{n}$ correspond to -encodings, the, $p(n)=q(n)$ characterized by the orthogonality
extreme encodings, $?n\neq n$ . For each Schatten decomposition of they form aconvex.
subset of -encodings with mixed commuting$\omega$$\omega$$\mathcal{G}\triangleright 0?t$$\mathcal{G}$$\rho$$\mathrm{t},1\mathrm{l}\mathrm{e}$$\sigma_{n}$

$B$

$0$

$\sigma_{n}=\eta_{n}\eta_{n}\dagger$$\mathrm{d}$$\sigma_{n},\sigma_{?1}=0$$\sigma$$\mathrm{d}$$\sigma_{n}$4. QUANTUM ENTROPY VIA ENTANGLEMENTS, which are:
As we have seen in the previous section, the encodings$A$usually described as in (17) with adiscrete Abelian , correspond to the case (12)when the general entanglement (7) is -encoding, with the diagonal couplingin the eigen-representation of adiscrete probability density on non-Abelian $A$ .
The true quantum entanglements with non-Abelian $A$ cannot be achieved by el-,and more general, encodings even in the case of discrete $A$ . The nonseparable,
called in [22] -compound states, can be achieved by qtrue entangled states
encodings the quantum-quantum nonseparable correspondences (6) which are notdiagonal. in the eigen-representation of .
As we shall prove in this section, the self-dual standard true entanglement, which is defined in (9), is the mostto the probe systemin the sense that it achieves the maximal
informative for aquantum systemwhen
is given in (10)mutual information in the coupled system $(A\propto)B$ ,$Aarrow B_{\mathrm{T}}$$\varpi$$d$$\pi=\varpi^{\mathrm{T}}$$\rho$$\mathrm{c}$$\omega$$q$$\rho$$\varpi_{q}=$$\varpi_{q}^{\mathrm{T}}$$(A^{0}, \rho_{0})=(\tilde{B},\tilde{\sigma})$$(B, \sigma)$$\omega)$$\omega$$=\omega_{q}$73t)$l^{\mathrm{v}}$ANTUM S EX AND MUTUAL$1\mathrm{N}\mathrm{F}()\Gamma \mathrm{t}$M ATlt)N

Let us consider entangled mutual information and quantum entropies of statesmeans of the above three types of compound states. To define the quantum
mutual entropy, we need to apply aquantum version of the relative entropy to$=A\infty B$ , called also the information divergency
compound state on Jhe algebra. The relative entropy wasof the state with respect to areference state on
defined in [20, 21, 24] even for most general von Neumann algebra A{, but for our
purposes we need the following its explicit description.
Let A4 be asemi-finite algebra with normal states and having the densitywith respect to the pairingandoperator$\rceil).\mathrm{Y}$$\mathcal{M}$$\mathcal{M}$$\omega$$\varphi$$\omega$$v^{1}v$$\varphi$$\phi\in \mathrm{A}4$$\langle II, v^{\dagger}v\rangle=(\overline{v}|\overline{\iota(\mathrm{J}I)}\overline{v})$,$M\in \mathcal{M}$,$v^{\dagger}v\in\overline{\mathcal{M}}$(notgiven by anormal faithful weight on the transposed algebra). Then thein (3) in the case ofnecessary decomposable asof the state with respect to is given by the formula
relative entropy$\overline{\mathcal{M}}=J\mathcal{M}J$$\tau$$\Lambda 4$$\tau=\tilde{\mu}\omega\tilde{\nu}$$\mathrm{R}(\omega;\varphi)$(21)$\mathrm{R}(\omega :$\omega$$B$$=A\mathrm{r}\aleph$$\varphi$\varphi)=\tau(v(\ln v^{\dagger}v-\ln\phi)v^{\dagger})=\tau(\omega(\ln\omega-\ln\phi))$.

(For the notational simplicity here and below we identify the state with its density
if the states are equally). It has apositive value
operatorno rmalized, say (as usually) $\tau(\omega)=1=\tau(\phi)$ , and it can be finite only if the stateis absolutely continuous with respect to the reference state , i.e. iff $\omega(E)=0$, $E\phi=0$ . Note that this definition
for the maximal null-0rthoprojector, and it can be extended also toweightsemi-finite
choiceofthedepends on the’the arbitrary normal and with unbounded self-adjoint density operators.
and$\omega$$v^{1^{1}}v$$\mathrm{R}(\omega :\varphi)\in[0, \infty]$$\omega$$\varphi$$E\in \mathcal{M}$$\tau$$v^{\acute{|}}\tau$$\omega$$\varphi$$\phi$The most important property of the information divergence is its monotonicity
after the application
property $[20, 25]$ , i.e. nonincrease of the divergency
to the:of tlte pre-dual of anormal completely positive unital map:
algebron
avon
Neumannstatesand$\mathrm{R}$$\mathrm{R}$$(\omega 0 :\varphi_{0})$$\mathcal{M}$$\mathrm{K}$$arrow \mathcal{M}^{0}$$\mathcal{M}^{0}$$\mathrm{a}$$\omega_{0}$$\varphi_{0}$(22)$\omega=\omega_{0}\mathrm{K}$,$\varphi=\varphi_{0}\mathrm{K}\Rightarrow \mathrm{R}(\omega :\varphi)\leq \mathrm{R}(\omega_{0} :\varphi_{0})$.

in acompound state
The mutual information I $(\pi)=|$$Aarrow
B_{*}$,
by:with the marginals
or
coupling :$(\pi^{*})$$B$ $arrow A_{*}$$\pi$$\omega$achieved by a$\pi^{*}$(A $00I$ )$\rho(A)=\omega$$=\langle A, \rho\rangle_{\mu}$, $\sigma(B)=\omega$ (I$\omega B$),$=\langle B, \sigma\rangle_{/}$

is defined as the relative entropy(23)

I$(\pi)=\tau(\omega (\ln\omega-111 (\rho\omega I)-\ln(I\triangleright j \sigma)))=\mathrm{R}(\omega :\rho\omega \sigma)$.

.
$=AwB$ with respect to the product stateforof the state on
This quantity, generalizing the classical mutual information corresponding to thease of Abelian $A$ , , describes an information gain in aquantum system $(A, \rho)$:. It, or in
via an entanglement
via the entanglement
is naturally treated as ameasure of the strength of the generalized entanglement.
having zero value only for completely disentangled states$\mathcal{M}$$\omega$$\tau=\tilde{\mu}\omega\tilde{\nu}$$\varphi=\rho \mathrm{t}\triangleleft\sigma$$B$$\mathrm{c}$$\varpi^{\mathrm{T}}=\pi$$(B, \sigma)$$\varpi$$\omega=\rho \mathrm{C}\triangleleft$$Aarrow B_{\mathrm{T}}$$\sigma$quantum system with norrmal faithful semifiProposition 4.1. Let$A^{0}arrow
B_{*}$
be
nor$mal$ coupling of the statenite weight, and:by the, defining an entanglement $\varpi=\pi^{*}-of$ $(A, \rho)$ toonto$Aarrow A^{0}$map
positive:
unital.
with a nomal completely
composition. In particular, for each norrmal coupling given, whereThen 1 $(\pi)\leq|$
there exists not less infor mative couplingby (15) such as$(A^{0}, \mu_{0})$$be$$A^{0}$$\sigma=l\iota$

