Additivity of classical capacity and related problems

Cite this problem as Problem 10.

Problem

For each quantum channel \!\,T (in the Schrödinger picture), define

\chi(T) = \sup\limits_{p,\rho}\left( H\left(\sum_i p_i T(\rho_i)\right) -\sum_i p_i H\left(T(\rho_i)\right) \right),

where the supremum is over all probability vectors \!\,p=(p_1,\ldots,p_n), and all collections of input states \{\!\,\rho_1,\ldots,\rho_n\}, and \!\,H denotes the von Neumann entropy.

Show that \!\,\chi (T_1\otimes T_2)=\chi (T_1)+\chi (T_2), or else give a counterexample.

The problem can be traced back to [1], see also [2].

Background

This problem can also be paraphrased as “Can entanglement between signal states help to send classical information on quantum channels?”.

Recall that the capacities of a memoryless channel are defined as the maximal transmission rate per use of the channel, with coding and decoding chosen for increasing number \!\,n of parallel and

independent uses of the channel

T^{\otimes n}=\underset{n}{\underbrace{T\otimes \dots \otimes T}}

such that the error probability goes to zero as \!\,n\rightarrow \infty . There are many different capacities, because one may consider sending different kinds (classical or quantum) information, restrict the admissible coding and decoding operations, and/or allow the use of additional resources. Here we only look at the transmission of classical information with no additional resources. Then one can distinguish four capacities [3], according to whether for each block length n we are allowed to use arbitrary entangled quantum operations on the full block of input (resp. output) systems, or if for each of the parallel channels we have to use a separate quantum coding (resp. decoding), and combine these only by classical pre (resp. post)-processing:

The equality in the lower right was established independently by several authors, see e. g. [4].

That \!\,C_{1\infty } on the left coincides with the quantity \!\,\chi given in the statement of the problem was shown in [5]. The inequality in the lower left is known to be strict sometimes [2], which means that entangling decodings indeed can increase the classical capacity. See [6] for investigation of the corresponding information gain.

The full capacity and \!\,\chi are connected by the limit formula

C_{\infty \infty}(T) = \mathrm{lim} _n (1/n) \chi(T^{\otimes n})

Since \!\,\chi is easily seen to be superadditive (i.\, e., \!\,\chi (T_{1}\otimes T_{2})\geq \chi (T_{1})+\chi (T_{2})), we immediately get \!\ C_{\infty \infty }\geq \chi . If additivity holds, then we will even have equality, i.e., “???” in the table can be replaced by “\!\,=” . While such a result would be very much welcome from a mathematical (and practical) point of view, giving a “single-letter” expression for the classical capacity, it would call for a physical explanation of strange asymmetry between the roles of entanglement in encoding and decoding procedures.

Partial Results

Validity of the additivity conjecture was established if one of the channels is

  • the identity channel [7], [8]
  • a unital qubit channel [9]
  • the depolarizing channel [10]
  • an entaglement breaking channel [2], [11] (both for “c-q/q-c” channels), [12] (general entanglement breaking channel).

Some further more recent partial results will be mentioned below. Whether the additivity holds “globally”, i.e. for all quantum channels, is still an open problem. No counterexample was found despite extensive numerical search by groups in IBM, IMaPh, see also [13]. If the conjecture is valid, then the additivity of \chi tentatively relies upon yet another hypothetical property of multiplicativity of norms of the completely positive mappings

T:\ell _{1}(\mathcal{H})\rightarrow \ell _{p}(\mathcal{H});\quad p\geq 1,

where

\ell _{p}(\mathcal{H})=\{X:X=X^{\ast },\quad \Vert X\Vert _{p}\equiv \left( \mathrm{Tr}|X|^{p}\right) ^{\frac{1}{p}}\}

is a noncommutative analog of the space \!\,\ell _{p} — the so called Schatten class. Namely, the conjecture [7] is that for \!\,p, sufficiently close to \!\,1

\Vert T_{1}\otimes T_{2}\Vert _{p}\overset{?}{=}\Vert T_{1}\Vert_{p}\Vert T_{2}\Vert _{p},                        (1)

where \!\,\Vert T\Vert _{p}=\max_{\rho }\Vert T(\rho )\Vert _{p}. Byletting \!\, p\downarrow 1 this implies additivity of the minimal output entropy

H_{\min }(T)=\min_{\rho }H\left( T\left( \rho \right) \right),

one of a whole number of properties equivalent, as it was shown in [14], to the additivity of \!\,\chi . The relation (1) can be re-expressed as the additivity of the minimal output Renyi

entropy of order \!\, p [15].

