arXiv:quant-ph/0602073v1 7 Feb 2006 fti yewsue n[]i osdrn rpriso nnt dimens infinite stat of separable properties of considering set the in of [5] and in channels used th entanglement-breaking a was example, to For type channels this systems. such of and with channels dealing dimensional inte an of infinite ch as advantages general CE-channels dimensional technical of use infinite class the and of consider one may features we essential So, time 2). (example same (proposit the channels dimensional at finite vealing chann of These properties analytical CE-channels. some called entropy, output continuous with iino t xsec a band nti ae ecnie spec a consider infi we unconstrained paper for measure this finite optimal In with an channels of mensional obtained. existence was of existence condition sufficie its the and of introduced was dition channels dimensional infinite strained vrg tts(oolr 1). (corollary states average uptotmlaeaesae(hoe ) ti hw hteach that shown is It 1). (theorem finite state with average optimal output iesoa uptsse,hvn close having system, output dimensional χ oeocpct i htflosthe follows finitene what by nontrivial (in characterized a capacity channels, consider Holevo We su quantum quite dimensional systems. differ quantum infinite channels finite of quantum and of infinite properties for the tially known, well is it As Introduction 1 cpct uha opcns fteotu e n xsec o existence and set output the of compactness as such -capacity ∗ n[]tento fa pia esr gnrlzdesml)frc for ensemble) (generalized measure optimal an of notion the [4] In h ls fcanl ihfinite with channels of class The h eut n[2 ml oegnrlpoete fcanl ihfinit with channels of properties general some imply [12] in results The tko ahmtclIsiue 191Mso,Russia Moscow, 119991 Institute, Mathematical Steklov ncanl ihfiieHlv capacity. Holevo finite with channels On χ cpct a euiomyapoiae ycanl ihfinite with channels by approximated uniformly be can -capacity M.E.Shirokov χ cpct term2 oolr ,rmr 1) remark 2, corollary 2, (theorem -capacity χ χ 1 cpct otissbls fchannels of subclass contains -capacity -capacity). χ cpcte n h uptoptimal output the and -capacities ∗ es. h unique the f l inherit els o ) re- 1), ion channel e rmediate so the of ss pproach channel iedi- nite tcon- nt annels bstan- ional class on- ial e and construct examples of channel with bounded output entropy, for which there exist no optimal measures (examples 1C and 2). An interesting question is a problem of extension of a quantum channel from the set of all quantum states (=normal states) on B( ) to the set of all states (=positive normalized linear functionals) on B( ), whichH is a compact H set in the -weak topology. By using a simple topological observation we obtain a condition∗ of existence of an extension of a channel to nonnormal states (proposition 3), which implies existence of such extension for arbitrary channel with finite χ-capacity (corollary 3). The entropic characteristics of a channel extension are considered as well as the condition of existence of an optimal measure (proposition 4), which shows the meaning of the conditions in theorem 2 (remark 2). We complete this paper by considering the class of infinite dimensional channel, for which the χ-capacity can be explicitly determined and there exists a simple condition of continuity of the output entropy (proposition 5). The CE-channels of this class demonstrate essential infinite dimensional features such as nonexistence of an input optimal average state and of an optimal measure, discontinuity of the χ-capacity as a function of a channel (example 2).
2 Preliminaries
Let be a separable Hilbert space, B( ) - the Banach space of all bounded H H operators in with the cone B+( ) of all positive operators, T( ) - the Banach spaceH of all trace-class operatorsH with the trace norm andH S( ) k·k1 H - the closed convex subset of T( ) consisting of all density operators in , which is complete separable metricH space with the metric defined by the traceH norm. Each density operator uniquely defines a normal state on B( ) [2]. H We will also consider the set S( ) of all normalized positive functionals on B( ), which is a compact subsetH of (B( ))∗ in the -weak topology. The H H ∗ set S( ) can be considered asb a -weak dense subset of S( ) [2]. Since the H ∗ H main attention in this paper is focused on S( ) rather then S( ) we will H H use the term state for elements of S( ) while the elementsb of S( ) S( ) will be called nonnormal states. H b H \ H Let , ′ be a pair of separable Hilbert spaces which we shallb call corre- H H spondingly input and output space. A channel Φ is a linear positive trace pre- serving map from T( ) to T( ′) such that the dual map Φ∗ : B( ′) B( ) H H H 7→ H 2 (which exists since Φ is bounded [3]) is completely positive. For arbitrary set let co( ) and co( ) be the convex hull and the convex A A A closure of the set correspondingly and let Ext( ) be the set of all extreme points of the set A [8]. A We will use theA following compactness criterion for subsets of states: a closed subset of states is compact if and only if for any ε > 0 there is a finite dimensionalK projector P such that TrρP 1 ε for all ρ [9],[4]. ≥ − ∈K Speaking about continuity of a particular function on some set of states we mean continuity of the restriction of this function to this set. Arbitrary finite collection ρ of states in S( ) with corresponding set { i} H of probabilities π is called ensemble and is denoted by π , ρ . The state { i} { i i} ρ¯ = i πiρi is called the average state of the ensemble. Following [4] we treat an arbitrary Borel probability measure µ on S( ) as generalized ensemble andP the barycenter of the measure µ defined by theH Pettis integral
ρ¯(µ)= ρµ(dρ)
S(ZH) as the average state of this ensemble. In this notations the conventional ensembles correspond to measures with finite support. For arbitrary closed subset of S( ) we denote by ( ) the set of all probability measures supportedA by theH set [14]. In whatP A follows an arbitrary ensemble π , ρ A { i i} is considered as a particular case of probability measure and is also denoted by µ. Consider the functionals
χ (µ)= H(Φ(ρ) Φ(¯ρ(µ)))µ(dρ) and Hˆ (µ)= H(Φ(ρ))µ(dρ). Φ k Φ Z Z In [4] (proposition 1 and the proof of the theorem) it is shown that both these functionals are lower semicontinuous on (S( )) = and P H P χ (µ)= H(Φ(¯ρ(µ))) Hˆ (µ) (1) Φ − Φ for arbitrary µ such that H(Φ(¯ρ(µ))) < + . If µ = π , ρ then ∞ { i i} n n χ ( π , ρ )= π H(Φ(ρ ) Φ(¯ρ)) and Hˆ ( π , ρ )= π H(Φ(ρ )). Φ { i i} i i k Φ { i i} i i i=1 i=1 X X 3 In what follows we will use the decrease coefficient dc(σ) of a state σ defined in [12] by dc(σ) = inf λ> 0 Trσλ < + [0, 1]. { | ∞} ∈ We will use the notion of the H-convergence of a sequence of states ρn to a state ρ defined by the condition lim H(ρ ρ ) = 0. { } 0 n→+∞ nk 0 In [12] the properties of the χ-capacity as a function of a set of states were explored and the notion of the optimal average state Ω( ) of a set with A A finite χ-capacity was introduced. The set of states with finite χ-capacity is called regular if one of the two following conditionsA hold: H(Ω( )) is finite and lim H(ρ ) = H(Ω( )) for arbitrary se- • A n→+∞ n A quence ρ of states in co( ) H-converging to the state Ω( ); { n} A A the function ρ H(ρ Ω( )) is continuous on the set . • 7→ k A A In a sense these conditions are the minimal continuity requirements which guarantee the ”good” properties of the χ-capacity (see theorems 2 and 3 in [12]). The simplest sufficient condition of regularity of a set is given by the inequality dc(Ω( )) < 1 (theorem 2E in [12]), but thisA condition is A not necessary and there exist regular sets, consisting of states with infinite entropy. The all notations used in this paper coincide with the notations accepted in [12].