$a$

$a$

$\pi_{0}$$\rho_{0}=\nu\circ\pi_{0}$$(B, \sigma)$$\circ\pi$$\pi^{*}=\pi_{0}\mathrm{K}$$\pi^{0}=\pi_{0}^{*}$$(\pi^{0})_{-}$$\pi$$\mathrm{K}$$=\varpi_{c}^{\mathrm{T}}$

$c$

$a$

$d$

$\pi^{0}=\varpi_{0}^{\mathrm{T}}$74
VIACHESLAV P BELAVKIN

with Abelian-couplingin this sense.

corresponding to the encoding,
to$\pi_{q}(B)=\sigma^{1/2}\tilde{B}\sigma^{1/2}$$\pi^{0}=\pi_{q}$

$q$

$P$

$A^{0}$

of$\varpi_{0}=\pi_{0}-$on$\rho_{0}=\tilde{\sigma}$, and the standard
is the maximal one$(B, \sigma)$$A^{0}=\overline{B}$roof. The first follows from the monotonicity property (22) applied to the ampli-).
) $=\mathrm{K}(A)00$ $B$ of the CP map
from $Aarrow A^{0}$ to A
than (A(I denotes the identity map $Barrow B$ ) is achieved
The compound state,, andcorrespondingby the entanglement
which is defined by. It corresponds to the couplingto, where:as$\alpha 1B$$\mathrm{K}$$\mathrm{K}$$Barrow A^{0}\ltimes$$\omega$$B$$\omega_{0}(\mathrm{K}(\mathfrak{p} \mathrm{I})$$\varpi$$\varphi_{0}=\rho_{0}\omega$$\mathrm{K}$’$=\varpi_{0}\mathrm{K}$$\varphi_{0}$$(\mathrm{K}\infty \mathrm{I})=\rho\triangleright)\sigma$$\rho=\rho_{0}\mathrm{K}$$\pi=\mathrm{K}^{*}\pi_{0}$$\sigma$$\mathrm{K}^{*}\tilde{\rho}_{0}=J(\mathrm{K}^{\mathrm{T}}\rho_{0})^{\uparrow}J$$A_{*}^{0}arrow A_{*}$,.
,’
This monotonicity property proves, in particular, that for any separable com,pound state (18) on A ) , which is prepared by the entanglement
having the same, orwith
there exists a entanglement,
even larger information gain (23). One can take even aclassical system
say the diagonal sublalgebrawith the state , induced byonmeasure $\mu=p$ , and consider the classical-quantum correspondence (encoding)$\forall A\in A$$\langle A, \mathrm{K}^{\mathrm{T}}\rho_{0}\rangle_{\mu}=\langle \mathrm{K}A, \rho_{0}\rangle_{\mu_{\mathrm{I}}}$

$y$

$B$

$\rho_{0}\in A_{\mathrm{T}}^{0}$$\pi_{c}=\varpi_{c}^{\mathrm{T}}$$\mathrm{c}$$(A^{0}, \rho_{0})$$\pi_{0}=\varpi_{0}^{\mathrm{T}}=\pi_{0}^{-}$$\mathrm{d}$$(A^{0}, \rho_{0})$$\mathcal{G}0=L_{p}^{2}$$A^{0}\simeq L_{p}^{\infty}$$\rho_{0}$$\mathrm{t},1_{1}\mathrm{e}$$\varpi_{0}(A^{0})=\int a(x)\sigma_{x}p(\mathrm{d}x)$,$A^{0}= \int^{\oplus}a(x)p(\mathrm{d}x)$,$a\in L_{p}^{\infty}$.
assigning the states, to the letters with the probabilities )In this case the state is described by the density $\rho=I$ the multiplication by iden) $H$ , and the mutual information
function in , is multiplication by . in(23) is given as$\sigma_{x}(B)=\langle B, \sigma_{x}\rangle,$

$x$

$l$

$(\subset 1.\iota\cdot)$$\rho$$L_{p}^{2}$$\mathrm{t},\mathrm{i}\mathrm{t}_{1}\mathrm{y}$

(24)

1

$L_{p}^{2}\alpha$$\sigma$$\omega$$( \pi^{0})=J^{\cdot}\grave{\nu}_{x}(\sigma_{x}(\ln\sigma_{x}-\ln\sigma))p(\mathrm{d}x)=\mathrm{S}(\sigma)-\int \mathrm{S}(\sigma_{x})p(\mathrm{d}x)$,( In ). The achieved information gain I
is larger than I
where
corresponding tobecause the entanglementin (15) is:represented as the compositionof the encodingwith the CP$\mathrm{S}(\sigma)=-\grave{|}\sqrt$$\sigma$$(\pi)$$(\pi^{0})$$\sigma$$\omega=\int\rho_{x}\omega$$\sigma_{x}p(\mathrm{d}x)$$\mathrm{c}$$\varpi_{c}$$\varpi\circ \mathrm{K}$$A^{0}arrow B_{\mathrm{T}}$$\varpi\circ$lnap$\mathrm{K}(A)=\int^{\oplus}\rho_{x}(A)p(\mathrm{d}x)$,$A\in A$

given by $a(x)=\rho_{x}(A)$ for each $A\in A$ . Hence$\pi^{*}(A)=\overline{\varpi(A)}=\overline{\varpi_{0}\mathrm{K}A}=\pi_{0}(\mathrm{K}A)$,$\forall A\in A$, and thus I, wherewhere.
Tlle inequality (22) can also be applied to the standard entanglement corresponding to tlte compound state (10) on 1300 . Indeed, any
entanglement$A$
as a CP map
on intocan be decomposed$(\pi^{0})\geq 1$ $(\mathrm{K}^{*}\pi^{0})=[(\pi)$$\pi_{0}=\varpi_{\overline{0}}$$\pi^{0}=\pi_{0}^{*}=\varpi_{0}^{\mathrm{T}}$$B$$-\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}$$Aarrow B_{\nu}$$B_{\mathrm{T}}$$\mathfrak{N}\mathrm{b}\varpi.(A)=l\iota$$(\tilde{\chi}^{\uparrow}(A \omega I) \tilde{x})$$l\iota(\tilde{x}^{\uparrow}(A\ltimes)I)\grave{x})=\sigma^{1/2}\mu(X^{\uparrow}(A \omega I)X)$$\sigma^{1/2}=\varpi_{0}(\mathrm{K}A)$,

where $\mathrm{K}A=\mu(X^{\uparrow}(A\ltimes.)I)X)$ is anormal unital CP map. It is uniquely
$X$
(
by
given
,
:
satisfying thean operator
with
$B’$
$X$
, and thus
condition
due to the commutativity ofwith $A’$ (A and with . Moreover, the partial weight ofi well-defined by$Aarrow\tilde{B}$$\mathcal{E}(\kappa H$$(I \mathrm{t}\triangleleft \sigma)^{1/2}=\tilde{x}$$B$$\sigma$$B$$arrow \mathrm{C}\mathcal{G}$$\aleph F$$\mathcal{E}=\mathcal{G}_{\rho}$$\mathcal{H}=\mathcal{F}_{\sigma}$$X\in A\mathrm{r}\aleph$$\tilde{\chi}$$X^{\uparrow}X$$\mu$$\mathrm{s}$75
QUANTUM SEX AND MUTUAL INF()RMATI$()\mathrm{N}$

as $\mu(X^{\uparrow}X)=I$ . Thus
, where is anormal
and
, and
unital CP map:. Hence the standard entanglement(coupling) (9) corresponds to the maximal mutual information, II. I$l^{\iota}(\tilde{x}^{1}\tilde{x})=\sigma$$\varpi$$Aarrow\tilde{B}$$\mathrm{K}$$\pi=\mathrm{K}^{*}\pi_{q}$$=\varpi_{q}\mathrm{K}$$B_{\mathrm{T}}=\tilde{B}_{*}arrow A_{*}$$\mathrm{K}^{*}$$(\pi_{q})\geq 1$$(\mathrm{K}^{*}\pi_{q})=$$(\pi)$

