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Russian Mathematical Surveys

Multiplicativity of p -norms of completely positive maps and the additivity problem in quantum information theory

Alexander Semenovich Holevo

Published Apr 30, 2006• 25 citations
Publisher

Abstract

The additivity problem is one of the most profound mathematical problems of quantum information theory. From an analytical point of view it is closely related to the multiplicative property, with respect to tensor products, of norms of maps on operator spaces equipped with the Schatten norms (non-commutative analogue of -norms). In this paper we survey the current state of the problem.

DOI
10.1070/rm2006v061n02abeh004313

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We prove additivity of the minimal conditional entropy associated with a quantum channel Phi, represented by a completely positive (CP), trace-preserving map, when the infimum of S(gamma_{12}) - S(gamma_1) is restricted to states of the form gamma_{12} = (I \\ot Phi)(| psi >< psi |). We show that this follows from multiplicativity of the completely bounded norm of Phi considered as a map from L_1 -> L_p for L_p spaces defined by the Schatten p-norm on matrices; we also give an independent proof based on entropy inequalities. Several related multiplicativity results are discussed and proved. In particular, we show that both the usual L_1 -> L_p norm of a CP map and the corresponding completely bounded norm are achieved for positive semi-definite matrices. Physical interpretations are considered, and a new proof of strong subadditivity is presented.