3 General properties
Let Φ : S( ) S( ′) be a channel with finite χ-capacity defined by H 7→ H C¯(Φ) = sup χ ( π , ρ ) = sup π H(Φ(ρ ) Φ(¯ρ)). (2) Φ { i i} i i k {πi,ρi} {πi,ρi} i X In [4] it is shown that
C¯(Φ) = sup χΦ(µ) = sup H(Φ(ρ) Φ(ρ(µ))µ(dρ). (3) µ∈P µ∈P k S(ZH) According to [10] a sequence of ensembles πn, ρn such that {{ i i }}n n n ¯ lim πi H(Φ(ρi ) Φ(¯ρn)) = C(Φ) n→+∞ k i X 4 is called approximating sequence for the channel Φ while any partial limit of the corresponding sequence of the average states ρ¯ = πnρn is called { n i i i } input optimal average state for the channel Φ. By the compactness argument for a finite dimensional channelP there exists at least one input optimal average state, coinciding with the average state of the optimal ensemble for this channel [13]. For an infinite dimensional channel with finite χ-capacity existence of an input optimal average state is a question depending on a channel (example 1), which is closely related to the question of existence of an optimal measure for this channel (theorem 2, corollary 2). By the definitions the χ-capacity of a channel Φ coincides with the χ- capacity of the output set Φ(S( )) of this channel. Thus theorems 1 and H 2D in [12] imply the following observation. Theorem 1. Let Φ: S( ) S( ′) be a channel with finite χ-capacity. The set Φ(S( )) is a relativelyH 7→ compactH subset of S( ′). H H There exists the unique state Ω(Φ) in S( ′) such that H H(Φ(ρ) Ω(Φ)) C¯(Φ), ρ S( ). k ≤ ∀ ∈ H The state Ω(Φ) lies in Φ(S( )). For arbitrary approximating sequence n n H of ensembles πi , ρi n the corresponding sequence Φ(¯ρn) n of images of their average{{ states H}}-converges to the state Ω(Φ). { } If there exists an input optimal average state ρ∗ for the channel Φ then Φ(ρ∗) = Ω(Φ). The χ-capacity of the channel Φ can be defined by the expression1
C¯(Φ) = inf sup H(Φ(ρ) σ) = sup H(Φ(ρ) Ω(Φ)). S ′ σ∈ (H ) ρ∈S(H) k ρ∈S(H) k
According to [10] the state Ω(Φ) is called the output optimal average state for the channel Φ. Below we consider examples of channels with finite χ-capacity with no input optimal average state, for which the output optimal average states are explicitly determined and play important role in studying of these channels. 1By corollary 9 in [12] the infinum in this expression can be over the subset of Φ(S( )) consisting of states invariant for all automorphism α of S( ′) such that α(Φ(S( )))H Φ(S( )). H H ⊆ H
5 For a finite dimensional channel Φ the state Ω(Φ) is an image of the average state of any optimal ensemble for this channel, which implies
Ω(Φ) Φ(S( )) and C¯(Φ) H(Ω(Φ)). ∈ H ≤ For an infinite dimensional channel Φ with finite χ-capacity these relations do not hold in general (see example 1C below). The first relation follows from existence of at least one input optimal average state for the channel Φ while a sufficient condition of the second one can be expressed in terms of the spectrum of the state Ω(Φ) (see proposition 2 below). The simplest examples of infinite dimensional channels with finite χ- capacity are channels from an infinite dimensional quantum system into a fi- nite dimensional one. Following [10] such channels will be called IF-channels. Theorem 1 implies, in particular, that each channel with finite χ-capacity can be uniformly approximated by IF-channels. Corollary 1. The channel Φ: S( ) S( ′) has finite χ-capacity H 7→ H ′ ′ ′ if and only if there exists a sequence Φn : S( ) S( n), n of IF-channels such that { H 7→ H H ⊆ H }
lim sup Φn(ρ) Φ(ρ) 1 =0 and sup C¯(Φn) < + . n→+∞ ρ∈S(H) k − k n ∞
The sequence Φ can be chosen in such a way that { n}
lim C¯(Φn)= C¯(Φ) and lim Ω(Φn) = Ω(Φ). n→+∞ n→+∞
Proof. If the above sequence Φn exists then the χ-capacity of the channel Φ is finite due to lower semicontinuity{ } of the χ-capacity as a function of a channel [10]. If the χ-capacity of the channel Φ is finite then by theorem 1 the set Φ(S( )) is relatively compact. Let Pn be a sequence of finite rank pro- H ′ { } jectors in strongly converging to I ′ . The compactness criterion implies H H limn→+∞ infρ∈S(H) TrΦ(ρ)Pn = 1. Hence the sequence of IF-channels