Note that the mutual information (23) is written as
I
where

)

$\varphi=\mu \mathrm{r}\cross$

$\nu$

(25)

,

$(\pi)=\mathrm{S}(\rho)+\mathrm{S}(\sigma)-\mathrm{S}(\omega/\varphi)$$\mathrm{S}(\rho)=\mathrm{S}(\rho/\mu)$

,

,

and

$\mathrm{S}(\sigma)=\mathrm{S}(\sigma/\nu)$$\mathrm{S}(\omega/\varphi)=-\tilde{\varphi}(v(\ln v^{\dagger}v)v^{\dagger})\equiv-\tilde{\varphi}(v^{\dagger}v\ln vv)\dagger$

denotes the entropy of the density operatorof the state with respect
to the weight on, coinciding. Note that the entropy(cf. with (21 in the case), is not in general positive, and may not even bebounded from below as afunction of . However in the case of irreducibleit
can always be made positive by the choice of the standard
on , inwhich case it is called the von Neumann entropy of the state, denoted
simply as:$v^{\uparrow}v\in\overline{\mathcal{M}}$$\omega$$\mathcal{M}$$\mathrm{S}(\mathrm{v}/\mathrm{t})$$\varphi$$\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}-\mathrm{R}(\omega :\varphi)$$\tau=\tilde{\varphi}$$\mathcal{M}$$\omega$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\tau=\mathrm{b}$$\mathcal{M}$$(=v^{\uparrow}v)$$\omega$$\mathrm{S}(\omega)$

(26)

.

$\mathrm{S}(\omega/\tau)=-\mathrm{H}\omega\ln\omega\equiv \mathrm{S}(\omega)$

In the following we shall assume that is adiscrete decomposition of the irreduciblewith theinduced on $B_{*}=B_{\tau}$ . The
entropyof the density operator for the normal state on canbe found in this case as the maximal informationachieved via:
all -encodingsof the systemsuch that,.
Indeed, as follows from the proposition above, is sufficient to find the maximum ofover all -couplings
1
mapping into Abelian $A$ with fixed $(I)=\sigma$ ,. to find maximum of (14) under the condition. Due to positivityof the -conditional entropy$B$$B_{i}=\mathcal{L}(H_{i})=\overline{B}_{i}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\nu=\mathrm{T}\mathrm{r}_{7\{}=\tilde{\nu}$$\mathrm{S}(\sigma)=\mathrm{S}(\sigma/\nu)$$\sigma$$\varpi$$\mathrm{c}$$(\pi)$$A\mapsto B_{\tau}$$(B, \sigma)$$\pi^{0}=\varpi^{\mathrm{T}}$$\mathrm{d}$$B$$\sigma$$\mathrm{S}(\sigma)=\sup 1$$(\pi_{c})$$(I)=\sigma\varpi^{\mathrm{T}}=\pi_{\overline{c}}$$\varpi$$B$$\varpi$$\int\sigma_{x}p(\mathrm{d}x)=\sigma$$\mathrm{i}.\mathrm{e}$

$d$

.Tr

(27)

$\mathrm{S}(\pi_{d})=-\int$$( \sigma_{x}\ln\sigma_{x})p(\mathrm{d}x)=\int \mathrm{S}(\sigma_{x})p(\mathrm{d}x)$

the information I $(\pi^{0})=1$
has the maximum
which is achieved on an
extreme -couplingwhen almost all
are zero, i.e. when almost allare
one-dimensional projectors
corresponding to pure states . One can takefor example, the maximal Abelian subalgebra
generated by $P_{n}=|n\rangle$$n|p(n)$ offor aSchatten decomposition. The maximal value
In rank of the von Neumann entropy is defined by the dimensionality rank flof the maximal Abelian subalgebra of the decomposable algebra , i.e. bydinl$\dim H$ .
However, if is not -coupling, the difference
can achievetlte negative value, and cannot serve as ameasure of conditional entropy in such$(\pi_{d})$$\mathrm{d}$$\mathrm{S}(\sigma)$$\pi_{d}^{0}$$\mathrm{S}(\sigma_{x})$$\sigma_{x}$$\sigma_{x}^{0}=P_{x}$$\sigma_{x}$$A^{0}\subseteq B$$\sigma$$= \sum_{n}|n\rangle\langle$$\langle n|\in B$$\sigma\in B_{\tau}$

$B$

$=$

$A^{0}$

$B$

$\pi$

$\mathrm{S}(\pi)=\mathrm{S}(\sigma)-1$ $(\pi)$$\mathrm{c}$case.

Definition 4.1. The supremum of the mutual infor mation(28)$\mathrm{H}(\sigma)=\sup\{1(\pi) :which is achieved on-coupling$q$$A–\overline{B}$$\pi_{q}(B)=\sigma^{1/2}\overline{B}\sigma^{1/2}$\mu 0\pi=\sigma\}=1$

for a fixed state$(\pi_{q})$,by the standard, is called -entropy of the state . The maximum$\sigma(B)=\mathrm{T}\mathrm{r}_{\mathcal{H}}B\sigma$$\sigma$$\mathrm{q}$$\mathrm{S}(\sigma)=\sup\{1(\pi_{c}) :\mu 0\pi_{c}=\sigma\}=1$$(\pi_{d}^{0})$76
VIACHESLAV P BELAVKIN

over all -couplings
extreme -coupling
$c$

$d$

corresponding to -encodings(15), which is achieved on an, is called -entropy of the state . The differences$c$$\pi_{c}$$\pi_{d}^{0}$$\mathrm{c}$$\sigma$,$\mathrm{H}(\pi)=\mathrm{H}(\sigma)-|$ $(\pi)$$\mathrm{S}(\pi)=\mathrm{S}(\sigma)-|$$(\pi)$

with respect to $A$ and theare called respectively, the -conditional entropy on$BarrowA$. A compound state is said(degree of) disentanglement for the coupling :, andbe essentially entangled if
for a -coupling $\pi=\pi_{c}$ is$A$called -conditional entropy on with respect to .
$B$$\mathrm{q}$$\pi$$\mathrm{S}(\pi)<0$$t()$$\mathrm{S}(\pi)\geq 0$

$c$

$B$

$\mathrm{c}$Obviously,
are both positive, do not depend unlike
and. The same is true foron the choice of the faithful weight on , and, wherehas always apositive valueandtlze conditional entropies$\mathrm{H}(\sigma)$$\mathrm{S}(\sigma)=\mathrm{S}(\sigma/|/)$$\mathrm{S}(\sigma)$