•Jun 23, 2005•78 citations
The additivity problem is one of the most profound mathematical problems of quantum information theory. From an analytical point of view it is closely related to the multiplicative property, with respect to tensor products, of norms of maps on operator spaces equipped with the Schatten norms (non-commutative analogue of -norms). In this paper we survey the current state of the problem.
We define 2-indexed (q,p)(q,p)(q,p)-Schatten quasi-norms for any q,p>0q,p > 0q,p>0 on operators on a tensor product of Hilbert spaces, naturally extending the norms defined by Pisier's theory of operator-valued Schatten spaces. We establish several desirable properties of these quasi-norms, such as relational consistency and the behavior on block diagonal operators, assuming that ∣1q−1p∣≤1|\frac{1}{q} - \frac{1}{p}| \leq 1∣q1​−p1​∣≤1. In fact, we show that this condition is essentially necessary for natural properties to hold. Furthermore, for linear maps between spaces of such quasi-norms, we introduce completely bounded quasi-norms and co-quasi-norms. We prove that the q→pq \to pq→p completely bounded co-quasi-norm is super-multiplicative for tensor products of quantum channels for q≥p>0q \geq p>0q≥p>0, extending an influential result of [Devetak, Junge, King, Ruskai, 2006]. Our proofs rely on elementary matrix analysis and operator convexity tools and do not require operator space theory. On the applications side, we demonstrate that these quasi-norms can be used to express relevant quantum information measures such as Rényi conditional entropies for α≥12α\geq \frac{1}{2}α≥21​ or the Sandwiched Rényi Umlaut information for α<1α< 1α<1. Our multiplicativity results imply a tensorizing notion of reverse hypercontractivity, additivity of the completely bounded minimum output Rényi-ααα-entropy for α≥12α\geq\frac{1}{2}α≥21​ extending another important result of [Devetak, Junge, King, Ruskai, 2006], and additivity of the maximum output Rényi-ααα entropy for α≥12α\geq \frac{1}{2}α≥21​.
Multiplicativity of certain maximal p -> q norms of a tensor product of linear maps on matrix algebras is proved in situations in which the condition of complete positivity (CP) is either augmented by, or replaced by, the requirement that the entries of a matrix representative of the map are non-negative (EP). In particular, for integer t, multiplicativity holds for the maximal 2 -> 2t norm of a product of two maps, whenever one of the pair is EP; for the maximal 1 -> t norm for pairs of CP maps when one of them is also EP; and for the maximal 1 -> 2t norm for the product of an EP and a 2-positive map. Similar results are shown in the infinite-dimensional setting of convolution operators on L^2(R), with the pointwise positivity of an integral kernel replacing entrywise positivity of a matrix. These results apply in particular to Gaussian bosonic channels.
We give a survey of the two remarkable analytical problems of quantum information theory. The main part is a detailed report of the recent (partial) solution of the quantum Gaussian optimizers problem which establishes an optimal property of Glauber's coherent states -- a particular instance of pure quantum Gaussian states. We elaborate on the notion of quantum Gaussian channel as a noncommutative generalization of Gaussian kernel to show that the coherent states, and under certain conditions only they, minimize a broad class of the concave functionals of the output of a Gaussian channel. Thus, the output states corresponding to the Gaussian input are "the least chaotic", majorizing all the other outputs. The solution, however, is essentially restricted to the gauge-invariant case where a distinguished complex structure plays a special role. We also comment on the related famous additivity conjecture, which was solved in principle in the negative some five years ago. This refers to the additivity or multiplicativity (with respect to tensor products of channels) of information quantities related to the classical capacity of quantum channel, such as (1→p)(1\rightarrow p)(1→p)-norms or the minimal von Neumann or Rényi output entropies. A remarkable corollary of the present solution of the quantum Gaussian optimizers problem is that these additivity properties, while not valid in general, do hold in the important and interesting class of the gauge-covariant Gaussian channels.
We introduce and study multi-indexed Schatten spaces which lift the commutative ℓq1 ⁣[ℓq2 ⁣[ℓq3 ⁣[… ]]]\ell_{q_1}\![\ell_{q_2}\![\ell_{q_3}\![\dots]]]ℓq1​​[ℓq2​​[ℓq3​​[…]]] -spaces to the quantum toolbox. They extend Pisier's operator-valued Schatten norms and the recently introduced two-indexed Schatten quasi-norms. For every ordered tuple Q=(q1,…,qn)\mathcal{Q}=(q_1,\dots,q_n)Q=(q1​,…,qn​) of non-zero indices we define a quasi-norm (for positive indices) or an anti-norm (for negative indices) on operators on an nnn-fold tensor product of Hilbert spaces. We do this for indices of the same sign satisfying the mutual compatibility condition ∣1qi−1qj∣≤1|\frac{1}{q_i}-\frac{1}{q_j}|\leq1∣qi​1​−qj​1​∣≤1. Beyond establishing desirable properties of these quasi- and anti-norms via factorization formulas that combine operator space and matrix-analytic techniques, we lift several central tools and results of the seminal [Devetak, Junge, King, Ruskai, CMP 2006] to the multi-indexed and quasi-normed setting. These include a general non-commutative Minkowski inequality relating swapped systems and two identity-removal equalities. As a consequence, complete hypercontractivity and complete reverse hypercontractivity tensorize for all compatible indices, including negative ones. Concerning quantum information theory we prove that the completely bounded minimal and the maximal output conditional entropies are additive under tensor products of quantum channels for all α≥12α\geq\frac12α≥21​. This extends the case α>1α>1α>1 of [Fawzi, Kochanowski, Rouzé, Van Himbeeck, CMP 2026] and the unconditional results of [Devetak, Junge, King, Ruskai, CMP 2006]. Beyond tensor products, we establish chain rules for H~α↑\tilde H^\uparrow_αH~α↑​, α∈[12,1)α\in[\frac12,1)α∈[21​,1), for compositions of possibly signaling channels, extending upon the smooth max-entropy chain rule used in device-independent entanglement certification to the whole interval α∈[1/2,1)α\in[1/2,1)α∈[1/2,1).
We prove additivity of the minimal conditional entropy associated with a quantum channel Phi, represented by a completely positive (CP), trace-preserving map, when the infimum of S(gamma_{12}) - S(gamma_1) is restricted to states of the form gamma_{12} = (I \\ot Phi)(| psi >< psi |). We show that this follows from multiplicativity of the completely bounded norm of Phi considered as a map from L_1 -> L_p for L_p spaces defined by the Schatten p-norm on matrices; we also give an independent proof based on entropy inequalities. Several related multiplicativity results are discussed and proved. In particular, we show that both the usual L_1 -> L_p norm of a CP map and the corresponding completely bounded norm are achieved for positive semi-definite matrices. Physical interpretations are considered, and a new proof of strong subadditivity is presented.