Φ (ρ)= P Φ(ρ)P + (Tr(I ′ P )Φ(ρ)) τ n n n H − n n ′ ′′ from S( ) into S(Pn( ) n), where τn is a pure state in some finite dimensionalH subspace H′′ of⊕′ H P ( ′), uniformly converges to the channel Hn H ⊖ n H Φ. Since for each n the channel Φn can be represented as a composition of
6 the channel Φ and the channel ρ PnρPn + (Tr(IH′ Pn)ρ) τn, the mono- tonicity property of the relative entropy7→ implies C¯(Φ−) C¯(Φ) for all n. n ≤ This and lemma 4 in [12] imply the limit expressions in the second part of the corollary. The above corollary shows that the class of channels with finite χ-capacity is sufficiently close to the class of IF-channels. Nevertheless channels of this class demonstrate many features of essential infinite dimensional channels (see examples 1 and 2 below). There exists a class of infinite dimensional channels with finite χ-capacity which is the most close to the class of IF-channels. Definition 1. A channel Φ: S( ) S( ′) is called CE-channel if the restriction of the quantum entropy toH the7→ set Φ(H S( )) is continuous. H The results in [12] provide different sufficient conditions of the CE-property of a channel (see proposition 2 below). In particular, proposition 10 in [12] implies that the class of CE-channels is closed under tensor products: If Φ: S( ) S( ′) and Ψ: S( ) S( ′) are CE-channels then the channelHΦ 7→Ψ: SH( ) S( K′ 7→′) isK a CE-channel. The χ-capacity⊗ CH⊗K¯(Φ) of7→ the channelH ⊗K Φ can be defined as the least upper bound of the χ-function of the channel Φ defined by [4]
χΦ(ρ) = sup πiH(Φ(ρi) Φ(ρ)) = sup H(Φ(σ) Φ(ρ))µ(dσ), k µ∈P k i πiρi=ρ i {ρ} S(ZH) P X where {ρ} is the set of all probability measures on S( ) with the barycenter ρ. P H It was shown in [11] that the convex closure of the output entropy is defined by
HˆΦ(ρ) = inf H(Φ(ρ))µ(dρ) + . µ∈P{ρ} ≤ ∞ S(ZH) In contrast to the χ-function the infinum over all measures in in the P{ρ} definition of the Hˆ -function does not coincide in general with the infinum over all measures in with finite support [11]. P{ρ} The χ-function and the Hˆ -function of an arbitrary channel are nonega- tive lower semicontinuous concave and convex functions correspondingly [11]. The following proposition shows, in particular, that the χ-function and the Hˆ -function of a CE-channel have properties similar to the properties of these
7 functions of a finite dimensional channel. Proposition 1. Let Φ be channel with finite χ-capacity. Then
χ (ρ) C¯(Φ) H(Φ(ρ) Ω(Φ)), ρ S( ). Φ ≤ − k ∀ ∈ H If Φ is a CE-channel then the functions χ and Hˆ are continuous on S( ). Φ Φ H Moreover
χΦ(ρ)= H(Φ(ρ)) HˆΦ(ρ) and HˆΦ(ρ) = inf πiH(Φ(ρi)). − πiρi=ρ i i P X Proof. The inequality for the χ-function follows from corollary 1 in [10]. The continuity assertion and the representations for the χ-function and the Hˆ -function are corollaries of proposition 7 and proposition 5 in [11]. The following proposition shows the special role of the output optimal average state. Proposition 2. Let Φ be a channel with finite χ-capacity. ¯ If dc(Ω(Φ)) < 1 then supρ∈S(H) H(Φ(ρ)) < + and C(Φ) H(Ω(Φ)). If dc(Ω(Φ)) = 0 then Φ is a CE-channel. ∞ ≤ Proof. Suppose dc(Ω(Φ)) < 1. By theorem 2E in [12] the entropy is bounded on the set Φ(S( )). By theorem 1 for arbitrary approximating sequence of ensembles πHn, ρn the corresponding sequence Φ(¯ρ ) of {{ i i }}n { n }n images of their average states H-converges to the state Ω(Φ). By proposition 2 in [12] the condition dc(Ω(Φ)) < 1 implies
¯ n n C(Φ) = lim χΦ( πi , ρi ) lim H(Φ(¯ρn)) = H(Ω(Φ)). n→+∞ { } ≤ n→+∞ Suppose dc(Ω(Φ)) = 0. By theorem 2E in [12] the entropy is continuous on the set Φ(S( )). The assertionsH of the above proposition are illustrated by the below ex- ample 1, in which the family of channels with different decrease coefficient of the output optimal average state is considered. According to [4] a measure µ in (ExtS( )), at which the supremum ∗ P H in definition (3) is achieved, is called optimal measure (optimal generalized ensemble) for the channel Φ. The notion of an optimal measure is a generalization of the notion of an optimal ensemble for a finite dimensional channel. In [13] it is shown that each optimal ensemble is characterized by the maximal distance property.