*In all cases listed above where the additivity conjecture is proved, the multiplicativity of \!\,p-norms (for all \!\,p\geq 1) also holds, moreover, it underlies the proof of additivity in [9], [10]. The multiplicativity of \!\,p-norms holds for arbitrary bounded maps of the classical spaces \!\,\ell _{p}, where its proof can be based on a Minkowsky inequality. Therefore quite intriguing is counterexample of the channel

T(\rho )=\frac{1}{d-1}\left[ I-\rho ^{T}\right],

for which (1) with \!\,T_{1}=T_{2}=T fails to hold for sufficiently large \!\,p (\!\,p\geq 4,7823 if \!\,d=\mathrm{dim}\mathcal{H}=3 [16]). Nevertheless, the additivity of \!\,H_{\min } and of \!\,\chi holds for such channels, as shown in [17], [18], [19]. The standing conjecture is that multiplicativity holds globally at least for \!\,1\leq p\leq 2, but  even the case \!\,p=2 is difficult, see [20], [21]. For some results concerning integer p, see [22].

In [23] it was shown that proving the multiplicativity would solve another important open problem — superadditivity of the entanglement of formation (EoF). Earlier [24] brought attention to a simple correspondence between \chi and EoF, and obtained several concrete results on additivity of EoF by using this correspondence. It was also remarked that superadditivity of EoF would imply additivity of \chi for channels with linear additive input constraints. By combining the MSW correspondence and the convex duality technique of [23] with an original and powerful channel extension technique, which allows to use effectively arbirariness of channels in question, [14] had shown equivalence of the global properties of additivity of the minimal output entropy, \chi, EoF and of superadditivity of EoF. The last equivalence for two fixed channels was also established in [25].

In [26] several equivalent formulations of the additivity conjecture for channels with arbitrarily constrained inputs, which formally is substantially stronger than additivity of the unconstrained \chi, were given. It was shown that the additivity conjecture for channels with constrained inputs holds true for certain nontrivial classes of channels, e. g. a direct sum mixture of the identity channel and an entaglement breaking channel (such as erasure channel). The channel extension technique was used to show that additivity for two fixed constrained channels can be reduced to the same problem for unconstrained channels, and hence, the global additivity for channels with arbitrary input constraints is equivalent to the global additivity without constraints.

The additivity problem is still open for the minimal dimension 2: it is not known if the additivity holds for all nonunital qubit channels, although a strong numerical evidence in favour of this was given in [27]. Nevertheless there are several reasons to consider the problem in infinite dimensions. There is a good chance that both the additivity and the multiplicativity for all p\geq 1 hold for important and interesting class of Gaussian channels that act in infinite dimensional Hilbert space. However the only instance where the additivity of \chi and the multiplicativity for integer p was proved is the pure loss channel, having the very special property H_{min}(T)=0 [15], [28].

It was observed recently that Shor’s proof of equivalence of different forms of the global additivity conjecture for finite dimensional channels is related to weird discontinuity of the \chi-capacity as a function of channel in infinite dimensions. This also calls for a mathematically rigorous treatment of the entropic quantities related to the classical capacity of infinite dimensional channels [29]. In particular it is possible to show that additivity for all finite dimensional channels implies additivity of the constrained \chi−capacity with constraints fulfilling finiteness of the output entropy [30].

References

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[25] A. A. Pomeransky, Strong superadditivity of the entanglement of formation follows from its additivity, [24] quant-ph/0305056 (2003)

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[27] M. Hayashi, H. Imai, K. Matsumoto, M. B. Ruskai, T. Shimono, Qubit channels which require four inputs to achieve capacity: implications for additivity conjectures, [27] quant-ph/0403176 (2004)

[28] V. Giovannetti, S. Lloyd, Additivity properties of a Gaussian channel, [28] quant-ph/0403075 (2004)

[29] A. S. Holevo, M. E. Shirokov, Continuous ensembles and the \chi-capacity of infinite-dimensional channels,, [29] quant-ph/0403072 (2004)

[30] M. E. Shirokov, The Holevo capacity of infinite dimensional channels, [30] quant-ph/0408009 (2004).