$B$

$\nu$

$\mathrm{H}(\sigma)\geq \mathrm{S}(\sigma)$$\mathrm{S}(\pi)$$\mathrm{S}(\pi)$$\mathrm{H}(\pi)$$\mathrm{S}(\pi)\geq \mathrm{S}(\pi^{0})\geq 0$

for anormal unital CP
in the case of a -coupling $\pi=\pi_{c}$ due to$\pi^{0}=\pi_{d}$$AarrowA^{0}$. But the,awith
Abelian-coupling
is
:
where
lllap
can also achieve the negative value
disentanglement$\pi_{c}^{*}=\pi_{d}^{*}\mathrm{K}$$\mathrm{c}$$A^{0}$$\mathrm{d}$$\mathrm{K}$$\mathrm{S}(\pi)$(29)$\inf\{\mathrm{S}(\pi) :\mu 0\pi=\sigma\}=\mathrm{S}(\sigma)-\mathrm{H}(\sigma)=-\sum_{i}x$$(i)\mathrm{S}(\sigma_{i})$as the following theorem states in the case of the discrete . Here thewithare the density operators of the normalized factor-states$(i)=\sigma(I^{i})$ , whereif
are the orthoprojectors onto . Note thatthe algebra is completely decomposable, i.e. Abelian. In this case the maximal
iscan be written as $\ln\dim$ Z3. The disentanglement
value In rank fl of$A$
as in the case of Abelian .
always positive in this case, and$B$$\sigma_{i}\in \mathcal{L}(H_{\tau})$$\sigma_{i}=x$$I^{i}$$(i)^{-1}\sigma|\mathcal{L}(H_{i})$$\mathrm{H}(\sigma)=\mathrm{S}(\sigma)$$\mathcal{H}_{i}$$\chi$$B$$\mathrm{S}(\pi)$$\mathrm{S}(\sigma)$$\mathrm{S}(\pi)=\mathrm{H}(\pi)$, withbe a discrete decomposable algebra on
Theorem 4.2. Let$with$$\sigma=(|)\sigma(i)$respect to the trace
normal state given by the density operator
induced by the
be its center with the state
on , and$(i)$is-entropy
given as the von. Then theprobability distribution
Neumann entropy (26) of the density operator , and the -entropy(28) is given$l)y$ the $f\dot{o}rmula$$B$$j\iota$$=\mathrm{n}_{\mathcal{H}}$$B$$\mathcal{H}=(\})_{i}H_{i}$$\mathrm{C}$$\subseteq B$$\chi$$\chi$$(i)=\mathrm{T}\mathrm{r}$$c$$\sigma$$\mathrm{H}(\sigma)=\sum_{i}$This can be written as
and($\chi$$\mathrm{H}_{B|C}(\sigma)=-2\sum_{i}x$$\mathrm{H}(\sigma)$a(i)).

, where $\mathrm{H}c(\sigma)=-\sum_{i}x$ $(i)\ln x$ $(i)$ ,$(i)\mathrm{b}_{\mathcal{H}},\sigma_{i}\ln\sigma_{i}=2\mathrm{S}_{B|C}(\sigma)$.
is finite iffwith
it is bounded, with the maximal value$\sigma_{i}=\sigma(i)\mathrm{x}(\mathrm{i})$$q$$(i)\ln x$ $(i)-2\mathrm{H}_{?\{;}\sigma(i)\ln$$\mathrm{H}(\sigma)=\mathrm{H}_{B|C}(\sigma)+\mathrm{H}c(\sigma)$$=\sigma|\mathrm{C}$$\mathrm{S}(\sigma)$$\sigma$(30)$\mathrm{S}(\sigma)<\infty$,, and if is finite-dimensional,
which is achieved for$B$$\sigma^{\mathrm{O}}=$$\mathrm{H}(\sigma^{\mathrm{o}})=\ln\dim B$$(|)\sigma_{i}^{\mathrm{o}}x^{\mathrm{o}}(i)$$\sigma_{i}^{\mathrm{o}}=(\dim H_{i})^{-1}I^{i}$,$\chi^{\mathrm{O}}(i)=\dim B(i)/\dim B$,

where $\dim B(i)=(\dim \mathcal{H}_{i})^{2}$ , $\dim B=\sum_{i}\dim B(i)$ .

Proof. We have already proven that$\mathrm{S}(\sigma)=\mathrm{S}(\sigma)$$\mathrm{S}(\sigma)=-\sum \mathrm{H}_{\mathcal{H}_{j}}\sigma(i)\ln$

a(i)

$a$

, where$=\mathrm{S}_{C}(\sigma)+\mathrm{S}_{B|C}(\sigma)$,

77$\mathrm{Q}l^{\tau}\mathrm{A}\mathrm{N}\mathrm{T}\mathrm{L}^{\cdot}.\backslash 4$

SEX AND MUTUAL$1\mathrm{N}\mathrm{F}()\mathrm{R}$M ATI$()\mathrm{N}$

,
with
.
The -entropy
is the supremum (28) of the mutual information (23) whichis achieved on the standard entanglement, corresponding to the density operator$\omega=(|)\omega(i, k)-$ with $\omega(i, k)=\chi$of the standard compound state(10) with $=B$ ,, where. Thus$\mathrm{S}_{B|C}(\sigma)=\sum x$ $(i) \mathrm{S}(\sigma_{i})=\frac{1}{2}\mathrm{H}_{B|C}(\sigma)$$\mathrm{S}_{C}(\sigma)=\mathrm{H}_{C}(\sigma)$$\mathrm{H}(\sigma)$$\mathrm{q}$$(i)|\sigma_{i}^{1/2})\langle\sigma_{i}^{1/2}|\delta_{k}^{i}$1$(\pi_{q})$$B$$\rho=\sigma$$=$

Tru ( $\ln\omega-\ln$ ($=$$\sum_{i}x$$\mathrm{H}(\sigma)=\mathrm{I}(\pi_{q})$$\sigma \mathrm{M}$$I)-\ln$ (I$(i)\ln x$ $(i)-2\mathrm{b}$Here we used that Tr$\omega$lnw$\omega$$\sigma)$)$=\mathrm{S}(\omega)-2\mathrm{S}(\sigma)$$\sigma\ln\sigma=-\sum_{i}x$$= \sum_{i}\chi$ $(i)\ln

x$ $(i)$$(i)(\mathrm{l}\mathrm{n}’\sigma(i)+2\mathrm{T}\mathrm{r}_{\mathcal{H}_{i}}\sigma_{i}\ln\sigma_{i})$due to$\omega\ln\omega=(\{)_{i,k}\omega(i, k)\ln\omega(i, k)=(\})_{i}\chi(i)|\sigma_{i}^{1/2})(\sigma_{i}^{1/2}|\ln x$ $(i)$and that Tra$\sigma\ln$$\sigma$$= \sum_{i}\chi$ $(i)(\ln x (i)-\mathrm{S}_{B_{i}}(\sigma_{i}))$$\ln rr$.

,

due to$=(|)_{i}\sigma(i)\ln\sigma(i)=(|’ i\chi$ $(i)\sigma_{i}(\ln x (i)+\ln\sigma_{i})$for the orthogonal decomposition a
Thus$=(|)_{i}x(i)\sigma_{i}$, where$\chi$$(i)=\mathrm{T}\mathrm{r}$ $\sigma(i)$.

, and it is bounded$\mathrm{H}(\sigma)=\mathrm{H}_{B|C}(\sigma)+\mathrm{H}_{C}(\sigma)=2\mathrm{S}_{B|C}(\sigma)+\mathrm{S}_{C}(\sigma)\leq 2\mathrm{S}(\sigma)$by$=$$\mathrm{C}_{B}$$\sup_{\chi}\sum_{i}x$$=$$(i)(2 \sup_{\sigma_{j}}\mathrm{S}_{B(i)}(\sigma_{i})-\ln x$$- \inf_{\chi}\sum_{i}x$ $(i)(\ln$(i))$

x (i)-2\ln\dim \mathcal{H}_{i})=\ln\dim B$.