8 The first part of the following theorem generalizes this observation to the case of infinite dimensional channel. Theorem 2. Let Φ: S( ) S( ′) be a channel with finite χ-capacity. H 7→ H If there exists an optimal measure µ∗ for the channel Φ then its barycenter ρ¯(µ∗) is an input optimal average state for the channel Φ and the following ”maximal distance property” holds
H(Φ(ρ) Ω(Φ)) = C¯(Φ) for µ almost all ρ in S( ). k ∗ − H
If there exists an input optimal average state ρ∗ for the channel Φ and the set Φ(S( )) is regular then there exists an optimal measure µ for the H ∗ channel Φ such that ρ¯(µ∗)= ρ∗. Proof. Since every probability measure can be weakly approximated by a sequence of measures with finite support lower semicontinuity of the functional χΦ implies that the barycenterρ ¯(µ∗) of any optimal measure µ∗ for the channel Φ is an input optimal average state for the channel Φ. The maximal distance property follows from the definition of an optimal measure and theorem 1. n n Suppose the set Φ(S( )) is regular and µn = πi , ρi n is an approxi- mating sequence of ensemblesH for the channel{ Φ such{ that}} the corresponding sequence ρ¯(µ ) converges to the state ρ . By convexity and lower semi- { n }n ∗ continuity of the relative entropy we may assume that each ensemble in this sequence consists of pure states. Since the sequence ρ¯(µ ) is relatively { n }n compact the sequence µn n is relatively weakly compact by proposition 2 in [4] and hence it contains{ } weakly converging subsequence. So, we may as- sume that the sequence µ weakly converges to a particular measure µ { n}n ∗ supported by pure states due to theorem 6.1 in [14]. −1 For given n let νn = µn Φ be the image of the measure µn under the mapping Φ, so that ν =◦ πn, Φ(ρn) . It is easy to see that ν is n { i i } { n}n an approximating sequence of ensembles for the set Φ(S( )). Since this set is regular by the condition the arguments from the proofH of theorem 3 and lemma 5 in [12] imply existence of a subsequence ν weakly converging { nk }k to an optimal measure ν∗ for the set Φ(S( )). But µnk µ∗ in the weak −1 −1 H → topology implies µnk Φ µ∗ Φ in the weak topology. So, we obtain −1 ◦ → ◦ µ∗ Φ = ν∗. Thus µ∗ is an optimal measure for the channel Φ. ◦Theorem 2 and theorem 2E in [12] imply the following sufficient condition of existence of an optimal measure. Corollary 2. Existence of an input optimal average state for the channel
9 Φ implies existence of an optimal measure for the channel Φ provided one of the following conditions holds:
dc(Ω(Φ)) < 1; • the channel Φ is a CE-channel. • Remark 1. The conditions in theorem 2 and in corollary 2 are essential as it is shown by the examples of channels with finite χ-capacity for which there exist no optimal measures (examples 1C and 2). The meaning of these conditions are considered in remark 2 below. The above general observations are illustrated by the following family of channels, depending on a sequence of positive numbers. ′ Example 1. Let n n∈N∪{0} be an orthonormal basis in and qn n∈N be a sequence of numbers{| i} in (0, 1], converging to zero. ConsiderH the set{ }1 S{qn} consisting of pure states
σ = (1 q ) 0 0 + q k k + sign(k) (1 q )q ( 0 k + k 0 ), k − |k| | ih | |k||| |ih| || − |k| |k| | ih| || || |ih | q indexed by the set Z 0 . This set can be considered as a sequence of states converging to the state\{ }0 0 and its properties were explored in subsection | ih | 5.1 in [12]. Consider the channel
Φ (ρ)= k ρ k σ , {qn} h | | i k k X where k k∈Z\{0} be an orthonormal basis in . {| i} H 1 This channel is entanglement-breaking [6],[5] and Φ q (S( )) = co( ) { n} H S{qn} S ′ 1 is a compact subset of ( ). Since the states in the sequence {qn} are ¯ H ∗ S pure C(Φ{qn}) = supρ∈S(H) H(Φ{qn}(ρ)). Let λ{qn} be the infinum of the set λ : exp λ < + if it is not empty and λ∗ = + otherwise. n − qn ∞ {qn} ∞ Letn o P +∞ 1 λ F{qn}(λ)= 1 exp qk − −qk k X=1 be a decreasing function on [λ∗ , + ) with the range (0, + ]. {qn} ∞ ∞ We will show that:
10 ∗ A) If λ{qn} =0 then Φ is a CE-channel; • {qn} there exist the unique optimal measure and the unique input optimal • average state for the channel Φ{qn};
Ω(Φ{qn}) Φ{qn}(S( )), • ∈ H ¯ dc(Ω(Φ{qn}))=0 and C(Φ{qn})= H(Ω(Φ{qn}));
∗ ∗ B) If 0 <λ < + and F q (λ ) 1 then {qn} ∞ { n} {qn} ≥ C¯(Φ ) < + , but Φ is not a CE-channel; • {qn} ∞ {qn} there exist the unique optimal measure and the unique input optimal • average state for the channel Φ{qn}; Ω(Φ ) Φ (S( )), • {qn} ∈ {qn} H 0 < dc(Ω(Φ )) 1 2 and C¯(Φ )= H(Ω(Φ )); {qn} ≤ {qn} {qn} ∗ ∗ C) If 0 <λ < + and F q (λ ) < 1 then {qn} ∞ { n} {qn} C¯(Φ ) < + , but Φ is not a CE-channel; • {qn} ∞ {qn} there exist no optimal measure and no input optimal average state • for the channel Φ{qn};
Ω(Φ{qn}) Φ{qn}(S( )) Φ{qn}(S( )), • ∈ H ¯ \ H dc(Ω(Φ{qn}))=1 and C(Φ{qn}) >H(Ω(Φ{qn}));
∗ D) If λ =+ then C¯(Φ q )=+ . {qn} ∞ { n} ∞ In the cases A and B the χ-capacity, the unique optimal measure, the unique input and output optimal average states are defined by the following expressions (correspondingly):
π1 ¯ 1 1 {qn} λn C Φ{qn} = λ{qn} log π{qn}, µ∗ = exp ; k k , − 2q|k| −q|k| | ih | ( )k∈Z\{0} 1 1 π{qn} λ{qn} ρ¯(µ∗)= exp k k , Z 2q|k| − q|k| ! | ih | k∈X\{0} 2 F λ∗ . In this case dc(Ω(Φ{qn})) = 1 if and only if {qn}( {qn})=1
