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arXiv:0709.0855 · Journal of Physics Conference Series

Notes on multiplicativity of maximal output purity for completely positive qubit maps

K M R Audenaert

Published Mar 1, 2008• 0 citations
arXivHTMLPDF

Abstract

A problem in quantum information theory that has received considerable attention in recent years is the question of multiplicativity of the so-called maximal output purity (MOP) of a quantum channel. This quantity is defined as the maximum value of the purity one can get at the output of a channel by varying over all physical input states, when purity is measured by the Schatten $q$-norm, and is denoted by $ν_q$. The multiplicativity problem is the question whether two channels used in parallel have a combined $ν_q$ that is the product of the $ν_q$ of the two channels. A positive answer would imply a number of other additivity results in QIT. Very recently, P. Hayden has found counterexamples for every value of $q>1$. Nevertheless, these counterexamples require that the dimension of these channels increases with $1-q$ and therefore do not rule out multiplicativity for $q$ in intervals $[1,q_0)$ with $q_0$ depending on the channel dimension. I argue that this would be enough to prove additivity of entanglement of formation and of the classical capacity of quantum channels. More importantly, no counterexamples have as yet been found in the important special case where one of the channels is a qubit-channel, i.e. its input states are 2-dimensional. In this paper I focus attention to this qubit case and I rephrase the multiplicativity conjecture in the language of block matrices and prove the conjecture in a number of special cases.

DOI
10.1088/1742-6596/104/1/012001

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A problem in quantum information theory that has received considerable attention in recent years is the question of multiplicativity of the so-called maximal output purity (MOP) of a quantum channel. This quantity is defined as the maximum value of the purity one can get at the output of a channel by varying over all physical input states, when purity is measured by the Schatten qqq-norm, and is denoted by νqν_qνq​. The multiplicativity problem is the question whether two channels used in parallel have a combined νqν_qνq​ that is the product of the νqν_qνq​ of the two channels. A positive answer would imply a number of other additivity results in QIT. Very recently, P. Hayden has found counterexamples for every value of q>1q>1q>1. Nevertheless, these counterexamples require that the dimension of these channels increases with 1−q1-q1−q and therefore do not rule out multiplicativity for qqq in intervals [1,q0)[1,q_0)[1,q0​) with q0q_0q0​ depending on the channel dimension. I argue that this would be enough to prove additivity of entanglement of formation and of the classical capacity of quantum channels. More importantly, no counterexamples have as yet been found in the important special case where one of the channels is a qubit-channel, i.e. its input states are 2-dimensional. In this paper I focus attention to this qubit case and I rephrase the multiplicativity conjecture in the language of block matrices and prove the conjecture in a number of special cases.
A class of problems in quantum information theory, having an elementary formulation but still resisting solution, concerns the additivity properties of various quantities characterizing quantum channels, notably the "classical capacity", and the "maximal output purity". All known results, including extensive numerical work, are consistent with the conjecture that these quantities are indeed additive (resp. multiplicative) with respect to tensor products of channels. A proof of this conjecture would have important consequences in quantum information theory. In particular, according to this conjecture, the classical capacity or the maximal purity of outputs cannot be increased by using entangled inputs of the channel. In this paper we state the additivity/multiplicativity problems, give some relations between them, and prove some new partial results, which also support the conjecture.
A class of problems in quantum information theory, having an elementary formulation but still resisting solution, concerns the additivity properties of various quantities characterizing quantum channels, notably the “classical capacity”, and the “maximal output purity”. All known results, including extensive numerical work, are consistent with the conjecture that these quantities are indeed additive (resp. multiplicative) with respect to tensor products of channels. A proof of this conjecture would have important consequences in quantum information theory. In particular, according to this conjecture, the classical capacity or the maximal purity of outputs cannot be increased by using entangled inputs of the channel. In this paper we state the additivity/multiplicativity problems, give some relations between them, and prove some new partial results, which also support the conjecture. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
We reformulate the Rényi entanglement of purification as a constrained minimum output Rényi entropy problem. Equivalently, for p>1p>1p>1, this formulation can be expressed in terms of a constrained maximal output Schatten ppp-norm. More precisely, for a completely positive map Ω:L(B′)→L(A)Ω:L(B')\to L(A)Ω:L(B′)→L(A), we consider the quantity υp(Ω)\upsilon_p(Ω)υp​(Ω) defined by optimizing ∥(Ω⊗idE)(σB′E)∥p\|(Ω\otimes \mathrm{id}_E)(σ^{B'E})\|_p∥(Ω⊗idE​)(σB′E)∥p​ over all bipartite states σB′Eσ^{B'E}σB′E whose B′B'B′-marginal is maximally mixed. We focus on the case p=2p=2p=2. First, we compute υ2\upsilon_2υ2​ for the transpose-depolarizing channel and prove that it is multiplicative under tensor powers. We then establish a general multiplicativity criterion: whenever a completely positive map N:L(B′)→L(A)N:L(B')\to L(A)N:L(B′)→L(A) satisfies N†∘N=a idA+b Tr[⋅] IdN^{\dagger} \mathbin{\circ} N=a\,\mathrm{id}_A+b\,\mathrm{Tr}[\cdot]\,I_dN†∘N=aidA​+bTr[⋅]Id​ for some constants a,b≥0a,b\ge 0a,b≥0, where N†N^{\dagger}N† denotes the Hilbert-Schmidt adjoint of NNN, the quantity υ2(N)\upsilon_2(N)υ2​(N) is multiplicative under tensor powers. Examples of channels satisfying this criterion include the transpose-depolarizing channel, the depolarizing channel, and their respective complementary channels. Furthermore, we show that, for every completely positive map ΩΩΩ, multiplicativity of υp(Ω)\upsilon_p(Ω)υp​(Ω) implies multiplicativity for its complementary map. This yields the corresponding additivity statements for the associated Rényi-2 entanglement of purification.
In this paper, we present a class of maps for which the multiplicativity of the maximal output p-norm holds for p=2 and p≥4. The class includes all positive trace-preserving maps from B(C3) to B(C2). In this sense, the result is a generalization of the corresponding result in the work of King and Koldan [“New multiplicativity results for qubit maps,” J. Math. Phys. 47, 042106 (2006)], where the multiplicativity was proved for all positive trace-preserving maps from B(C2) to B(C2) with p=2 and p≥4. Interestingly, by contrast, the multiplicativity of p-norm was investigated in the context of quantum information theory and shown not to hold, in general, for high dimensional quantum channels [Hayden, P. and Winter, A., “Counterexamples to the maximal p-norm multiplicativity conjecture for all p>1,” Commun. Math. Phys. 284, 263 (2008)]. Moreover, the Werner–Holevo channel, which is a map from B(C3) to B(C3), is a counterexample for p>4.79 [Werner and Holevo, J. Math. Phys. 43, 4353 (2002).].