Here we used the fact that the supremum of von Neumann entropies
for the$\dim
B(i)=(\dimH_{i})^{2}<\infty$$(i)=\mathcal{L}(H_{i})$simple algebras
withis achieved on the, and the infimum of the relative
tracial density operators$\mathrm{S}(\sigma_{i})$$B$$\sigma_{i}=(\dim 7\{_{i})^{-1}I^{i}\equiv\sigma_{i}^{\mathrm{o}}$entropy$\mathrm{R}(x :where$\chi^{\mathrm{O}}(i)=\mathrm{d}\mathrm{i}\mathrm{m}B$!)$$= \sum_{i}x$$(i)$( $\ln x$ $(i)-\ln$ ? (i)),$(i)/\dim$ Z{, is zero, achieved at$\chi$$=pt^{\mathrm{O}}$.

$\mathrm{I}$5. QUANTUM CHANNEL AND ENTROPIC CAPACITIESLet
be aHilbert space describing aquantum input system anddescribe
its output Hilbert space. Aquantum channel is an affine operation sending each
input state defined on
to an output state defined on $H$ such that the mixtures ofstates are preserved. Adeterministic quantum channel is given by alinear isometry
with $U\dagger U=I^{1}$ ( is the identify operator in) such that each
, $||\eta_{1}||=1$ , is transmitted into an output state vector
input state vector$1(n)$ of pure input states$/7=U\eta_{1}\in H$ , $||\eta||=1$ . The orthogonal sums
Bl. (n)are sent into the orthogonal sums $\sigma=(|)\sigma(n)$ of purestates oncorresponding to the orthogonal state vectors $\eta(n)=U\eta_{1}(n)$ .
Anoisy quantum channel sends pure input states
on an algebra
given by the composition with anormal completely positive
into mixed ones
(as well as ) is equipped withunital map : $Barrow B^{1}$ . We shall assume thatdefining the pairing
ano rmal faithful semiflnite weight$\mathcal{H}$$\mathcal{H}_{1}$$H_{1}$$I^{1}$$U:\mathcal{H}_{1}arrow \mathcal{H}$$\mathcal{H}_{1}$$\eta_{1}\in \mathcal{H}_{1}$$\sigma_{1}=(\mathrm{I}$$\backslash$$1(B_{\backslash }7\iota)$$=\eta_{1}(n)^{\dagger}$$B$$=\mathcal{L}(H)$$B^{1}\subseteq \mathcal{L}(\mathrm{H})$$\sigma_{1}$$\sigma=\sigma_{1}\Lambda$$B^{1}$$\Lambda$$\nu_{1}$$B$$\langle B, u^{\uparrow}u\rangle_{1}=\nu_{1}(\tilde{u}^{\uparrow}B\tilde{u})$78
VIACHESLAV P B ELAVKIN. Then the input-0utput state transformations are described byof
and:tlle transposed map$B^{1}$$B_{\mathrm{T}}^{1}=\overline{B_{*}^{1}}$$\Lambda^{\mathrm{T}}$$B_{\mathrm{T}}^{1}arrow B_{\mathrm{T}}$(31),$\langle B, \Lambda^{\mathrm{T}}(\sigma_{1})\rangle=(\mathrm{A}$,$\sigma_{1}\rangle_{1}$$B\in B$,$\sigma_{1}\in B_{\mathrm{T}}^{1}$for any input normal state
defining the output density operators. Without loss of generality the input algebra
can be assunied being the smallest decomposable algebra generated by the rangeofis Abelian if
the channel map
consists of only commuting operators on$\sigma=\Lambda^{\mathrm{T}}(\sigma_{1})$$B^{1}$$\sigma_{1}(B)=\langle B, \sigma_{1}\rangle_{1}$$\Lambda(B)$$\Lambda(B^{1}$$\Lambda(B)$$\gamma\{_{1})$, including encodings of the:
The input generalized entanglements, will be defined by the couplings
state with the density:. Here :asis anormal TCP map defining the state
by $\kappa^{-}(A)=J\kappa$, andof aprobe system $(A, \mu)$ which is entangled tois defined as usually by
the adjoint map$\varpi^{1}$$Aarrow B_{\mathrm{T}}^{1}$$\sigma_{1}=\varpi^{1}(I)$$\sigma_{1}$$\varpi^{1}=\kappa^{-}$$\kappa$$\kappa^{*}$$Aarrow B_{*}^{1}$$B^{1}arrow A_{*}$$\rho=\nu_{1}\circ\kappa$$(A\dagger)J$$(B^{1}, \sigma_{1})$$\kappa^{*}$$\langle A|\kappa^{*}(B)\rangle_{\mu}=\omega_{1}(A^{\uparrow}\mathrm{t}\triangleleft B)=\langle\kappa(A)|B\rangle_{1}$$\forall A\in A$:, $B\in B_{1}$ ,.
is the corresponding compound state on A
where
These (generalized) entanglements describe the quantum-quantum correspondences (q-, c-, or -encodings) of the probe systems $(A, \rho)$ with the density oper, to the input
stateof the channel . In particular, the mostinformative standard input entanglement:
is the entanglement of
the transposed input systemcorresponding to the TCP map$\omega$$\omega_{1}$$B^{1}$$0$$(I^{1} )$$\rho=\kappa^{\mathrm{T}}$$(B^{1},\sigma_{1})$$\Lambda$$\overline{B^{1}}arrow B_{\mathrm{T}}^{1}$$\varpi_{q}^{1}$$(A^{0}, \rho_{0})=(\overline{B^{1}},\tilde{\sigma_{1}})$. In the case of discrete decomposabl
with the density operator
this extreme input -encoding defines the$A^{0}=\overline{B^{1}}=B^{1}$$h_{l}.l(A)=J\sigma_{1}^{1/2}A^{\uparrow}\sigma_{1}^{1/2}J$$\mathrm{e}$$\sigma_{1}=(|)_{i}\sigma_{1}(i)$$\mathrm{q}$following density operator(31)$\omega_{q}=(\mathrm{I}\mathrm{t}\aleph \Lambda^{\mathrm{T}})(\omega_{q1})$,$\omega_{q1}=(|)_{i}|\sigma_{1}(i)^{1/2})(\sigma_{1}(i)^{1/2}|$$B=B^{1}\omega B$of the input-0utput compound stateon.
other extreme case of the generalized input entanglements, the pure cencodings corresponding to (12), are less informative then the pure d-encodings
given by the decompositions
with pure states $\sigma_{1}(B, n)=$on . They define the density operators$A^{0}\omega$$\omega_{q1}\Lambda$$\mathrm{T}11\mathrm{e}$$\varpi_{d}^{1}=\kappa_{d}$$\kappa_{d}^{*}=\sum|n\rangle$$\eta(n)^{\uparrow}B\eta(n)$(32)$n|\sigma_{1}(n)$$B_{1}$,$\omega_{d}=(\mathrm{I}\omega \Lambda^{\mathrm{T}})(\omega_{d1})$)
compound stateof thecompound states$B^{1}\alpha$$\langle$$B$$\eta_{1}(n)\eta_{1}(n)^{\uparrow}$$\omega_{d1}=\sum_{n}|n\rangle\langle n|\omega$$\omega_{d1}\Lambda=\omega_{d1}\circ$(I 00