11 +∞ λ1 1 1 {qn} Ω Φ{qn} = π{qn} 0 0 + π{qn} exp n n , | ih | − qn | ih | n=1 ! X 1 where λ{qn} is the unique solution of the equation F{qn}(λ)=1 and
−1 −1 +∞ λ1 +∞ 1 λ1 π1 = 1+ exp {qn} = exp {qn} [0, 1]. {qn} − q q − q ∈ n=1 n !! n=1 n n !! X X In the case C the χ-capacity and the output optimal average state are defined by the following expressions (correspondingly): C¯ Φ = λ∗ log π∗ , {qn} {qn} − {qn} +∞ λ∗ ∗ ∗ {qn} Ω Φ{qn} = π{qn} 0 0 + π{qn} exp n n , | ih | − qn | ih | n=1 X where +∞ ∗ −1 λ q π∗ = 1+ exp { n} [0, 1]. {qn} − q ∈ n=1 n ! X ∗ 1 If λ = + then C¯(Φ q ) = C¯( )=+ by the observation in {qn} ∞ { n} S{qn} ∞ subsection 5.1 in [12]. Suppose λ∗ < + . Without loss of generality we may assume that {qn} ∞ the sequence q is nonincreasing. Let { n} +∞ H =0 0 0 + q−1 n n (4) | ih | n | ih | n=1 X H ′ ∗ be a -operator in so that ic(H) = λ{qn} (see section 3 in [12]). It Hε is easy to see that {qn} H,1 and hence proposition 1b in [12] implies ¯ ¯ 1 S ¯⊆ K C(Φ{qn}) = C( {qn}) C( H,1) = supρ∈KH,1 H(ρ) < + . We will show 1 S ≤ K ∞ that C¯( )= C¯( H,1). In the proof of proposition 1a in [12] the sequence S{qn} K ρn of states defined by formula (9) was constructed and it was shown that { } H limn→+∞ H(ρn) = supρ∈KH,1 H(ρ). For the -operator H defined by (4) the states of the above sequence have the form
n −1 n λn λn ρn = 1+ exp 0 0 + exp k k (5) − qk | ih | − qk | ih | k ! k ! X=1 X=1 12 for sufficiently large n N, where λ is the unique solution of the equation ∈ n n n λ 1 λ 1+ exp = exp . (6) −qk qk −qk k k X=1 X=1 The sequence of states defined by (5) lies in co( 1 ). Indeed, for given S{qn} n let
n −1 n 2 λn 1 λn Z πk = exp exp , k 0 , q k −q k q k −q k ∈ \{ } k | | | | ! | | | | X=1 be a probability distribution. By using equality (6) with λ = λn it is easy to n 1 see that π σk = ρn and hence ρn co( ). This, proposition 1b and k k ∈ S{qn} theorem 2C in [12] imply P C¯(Φ )= C¯( 1 )= C¯( )= λ∗ log π∗ (7) {qn} S{qn} KH,1 − and
+∞ λ∗ Ω(Φ )=Ω( 1 )=Ω( )= π∗ 0 0 + π∗ exp n n , (8) {qn} S{qn} KH,1 | ih | − q | ih | n=1 n X +∞ λ∗ where π∗ =1+ exp and λ∗ is the unique solution of the equation − q n=1 n X +∞ λ +∞ 1 λ 1+ exp = exp (9) −qk qk −qk k k X=1 X=1 if
+∞ 1 λ∗ exp {qn} qk − qk TrH exp( ic(H)H) k=1 h∗(H)= − = X h =1 (10) Tr exp( ic(H)H) +∞ λ∗ ≥ − 1+ exp {qn} − qk k X=1 ∗ ∗ and λ = λ{qn} otherwise. It is easy to see that (9) is equivalent to the ∗ equation F q (λ) = 1 while (10) means the inequality F q (λ ) 1. { n} { n} {qn} ≥ The assertion concerning existence of the unique optimal measure µ∗ for the channel Φ{qn} in the cases A and B and the expression for this measure
13 follows from the observation in subsection 5.1 in [12] (with ε = 1). The barycenter of the measure µ∗ is the unique input optimal average state for the channel Φ{qn}. Uniqueness of the input optimal average state follows from uniqueness of the optimal measure, since proposition 1b in [12] and (8) imply regularity of the set Φ (S( )) in the cases A and B and {qn} H ⊆ KH,1 hence theorem 2 shows that each input optimal average state is a barycenter of at least one optimal measure.
In the cases A and B the state Ω(Φ{qn}) is an image of the barycenter of the optimal measure and hence it lies in Φ(S( )). To prove nonexistence H of an input optimal average state for the channel Φ{qn} in the case C it is sufficient to show that in this case the state Ω(Φ{qn}) does not lie in Φ(S( )). 1 H The set Φ(S( )) is a σ-convex hull of the set = σk k Z , so H S{qn} { } ∈ \{0} that the assumption Ω(Φ ) Φ(S( )) implies existence of a probability {qn} ∈ H distribution π Z such that Ω(Φ ) = π σ . By using the { k}k∈ \{0} {qn} k∈Z\{0} k k expression for the state Ω(Φ ) in the case C we obtain from this decom- {qn} P position that λ∗ exp {qn} − qn qn(π−n + πn)= , n N, +∞ λ∗ ∀ ∈ 1+ exp {qn} − qk k X=1 and hence
+∞ 1 λ∗ +∞ λ∗ exp {qn} =1+ exp {qn} . qk − qk − qk k k X=1 X=1 But this equality means that the equality holds in inequality (16) in [12] for the H-operator H defined by (4). By the observation in [12] equality in inequality (16) means inequality h∗(H) h = 1 equivalent to the inequality ∗ ≥ F{qn}(λ{qn}) 1, which contradicts to the definition of the case C. ∗ ≥ If λ{qn} = ic(H) = 0 then the above observation implies dc(Ω(Φ{qn})=0 and by proposition 2 Φ{qn} is a CE-channel. Suppose the entropy is continuous on the set co( 1 ). Consider the S{qn} sequence of states
n n ρ = 1 q π 0 0 + q π k k , n − k k | ih | k k| ih | k ! k X=1 X=1 14 −1 n −1 −1 n where πk = qk ( k=1 qk ) k=1 is a probability distribution. It is easy to see that{ this sequence lies in co(} 1 ) and converges to the pure state 0 0 . P S{qn} | ih | By the continuity assumption limn→+∞ H(ρn) = 0, which implies
n −1 lim n(qkπk( log(qkπk))) = lim nf qk =0, (11) n→+∞ − n→+∞ k ! X=1 where f(x) = log x/x. Since the function f(x) is decreasing for large x the n −1 −1 obvious inequality k=1 qk nqn and (11) imply limn→+∞ νn = 0, where −1 −1 ≤ νn = nf(nq )= qn log(nq ). Hence for arbitrary λ> 0 we have n P n λ q νn λ n = exp , n −qn ∗ which implies λ{qn} = 0. The most interesting case of the above example is the case C, in which the channel Φ{qn} demonstrates essential infinite dimensional features. The example of a sequence q corresponding to this case can be found in [12]. { n} Example 1 can be generalized by considering the set ε with arbitrary S{qn} ε in [0, 1] instead of 1 [12]. S{qn} 4 On extension to nonnormal states
