Comparison of noisy channels and reverse data-processing theorems

Buscemi, Francesco

doi:10.1109/itw.2017.8278038

Cited by 12 publications

(14 citation statements)

References 31 publications

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“…We call a (possibly infinite) family of monotones a complete set of monotones if it fully characterizes the necessary and sufficient conditions for the existence of a free transformation. Such sets of monotones were discussed in several specific settings [25,26,32,[121][122][123][124][125][126][127][128][129][130][131], but no set of general conditions with a clear operational meaning was previously known to provide a comprehensive characterization of transformations in general resource theories.…”

Section: Complete Sets Of Monotonesmentioning

confidence: 99%

General Resource Theories in Quantum Mechanics and Beyond: Operational Characterization via Discrimination Tasks

2019

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We establish an operational characterization of general convex resource theories -describing the resource content of not only states, but also measurements and channels, both within quantum mechanics and in general probabilistic theories (GPTs) -in the context of state and channel discrimination. We find that discrimination tasks provide a unified operational description for quantification and manipulation of resources by showing that the family of robustness measures can be understood as the maximum advantage provided by any physical resource in several different discrimination tasks, as well as establishing that such discrimination problems can fully characterize the allowed transformations within the given resource theory.Specifically, we introduce a quantifier of resourcefulness of a measurement in any GPT, the generalized robustness of measurement, and show that it admits an operational interpretation as the maximum advantage that a given measurement provides over resourceless measurements in all state discrimination tasks. In the special case of quantum mechanics, we connect discrimination problems with single-shot information theory by showing that the generalized robustness of any measurement can be alternatively understood as the maximal increase in one-shot accessible information when compared to free measurements. We introduce two different approaches to quantifying the resource content of a physical channel based on the generalized robustness measures, and show that they quantify the maximum advantage that a resourceful channel can provide in several classes of state and channel discrimination tasks. Furthermore, we endow another measure of resourcefulness of states, the standard robustness, with an operational meaning in general GPTs as the exact quantifier of the maximum advantage that a state can provide in binary channel discrimination tasks. Finally, we establish that several classes of channel and state discrimination tasks form complete families of monotones fully characterizing the transformations of states and measurements, respectively, under general classes of free operations. Our results establish a fundamental connection between the operational tasks of discrimination and core concepts of resource theories -the geometric quantification of resources and resource manipulation -valid for all physical theories beyond quantum mechanics with no additional assumptions about the structure of the GPT

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Section: Complete Sets Of Monotonesmentioning

confidence: 99%

General Resource Theories in Quantum Mechanics and Beyond: Operational Characterization via Discrimination Tasks

2019

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“…We used the fact that Θ is a superchannel iff (Θ T ) * is a superchannel. To see why, note that from (15) and (52) it follows that J AB (Θ T ) * = J AB Θ , and therefore J AB (Θ T ) * satisfies the conditions (19) of a superchannel. For the last equality we used the definition (2) of an inner product between channels, and took an orthonormal basis {X a } of B(HB 0B0 ), whose first element X 1 = 1…”

Section: A the Extended Conditional Min-entropymentioning

confidence: 99%

“…More recently, it was solved for pure states in [9], characterized in [10], [11], [12], [13], and finally, in [14] it was fully solved (for finite dimensions) with semidefinite programming. In [14], [15] (see also references therein) it was also shown that this pre-order can be characterized completely in terms of a family of distinguishability measures that are given in terms of the conditional min-entropy [16], [17].…”

Section: Introductionmentioning

confidence: 99%

Comparison of Quantum Channels by Superchannels

Gour

2019

IEEE Trans. Inform. Theory

123

156

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We extend the definition of the conditional minentropy from bipartite quantum states to bipartite quantum channels. We show that many of the properties of the conditional min-entropy carry over to the extended version, including an operational interpretation as a guessing probability when one of the subsystems is classical. We then show that the extended conditional min-entropy can be used to fully characterize when two bipartite quantum channels are related to each other via a superchannel (also known as supermap or a comb) that is acting on one of the subsystems. This relation is a preorder that extends the definition of "quantum majorization" from bipartite states to bipartite channels, and can also be characterized with semidefinite programming. As a special case, our characterization provides necessary and sufficient conditions for when a set of quantum channels is related to another set of channels via a single superchannel. We discuss the applications of our results to channel discrimination, and to resource theories of quantum processes. Along the way we study channel divergences, entropy functions of quantum channels, and noise models of superchannels, including random unitary superchannels, and doubly-stochastic superchannels. For the latter we give a physical meaning as being completely-uniformity preserving.

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“…Mathematically, the question is to determine, for fixed states ρ, σ, τ , and ω, whether there exists a completely positive and trace-preserving map N such that N (ρ) = τ and N (σ) = ω. As it turns out, various instantiations of this question have been studied considerably in prior work [Bla53, AU80, CJW04, MOA11, Bus12, HJRW12, BDS14, BaHN + 15, Ren16, BD16, Bus16, GJB + 18, Bus17,BG17], and a variety of results are known regarding it. In this paper, we offer a fresh resource-theoretic perspective on this matter.…”

Section: Resource Theory Of Asymmetric Distinguishabilitymentioning

confidence: 99%

Resource theory of asymmetric distinguishability

Wang

Wilde

2019

Phys. Rev. Research

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This paper systematically develops the resource theory of asymmetric distinguishability, as initiated roughly a decade ago [K. Matsumoto, arXiv:1006.0302 (2010]. The key constituents of this resource theory are quantum boxes, consisting of a pair of quantum states, which can be manipulated for free by means of an arbitrary quantum channel. We introduce bits of asymmetric distinguishability as the basic currency in this resource theory, and we prove that it is a reversible resource theory in the asymptotic limit, with the quantum relative entropy being the fundamental rate of resource interconversion. The distillable distinguishability is the optimal rate at which a quantum box consisting of independent and identically distributed (i.i.d.) states can be converted to bits of asymmetric distinguishability, and the distinguishability cost is the optimal rate for the reverse transformation. Both of these quantities are equal to the quantum relative entropy. The exact one-shot distillable distinguishability is equal to the min-relative entropy, and the exact oneshot distinguishability cost is equal to the max-relative entropy. Generalizing these results, the approximate one-shot distillable distinguishability is equal to the smooth min-relative entropy, and the approximate one-shot distinguishability cost is equal to the smooth max-relative entropy. As a notable application of the former results, we prove that the optimal rate of asymptotic conversion from a pair of i.i.d. quantum states to another pair of i.i.d. quantum states is fully characterized by the ratio of their quantum relative entropies.

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Comparison of noisy channels and reverse data-processing theorems

Cited by 12 publications

References 31 publications

General Resource Theories in Quantum Mechanics and Beyond: Operational Characterization via Discrimination Tasks

General Resource Theories in Quantum Mechanics and Beyond: Operational Characterization via Discrimination Tasks

Comparison of Quantum Channels by Superchannels

Resource theory of asymmetric distinguishability

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