in the case$\omega_{o}=\omega_{\mathit{0}1}\Lambda[10]$$\sigma_{1}(n)=\eta_{1}^{o}(n)\eta_{1}^{o}(n)^{\uparrow}$,$\Lambda$,). They are known as the Ohya$\eta_{1}^{o}(n)^{\dagger}\eta_{1}^{o}(m)=p_{1}(n)\delta_{n}^{m}$,density operatorsof orthogonality ofnormalized to the eigen-valuesof . The -compound states are achieved by pure encodings
described
by the couplings
with corresponding to . The input-0utput
density operator$\sigma_{1}(n)$$\mathrm{t}\mathrm{l}\mathrm{l}\mathrm{e}$$\sigma_{1}$$0$$\kappa_{o}=\sum|7l\rangle$(33)$p_{\rceil}(n)$$\varpi_{d}^{1}=\kappa_{\overline{o}}$$\mathrm{o}$$\langle$$n|\sigma_{1}^{o}(n)$$\omega_{o}=(\mathrm{I}\ltimes\}\Lambda^{\mathrm{T}})\omega_{\mathit{0}1}$,$\sigma_{1}^{o}$$\eta_{1}^{o}$$\eta_{1}^{o}(n)\eta_{1}^{o}(n)^{\dagger}$$\omega_{\mathit{0}1}=\sum_{n}|n\rangle\langle n|\mathrm{W}$79
QUANTUM SEX AND MUTUAL INFORMATIONof the output
is achieved by the couplingof the Ohya compound stateas the composition of
to the extreme probe systemand the channel .
is anormal completely positive unital map
If$\lambda=\kappa^{*}\Lambda$$\omega_{o}$$(A^{0}, \rho_{0})=(B^{1}, \sigma_{1})$$(B, \backslash )$$\kappa^{*}$$\Lambda$$\mathrm{K}:Aarrow A^{0}$$\mathrm{K}(A)=\mathrm{H}_{F_{-}}\overline{X}A\overline{X}^{\uparrow}$where $X$ is abounded operator$h$.

$=\mathrm{K}\mathrm{o}\mathrm{K}$, vr$F_{-}\omega \mathcal{G}0arrow \mathcal{G}$,$A\in A$with,$\mathrm{b}_{F_{-}}X\dagger X=I^{0}$, the compositions. describe the entanglements of the probe system $(A, \rho)$ to the$=\Lambda^{*}\kappa$channel input
is given by$(B^{1}, \sigma_{1})$and the output$(\mathrm{S}, \sigma)$

via this channel respectively. The state$\llcorner 0=\rho_{0}\mathrm{K}$$\mathrm{K}^{\mathrm{T}}(\rho_{0})=X(I^{-}\infty \rho_{0})X^{\dagger}\in A_{*}$

for each density operator
resulting entanglement$A$with$\omega$is the identity operator in
defines the compound state$\rho_{0}\in A_{*}^{0}$$\pi=\lambda^{*}\mathrm{K}$, where$I^{-}$$\omega$$=\omega 01\circ$$\mathcal{F}_{-}$

. The
on

$(\mathrm{K}\infty \Lambda)$$B$$\omega_{01}$$(A^{0}\omega B^{1})=\mathrm{b}\tilde{A}^{0}\kappa_{0}^{*}(B^{1})=\mathrm{T}\mathrm{r}$$\tilde{v}_{01}^{\dagger}(A^{0}\mathrm{c}\triangleleft B^{1})\tilde{v}_{01}$$H_{1}arrow F_{01}$ is the amplitude operator uniquely defined:
operator
densitycompoundIvy tlte input
uP to aunitary operatoron$A^{0}\omega$$B^{1}$. Here$v_{01}$$\mathcal{G}_{0}\omega$$\omega_{01}\in A_{\mathrm{T}}^{0}\omega$$B_{\mathrm{T}}^{1}$. The effect of the input entanglement and the output channelonbe written in terms of the amplitude operator of the state as$U^{0}$$\kappa$$\mathcal{F}_{01}$$\Lambda$

can$\omega$$v=(X\alpha)Y)(I^{-}\infty v_{01}\mathrm{C}\triangleleft I^{+})U$

to aunitary operator $U$ in $F$
the input-0utput compound state
$n1)$

(34)

$=\mathcal{F}_{-}$$\omega$C4$F_{+}$$F_{01}\mathrm{C}\mathrm{A}$is given by$\omega_{01}$. Thus the density operator ofwith the density$(\mathrm{K}\mathrm{t}\triangleleft \Lambda)$$(\mathrm{K}\triangleright 0 \Lambda)^{*}(\omega_{01})=(X\mathrm{c}\triangleleft Y)\omega_{01}(X\infty Y)^{\dagger}$

,

.
where
with any probe algebra $A$
be the set of all normal TCP maps :
Let
.
.
with
$(I)=1$ , andbe the subset of all
normalized as.
:
defines the standard, wherecan be decomposed as
Each$\omega 01$$=v_{01}v_{01}^{\lceil}.$$\mathrm{T}\mathrm{r}\kappa$$h$$Aarrow B_{*}^{1}$$\kappa$$\mathcal{K}_{q}^{1}$$A^{0}arrow B^{1}$$\kappa_{q}\mathrm{K}$$\in \mathcal{K}_{q}^{1}$$\kappa(I)=\sigma_{1}$$\kappa\in \mathcal{K}_{q}^{1}$$\mathcal{K}_{q}(\sigma_{1})$$\kappa_{q}$.
, andis anormal unital CP map
input entanglementbythecombinations
maps
described
set
be
the
of
all
CP-TCP
Further let$Aarrow\overline{B^{1}}$$\mathrm{K}$$\varpi_{q}^{1}=\kappa_{\overline{q}}$$\mathcal{K}_{c}^{1}$$\kappa$(35)$\kappa$$(A)= \sum_{n}\rho_{n}(A)\sigma_{1}(n)$of the primitive maps $A\mapsto\rho_{n}(A)\sigma_{1}(n)$ , andentanglements , . the decompositions$\kappa$$\mathcal{K}_{d}^{1}$be the subset of the diagonalizing$\mathrm{i}.\mathrm{e}$(36)$\kappa(A)=\sum_{n}\langle n|A|n\rangle\sigma_{1}(n)$.

denote the subsets corresponding to afixedandAs in the first case$(I)=\sigma_{1}$ .
, wherecan be represented as the composition
Eachfor aproperofdescribes apure -encoding.normalized to
choice of the CP map : $Aarrow B^{1}$ .
) be the subset of all decompositions (36) with(andFurthermore let):(and fixedorthogonal$\mathcal{K}_{c}(\sigma_{1})$$\mathcal{K}_{d}(\sigma_{1})$$\mathcal{K}_{c}(\sigma_{1})$$t\backslash \cdot$$\kappa$$\mathrm{d}$$h_{d}$$\sigma_{1}$$\varpi_{d}^{1}=\kappa_{d}$$\mathrm{K}$$\mathcal{K}_{o}^{1}$$\sigma_{1}(n)$$\mathcal{K}_{o}(\sigma_{1})$$\sum_{n}\sigma_{1}(n)=\sigma_{1}$$\sigma_{1}(?n)\sigma_{1}(n)=0$, $m\neq n$ .