Let S( ) be the set of all normalized positive functionals on B( ), so that H H S( ) (B( ))∗. It is known that S( ) is compact in the -weak topology H ⊂ H H ∗ and thatb S( ) can be considered as a -weak dense subset of S( ) [2]. So, it H ∗ H isb natural to explore a possibility tob extend an arbitrary channel from S( ) H to S( ). The following proposition provides a simple characterizationb of the H class of channels, which has natural extension to S( ). bProposition 3. 3 A channel Φ: S( ) S( ′)Hcan be extended to the H 7→ H mapping Φ: S( ) S( ′) continuous with respectb to the -weak topology H 7→ H ∗ on the set S( ) and the trace norm topology on the set S( ′) if and only if the set Φ(bS( Hb)) is relatively compact. H H If the aboveb extension Φ exists then Φ(S( )) = Φ(S( )). H H 3This simple observation seems to be a corollary of a general result in the functional analysis. The author would beb grateful for any references. b b
15 Proof. If Φ is the above extension of the channel Φ then Φ(S( )) is compact as an image of a compact set under a continuous mapping.H Since S( ) is a -weakb dense subset of S( ) we have Φ(S( )) = Φ(Sb(b )). HSuppose∗ the set Φ(S( )) is compact.H Since B( )∗ =H T( )∗∗ thereH exists H H H the linear mapping Φ∗∗ : B( )∗ b B( ′)∗ continuous withb respectb to the H 7→ H ∗∗ -weak topologies on the both spaces and such that Φ S = Φ. By using ∗ | (H) the compactness argument and -weak density of the set S( ) in S( ) it is ∗ H H easy to show that Φ∗∗(S( )) Φ(S( )). Since each one-to-one continuous H ⊆ H mapping from compact topological space onto Hausdorff topologicab l space is a hemeomorphism [7]b we can conclude that the identity mapping from Φ(S( )) with the trace norm topology onto itself with the -weak topology H ∗ has a continuous converse. This and the previous observation imply that Φ∗∗ is continuous with respect to the -weak topology on S( ) and the trace ∗ H norm topology on S( ′). So, the mapping Φ∗∗ has all the properties of H |S(H) b the extension Φ, stated in the proposition. b Definition 2. For an arbitrary channel Φ with relatively compact output the mapping Φbintroduced in proposition 3 is called its channel extension. Proposition 3 and theorem 1 imply the following observation. Corollaryb 3. If Φ is a channel with finite χ-capacity then it has the channel extension Φ. By this corollary for given channel Φ with finite χ-capacity it is possible to consider the entropicb characteristics of its channel extension Φ such as the minimal output entropy Hmin(Φ) = infρˆ∈S(H) H(Φ(ˆρ)) and the χ-capacity b b C¯(Φ) = sup b π H(Φ(ˆρ ) Φ(b π ρˆ )), (12) i i k j j j {πi,ρˆi} i X b b b P where the supremum is over all finite ensembles πi, ρˆi of states in S( ). By proposition 3 and theorem 2B in [12] we have{ } H b C¯(Φ) = C¯(Φ(S( ))) = C¯(Φ(S( ))) = C¯(Φ). (13) H H This means thatb one can not increase the χ-capacity by using nonnormal states. In the same way as for the initial channel Φ we may define the notions of an approximating sequence of ensembles and of an input optimal average state for the channel extension Φ. In contrast to the case of the initial channel Φ compactness of the set S( ) guarantees existence of at least one H b b 16 input optimal average state for the channel extension Φ. By using lower semicontinuity of the relative entropy and (13) it is possible to show that each normal input optimal average state for the channelb extension Φ is an input optimal average state for the initial channel Φ and vice versa. For the channel extension Φ it is possible to prove the analog of theoremb 1, in particular, to show that the image of any input optimal average state for the channel extension Φ coincidesb with the output optimal average state Ω(Φ). In contrast to the χ-capacity the minimal output entropies for the channel Φ and for its extensionb Φ may be different as it follows from the below simple +∞ ′ example. Let ρn n=1 be a sequence of states in S( ) with infinite entropy, { } ¯ +H∞ converging to pure stateb ρ0 and such that C( ρn n=1) < + . Consider the +∞ { } ∞ channel Φ(ρ) = n=1 n ρ n ρn, where n is some orthonormal basis of . Since an arbitraryh state| | i in Φ(S( )){| majorizesi} at least one trace class H H operator with infiniteP entropy it has infinite entropy as well [15]. So we have H (Φ) = + . But by proposition 3 we have Φ(S( )) = co( ρ +∞ ) and min ∞ H { n}n=1 hence there exists a state ρ in S( ) such that Φ(ρ ) = ρ . This implies 0 H 0 0 Hmin(Φ) = 0. b b Let be the set of all regularb b Borel measuresb onb the set S( ) endowed P H with theb -weak topology. In contrast to the set the set is not metrizable but it isb compact∗ in the weak topology [1]. P P b In the same way 4 as in [4] it is possible to show that b
C¯(Φ) = sup H(Φ(ˆρ) Φ(¯ρ(ˆπ)))ˆπ(dρˆ) πˆ∈P k S(ZH) b b b b b where the supremum is over all probability measures on S( ) andρ ¯(ˆπ) is a H barycenter of the measureπ ˆ. The measureπ ˆ∗ at which the above supremum is achieved (if it exists) is called optimal for the channelb extension Φ. By using lower semicontinuity of the relative entropy, equality (13) and theorem 1 it is possible to prove that any optimal measureπ ˆ for theb channel extension Φ has the generalized maximal distance property:
H(Φ(ˆρ) Ω(Φ)) = C¯(Φ) forˆπ -almost allρ ˆ in S( ). b k ∗ H 4The only difference in the argumentation is a necessity to use proposition I.2.3 in [1] instead of lemmab 1 in [4] since the setbS( ) is not complete separableb metric space, but it is compact. H b
17 The following proposition shows, in particular, that the condition of ex- istence of an optimal measure for the channel extension Φ is substantially weaker than the condition of existence of an optimal measure for the initial channel Φ. b Proposition 4. Let Φ: S( ) S( ′) be a channel with finite χ- H 7→ H capacity and Φ: S( ) S( ′) be its channel extension. The following propertiesH 7→ areH equivalent: b b there exists an optimal measure for the set Φ(S( )); • H there exists an optimal measure for the channel extension Φ. • The above equivalent properties hold if the set Φ(S( ) contains regular H b subset with the same χ-capacity. Proof. Ifµ ˆ∗ is an optimal measure for the channel extension Φ then its imageµ ˆ Φ−1 corresponding to the mapping Φ is an optimal measure for ∗ ◦ the set Φ(S( )). b H n n Let ν∗ beb an optimal measure for the set Φ(S(b )) and let νn = πi , ρi n be a sequence of measures in (Φ(S( ))) withH finite support{ weakly{ con-}} P H verging to the measure ν∗. Since Φ(S( )) = Φ(S( )) for each n and i there exists a stateρ ˆn in Sˆ ( ) such thatH Φ(ˆˆ ρn) =Hρn. The sequence i H i i µˆ = πn, ρˆn of measures in the weakly compactb b set (S( )) has a { n { i i }}n P H weak limit pointµ ˆ∗. Since the mapping Φ is continuous with respect to the -weak topology on S( ) and the trace norm topology on S( b′) the image ∗ H H µˆ Φ−1 of the weak limit pointµ ˆ of theb set µˆ is a weak limit point ∗ ◦ ∗ { n}n of the set ν =µ ˆ bΦ−1 , so thatµ ˆ Φ−1 = ν . Thusµ ˆ is an optimal { n n ◦ }n ∗ ◦ ∗ ∗ measureb for the channel extension Φ. The last assertionb of the proposition followsb from the previous one and theorem 3 in [12]. b Remark 2. It is clear that each optimal measure for the channel Φ is an optimal measure for the channel extension Φ.ˆ Proposition 4 shows the meaning of the conditions of existence of an optimal measure for the channel Φ in theorem 2. Namely, the regularity condition implies that the set of optimal measures for the channel extension Φ is not empty while the condition of existence of an input optimal average state for the channel Φ implies that this set contains at least one optimalb measure for the channel Φ. This observation is illustrated by the example in the next section.
18 5 On a class of channels with finite χ-capacity
In this section we consider nontrivial class of entanglement-breaking channels generalizing the example considered in [5]. For channels of this class the χ- capacity and the minimal output entropy can be explicitly calculated. There also exists simple necessary and sufficient of continuity of the output entropy for these channels. ′ Let G be a compact group, Vg be unitary representation of G on , M(dg) be a positive operator-valued{ } measure (POVM) on G, such that theH set of probability measures TrρM( ) ρ∈S(H) is weakly dense in the set of all probability measures on G. { · } For arbitrary state σ in S( ′) consider the channel H
∗ Φσ(ρ)= VgσVg TrρM(dg). ZG ∗ Let ω(G,Vg, σ)= G VgσVg µH (dg), where µH is the Haar measure on G. It follows from the assumption of weak density of the set TrρM( ) S in R { · }ρ∈ (H) the set of all Borel probability measures on G that Φ (S( )) = co V σV ∗ . σ H { g g }g∈G Thus proposition 12 in [12] implies the following observation. Proposition 5. The χ-capacity of the channel Φσ is equal to
C¯(Φ )= H(σ ω(G,V , σ)). σ k g
If this capacity is finite then Ω(Φσ)= ω(G,Vg, σ) and Hmin(Φσ)= H(σ). The channel Φ is a CE-channel if and only if H(ω(G,V , σ)) < + . In σ g ∞ this case C¯(Φ )= H(ω(G,V , σ)) H(σ)= H(Ω(Φ )) H (Φ ). σ g − σ − min σ Since the set Φ (S( )) = co V σV ∗ is compact for arbitrary σ σ H { g g }g∈G proposition 3 implies existence of the extension Φσ of the channel Φσ to the set S( ) such that Φ (S( )) = Φ (S( )) = co V σV ∗ . H σ H σ H { g g }g∈G Proposition 4 and proposition 12 in [12] showb existence of an optimal measureb for the channelb extensionb Φ provided C¯(Φ )= C¯(Φ ) < + . The σ σ σ ∞ support of any optimal measure for the channel extension Φσ consists of statesρ ˆ in S( ) such that Φ (ˆρ) =b V σV ∗ for someb g G. As it is shown H σ g g ∈ by the following example we can not assert existence of an optimalb measure for the channelb Φσ even in theb case when Φσ is a CE-channel. Example 2. Consider the case G = T, which can be identified with [0, 2π). In this case the Haar measure µH is the normalized Lebesgue mea- sure dx . Let V is the group of shifts in ′ = (T) and M(dg) is the 2π g H L2 19 spectral measure of the operator of multiplication by