$=\kappa_{d}\mathrm{K}$$(B^{1}, \sigma_{1})$80
VIACHESLAV P BELAVKIN

describing the pure
can also be represented as, withEach.
ofoencodingquantum channel
Now, let us maximize the entangled mutual entropy for a$B^{1}=B^{1}$) by means of the(and afixed input stateon the decomposable(23)
was defined in the.
mutualinformation
entanglements
The
types
above four
previous section by tlle density operators of the corresponding compound stateof the marginals , for . In each caseand the product-stateon$\kappa\in \mathcal{K}_{o}(\sigma_{1})$$\kappa$$=\mathrm{K}\mathrm{O}\mathrm{K}$$\kappa_{o}$$(B^{1}, \sigma_{1})=(A^{0}, \rho_{0})$$\varpi_{o}^{1}=\kappa_{\overline{O}}$$\mathrm{g}\mathrm{i}\underline{\mathrm{v}\mathrm{e}}\mathrm{n}$$\Lambda$$\sigma_{1}$$\kappa$$\omega$$A\ltimes lB$$\varphi=\rho\omega$$\omega$$=\omega_{01}(\mathrm{K}\triangleright)\Lambda)$,$\sigma$$\omega$$\sigma$$\rho$,$\varphi=\varphi_{01}(\mathrm{K}\omega \Lambda)$

is one of the corresponding extreme
is aCP map $Aarrow A^{0}=B^{1}$ ,
where
, and
. The density,,
oncompound states
can be written as(\^uoi) is written in (34), andoperator$\mathrm{K}$$\omega_{01}$$\omega$$B^{1}\omega$ $B^{1}$$\omega_{c1}=\mathrm{W}\mathrm{d}\mathrm{i}$$\omega_{q1}$$\phi=\rho\omega$$=(\mathrm{K}(\cross|\Lambda)^{\mathrm{T}}$$\phi=\kappa^{\mathrm{T}}(I)\mathrm{W}$where$\varphi_{01}=\rho_{0}\omega$$\omega_{\mathit{0}1}$$\lambda^{\mathrm{T}}=\Lambda^{\mathrm{T}}\pi_{1}^{0}$$\lambda^{\mathrm{T}}(I)$$\sigma_{1}$$\sigma$,. This proves the following proposition.

Proposition 5.1. The entangled mutual informations achieve the following maximal values(37)

1$\sup$$(\kappa^{*}\Lambda)=1_{q}(\sigma_{1}, \Lambda):=1$,$(\kappa_{q}^{*}\Lambda)$$\kappa\in \mathcal{K}.,(\sigma_{1})$$(\kappa_{d}^{*}\Lambda)\equiv 1_{d}(\sigma_{1}, \Lambda)$$( \kappa^{*}\Lambda)=\sup_{\kappa_{l}}.1$$1_{c}( \sigma_{1}, \Lambda):=\sup_{\kappa\in \mathcal{K}..(\sigma_{1})}1$(38)$. \sup_{\kappa\in \mathcal{K}||(\sigma_{1})}1$.1$(\kappa_{o}^{*}\Lambda)$$( \kappa^{*}\Lambda)=1_{o}(\sigma_{1}, \Lambda):=\sup_{\kappa}‘$

where . are the corresponding extremal input couplings
They are ordered as$\kappa$(39),$A^{0}arrow B_{*}^{1}$$1_{q}(\sigma_{1}, \Lambda)\geq 1_{c}(\sigma_{1},\Lambda)=1_{d}(\sigma_{1}, \Lambda)\geq 1_{o}(\sigma_{1}, \Lambda)$

In the following definition the maximal informations.
simply denoted as

,

with

.

$\mu\circ\kappa^{*}$$=\sigma_{1}$.

.

$1_{c}(\sigma_{1}, \Lambda)=1_{d}(\sigma_{1}, \Lambda)$

is$1_{1}(\sigma_{1}, \Lambda)$

Definition 5.1. The suprema$\mathrm{C}_{q}(\Lambda)=\sup 1$$( \kappa^{*}\Lambda)=\sup_{\sigma_{1}}1_{q}(\sigma_{1}, \Lambda)$,$\kappa\in \mathcal{K}^{1}‘$

’

(40)

Sllp I:$( \kappa^{*}\Lambda)=\mathrm{C}_{1}(\Lambda):=\sup_{\sigma_{1}}1_{1}(\sigma_{1}, \Lambda)$$\kappa.\in \mathcal{K}_{l}^{\mathrm{I}}$‘$\mathrm{C}_{o}(\Lambda)=.\sup_{\kappa\in \mathcal{K}_{1}’}.1$$( \kappa^{*}\Lambda)=\sup_{\sigma_{1}}1_{o}(\sigma_{1},\Lambda)$,

are called the q-, c- or d-, and -capacities respectively for the quantum channel.
defined by a normal unital $CP$ map$0$$\mathrm{A}:Barrow B^{1}$

Obviously, the capacities (40) satisfy the inequalities$\mathrm{C}_{o}(\Lambda)\leq \mathrm{C}_{1}(\Lambda)\leq \mathrm{C}_{q}(\Lambda)$.

81
ANTUM SEX AND MUTUAL lNFt)R M ATI$\mathrm{Q}\mathrm{L}^{\gamma}$

Theorem 5.2. Let $\Lambda(B)=U\dagger BU$ be a unital $CP$ map
quantum deterministic channel. Then
$1_{1}(\sigma_{1}, \Lambda)=\mathrm{I}_{o}(\sigma_{1}, \Lambda)=\mathrm{S}(\sigma_{1})$where$\mathrm{S}_{q}(\sigma_{1})=\mathrm{H}(\sigma_{1})$,$()\mathrm{N}$$B$$arrow B^{1}$$1_{q}(\sigma_{1}, \Lambda)=\mathrm{S}_{q}(\sigma_{1})$,$\mathrm{C}_{q}(\Lambda)=\ln\dim B^{1}$.

describing$a$, and thus in this case$\mathrm{C}_{1}(\Lambda)=\mathrm{C}_{o}(\Lambda)=\ln$rank$B^{1}$,

Proo . It was proved in the previous section for the case of the identity channel, and thus it is also valid for any isomorphism
describing thestate transformations:
by aunitary operator. In the case of
non-unitary $Y$ we can use the identity$.f\cdot$$\mathrm{A}:B\mapsto U^{\uparrow}BU$$\Lambda=\mathrm{I}$$\sigma\mapsto Y\sigma Y^{\uparrow}$$\Lambda^{\mathrm{T}}$

Tr

$Y$

$S(\sigma_{1}\mu I^{+})\ln S(\sigma_{1}\alpha)I^{+})$$(\sigma_{1} (n I^{+})Y^{\uparrow}\ln Y(\sigma_{1}\alpha)I^{+})Y^{\uparrow}=\mathrm{T}\mathrm{t}$Due to thiswhere $S=Y\dagger Y$ .
$\mathrm{S}$$U=\overline{Y}$$\mathrm{S}(\sigma_{1}\Lambda)=$-Tr$S(\sigma_{1}\infty I^{+})\ln S(\sigma_{1}\mathrm{C}\triangleleft I^{+})$

,

, and

$(\omega_{01}(\mathrm{K}\mathrm{x}) \Lambda))=$-Tr $(R\ltimes)S)(I^{-}\omega \omega_{01}\omega I^{+})\ln(R\omega S)(I^{-}\omega \omega_{01}\omega I^{+})$ ,
where
, and

$R=X^{\mathrm{J}}|X$. Tlu$\mathrm{S}$$\iota \mathrm{s}$(ci)$=\mathrm{S}$(ci)$\mathrm{S}(\omega_{01} (\mathrm{K}\omega \Lambda))=\mathrm{S}$(ci)$(\mathrm{K}\triangleright\iota \mathrm{I}))$

if $Y\dagger Y=$

$I$

1
.

$((\pi_{1}\Lambda))$$=$$\mathrm{S}(\rho_{0}\mathrm{K})+\mathrm{S}(\sigma_{1})-\mathrm{S}(\omega_{01}(\mathrm{K}\omega \mathrm{I}))$$\leq$$\mathrm{S}(\rho_{0})+\mathrm{S}(\sigma_{1})-\mathrm{S}(\omega_{01})=1$$(\omega_{01})$with any normal unital CP map: $Aarrow A^{0}$ and acompound
stateon. The supremum (37), which is achieved at the standard
entanglement, corresponding to, coincides with -entropy, andtlte supremum (??), coinciding with, is achieved for apure 0-entanglement,
corresponding togiven by any Schatten decomposition for . Moreover,the entropy
is also achieved by any pure entanglement, corresponding togiven by any extreme decomposition for, and thus is the maximal
mutual informationin the case of deterministic . Thus the capacityof the deterministic channel is given by the maximumof theNeumann entropy , and the capacity
is equal.
$\mathrm{f}_{\circ 1}\kappa$$=h_{0}.\mathrm{K}$$\mathrm{K}$$A^{0}\omega$$\omega_{01}$$B^{1}$$\omega_{01}=\omega_{q1}$$\mathrm{H}(\sigma_{1})$$\mathrm{q}$$\mathrm{S}(\sigma_{1})$$\omega_{01}=\omega_{\mathit{0}1}$$\sigma_{1}$$\mathrm{H}(\sigma_{1})$$\mathrm{d}$$\omega_{01}=\omega_{d1}$$\sigma_{1}$$1_{1}(\sigma_{1}, \Lambda)$$\Lambda$$\mathrm{C}_{o}=\ln\dim 7\{_{1}$$\mathrm{C}_{[perp]}(\Lambda)$$\mathrm{C}_{B^{1}}=\ln\dim B^{1}$$\mathrm{S}$$\backslash \gamma \mathrm{o}\mathrm{n}$$\mathrm{C}_{q}(\Lambda)$$\mathrm{q}$$\mathrm{I}$In tlle general case, -entanglements can be more informative than 0-entanglements
as can be shown by an example of aquantum noisy channel for which$\mathrm{d}$$\mathrm{I}_{1}(\sigma_{1}, \Lambda)>1_{o}(\sigma_{1}, \Lambda)$,$\mathrm{C}_{1}(\Lambda)>\mathrm{C}_{o}(\Lambda)$.

The last equalities of the above theorem will be related to the work on entropy by
Voiculescu [26],

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[10]$\mathrm{R}\dot{‘}\iota$($1\mathrm{i}_{1}$$\mathrm{F}()\prime 111(l$$[()\backslash ]$$\mathrm{T}11‘.{}^{\mathrm{t}}(’ 1\mathrm{y},$$\mathrm{N}\prime 111()$$\mathrm{C}\mathrm{i}_{111(11}.\mathrm{t}\mathrm{t})$$” \mathrm{C}\iota$$\dot{c}\mathrm{t}11$$\prime 1111i\iota 1$$j11^{)}1)1.i1$$1111)(’ 11\downarrow(1$$\mathrm{M}i\iota \mathrm{t}111.11\downarrow j\iota\dagger \mathrm{i}1^{\cdot}\aleph\dot{e}11\downarrow(1$$11\downarrow \mathrm{f}(’ 111\mathrm{l}\mathrm{i}\iota \mathrm{t}\mathrm{i}\mathrm{t})11$$111\downarrow(1\mathrm{S}\mathrm{t}\mathrm{i}\iota \mathrm{t}\mathrm{t}!\mathrm{s}$$‘.\mathfrak{l}1\downarrow \mathrm{t}1$$\iota\iota \mathrm{u}\mathrm{m}$$\mathfrak{l}$$\mathrm{A}1‘.\iota 111()_{1}\backslash \mathrm{A}_{1\mathrm{C}}\cdot 1\downarrow \mathrm{i}\mathrm{v}(., \mathrm{t})\iota i111\mathrm{t}- \mathrm{P}1\downarrow/(](\}(]4\mathrm{t}\mathfrak{l}6^{(}\mathrm{J}, 1)])$$\mathrm{T}11\mathrm{t}^{\backslash }\mathrm{t})1^{\cdot}\mathrm{v}$$\mathrm{M}:\iota \mathrm{t}\mathrm{l}\mathrm{l}\mathrm{l}\cdot 111:\iota \mathrm{t}\mathrm{i}\mathrm{t}$$\mathrm{N}(’ 1\mathrm{l}\mathrm{t}\cdot \mathrm{t}’ 111111’\iota \mathrm{t}\iota \mathrm{t}.\mathrm{i}\mathrm{v}$$\mathrm{C}j\iota 111\mathrm{t}^{\backslash }.\mathrm{r}\mathrm{i}11\mathfrak{l}$$\mathrm{w}i\iota \mathrm{t},\dot{e}111i11$$\mathrm{t}^{\mathrm{J}}..$$\mathrm{A}111\mathfrak{l}!\mathrm{r}$$\mathrm{P}\mathrm{l}\iota \mathrm{y}\mathrm{s}$$\mathrm{M}‘.\iota \mathrm{t}_{1}11$$i11\downarrow(1.\mathrm{I}$$\mathrm{P}1_{1}\mathrm{y}_{8}.$$\mathrm{C}(’ 1111\mathrm{U}$$\mathrm{P}’\iota \mathfrak{l},\mathrm{l}\mathrm{i}\mathrm{t}.:\iota \mathrm{t}\mathrm{i}$$8\mathrm{t}\mathfrak{l}^{(}\mathrm{J}8^{\cdot}\mathrm{d}.\mathrm{d}$$\mathrm{U}11\mathrm{i}\mathrm{v}\mathrm{c}^{\backslash }.\mathrm{r}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{t}.\mathrm{y}$$()11.\mathrm{v}i\mathrm{t}$$’\iota:\iota 11\mathrm{t}_{1}\prime 1111$$()1\downarrow \mathrm{y}\dot{.}\iota$$\mathrm{U}1111.\ltimes’:\iota \mathrm{k}\mathrm{i}$$\mathrm{U}1111\downarrow\iota.\cdot\iota 1111$$_{()}\mathrm{i}\mathrm{t}.’\iota 1\mathrm{t}.t\dagger$$\mathrm{D}\mathrm{E}\mathrm{P}\mathrm{A}\mathrm{R}\mathrm{T}\mathrm{A}\backslash \mathrm{I}\mathrm{E}\mathrm{N}\mathrm{T}$$\mathrm{I}\dot{\mathrm{t}}()$$1:\iota \mathrm{i}$$\mathrm{I}\dagger \mathrm{t}:1$$1^{(}\mathrm{J}\mathrm{b}.2$$\mathrm{C}\{’ 111111\prime 111$$\mathrm{h}4\dot{‘}\iota \mathrm{t}\mathrm{l}\mathrm{l}$$1)24^{(}\mathrm{J}- 281$

OF MATHEMATICS, UNIVERSITY OF NOTTINCHAM, NG72RD NOTTINCHA$\Lambda 1$. UK