an independent vari- able in = (T). It is easy to see that the above density assumption H L2 is valid in this case and hence all the above results hold for the corre- sponding channel Φ with arbitrary σ S( (T)). Suppose σ = ϕ ϕ , σ ∈ L2 | ih | where ϕ be an arbitrary function in 2(T) with unit norm. This implies 2π L ω(T,V , ϕ ϕ )=(2π)−1 ϕ ϕ dx, where ϕ (t)= ϕ(t x). This chan- g | ih | 0 | xih x| x − nel was originally used in [5] as an example of entanglement-breaking channel R which has no canonical representation with purely atomic POVM and hence has no Kraus representation with operators of rank 1. Here we will use this channel as an example of CE-channel, which demonstrates the essential features of infinite-dimensional channels. By using proposition 5 and direct calculation it is easy to obtain (cf.[5]) that
+∞ C¯ Φ = H(ω(T,V , ϕ ϕ )) = ϕ 2 log ϕ 2, (14) |ϕihϕ| g | ih | − | k| | k| k=−∞ X −1 2π −ixk Z where ϕk = (2π) 0 e ϕ(x)dx k∈ is the sequence of the Fourier coeffi- cients of{ the function ϕ, and that finiteness} of C¯(Φ ) implies CE-property R |ϕihϕ| of the channel Φ|ϕihϕ| and
+∞ Ω Φ = ω(T,V , ϕ ϕ )= ϕ 2 τ τ , |ϕihϕ| g | ih | | k| | kih k| k=−∞ X where τ (x) = exp(ikx) Z is the trigonometric basis in (T). It is inter- { k }k∈ L2 esting to note that the properties of the channel Φ|ϕihϕ| are determined by the rate of vanishing of the Fourier coefficients of the function ϕ. Suppose series (14) is finite for a function ϕ and hence Φ|ϕihϕ| is a CE- channel. By proposition 5 Hmin(Φ|ϕihϕ|) = 0 in spite of the fact that the set Φ (S( )) does not contain pure states. The set ϕ ϕ T of |ϕihϕ| H {| xih x|}x∈ pure states in Φ (S( )) is contained in the output set of the channel |ϕihϕ| H extension Φ|ϕihϕ| and corresponds to a particular subset of nonnormal states S ˆ in ( ). By proposition 1 the functions χΦ|ϕihϕ| and HΦ|ϕihϕ| are bounded and continuousH b on S( ). Nevertheless there exists no optimal measure and H henceb there exists no input optimal average state for the channel Φ|ϕihϕ| (since otherwise corollary 2 implies existence of an optimal measure). The ˆ continuous functions χΦ|ϕihϕ| and HΦ|ϕihϕ| do not achieve their finite supremum
20 C¯(Φ|ϕihϕ|) and infinum Hmin(Φ|ϕihϕ|) = 0 correspondingly on noncompact set S( ). H The family of channels Φ|ϕihϕ| ϕ∈L2(T) provides an another example show- ing that in the infinite dimensional{ } case the χ-capacity is not continuous but only lower semicontinuous function of a channel [10]. It is easy to see that for arbitrary unit vector ϕ∗ in 2(T) and for arbitrary sequence ϕ of unit vectors in (T) converging| i L to the vector {| ni} L2 ϕ the sequence sup S Φ (ρ) Φ (ρ) tends to zero. It | ∗i ρ∈ (H) k |ϕnihϕn| − |ϕ∗ihϕ∗| k1 means that ϕ Φ|ϕihϕ| is a continuous mapping from 2(T) into the set of all channels endowed7→ with the topology of uniform convergence.L Let ϕn be the sequence of function in 2(T) with the following Fourier coefficients{ } L √1 q , k =0 − n (ϕ ) = q /n, 0
C¯ Φ = q log q (1 q ) log(1 q )+ q log n, n N, |ϕnihϕn| − n n − − n − n n ∀ ∈ ¯ and hence limn→+∞ C Φ|ϕnihϕn| = C > 0. Acknowledgments. The author is grateful to A. S. Holevo for the help in preparing of this paper. The work was partially supported by the program ”Modern problems of theoretical mathematics” of Russian Academy of Sciences.
References
[1] Alfsen E. ”Compact convex sets and boundary integrals”, Springer- Verlag, 1971.
[2] Bratteli O., Robinson D.W. ”Operators algebras and quantum statistical mechanics”; Springer-Verlag, New York-Heidelberg-Berlin, vol.I, 1979.
21 [3] Holevo A.S. ”Statistical structure of quantum theory”, Springer-Verlag, 2001. [4] Holevo, A.S., Shirokov M.E. ”Continuous ensembles and the χ-capacity of infinite dimensional channels”, Probability Theory and Applications, 50, N.1, 98-114, 2005, e-print quant-ph/0408176; [5] Holevo A.S., Shirokov M.E., Werner R.F. ”On the notion of entanglement in Hilbert space”, Russian Math. Surveys, 60, N.2, 153-154, 2005, e-print quant-ph/0504204; [6] Horodecki M., Shor P.W., Ruskai, M.B. ”General Entanglement Breaking Channels”, Rev. Math. Phys. 15, 629-641, 2003, e-print quant-ph/0302031; [7] Kolmogorov, A.N., Fomin, S.V. ”Elements of function theory and func- tional analysis”, Moscow, Nauka, 1989 (In Russian); [8] Joffe, A. D., Tikhomirov, W. M. ”Theory of extremum problems”, AP, NY, 1979; [9] Sarymsakov, T.A. ”Introduction to quantum probability theory” (FAN, Tashkent, 1985), (In Russian); [10] Shirokov, M.E. ”The Holevo capacity of infinite dimensional channels and the additivity problem”, Commun. Math. Phys. 262, N.1, 137-159, 2006, e-print quant-ph/0408009; [11] Shirokov M.E. ”On entropic quantities related to the classical capacity of infinite dimensional quantum channels”, e-print quant-ph/0411091, 2004; [12] Shirokov M.E. ”Entropic characteristics of subsets of states”, e-print quant-ph/0510073, 2005; [13] Schumacher, B., Westmoreland, M. ”Optimal signal ensemble”, Phys. Rev. A 63, 022308, 2001, e-print quant-ph/9912122; [14] Parthasarathy, K. ”Probability measures on metric spaces”, Academic Press, New York and London, 1967; [15] Wehrl, A. ”General properties of entropy”, Rev. Mod. Phys. 50, 221-250, 1978.
22