Strong Converse Exponents for a Quantum Channel Discrimination Problem and Quantum-Feedback-Assisted Communication

Cooney, Tom; Mosonyi, Milán; Wilde, Mark M.

doi:10.1007/s00220-016-2645-4

Cited by 142 publications

(123 citation statements)

References 69 publications

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“…The sandwiched Rényi relative entropy [MLDS + 13, WWY14] is another variant of the Rényi relative entropy which has found a number of applications recently in the context of strong converse theorems [WWY14,MO15,GW15,CMW14,TWW14]. It is defined for α ∈ (0, 1)∪(1, ∞) as follows:…”

Section: Rényi Entropies and Information Measuresmentioning

confidence: 99%

Rényi squashed entanglement, discord, and relative entropy differences

Seshadreesan¹,

Berta²,

Wilde³

2015

J. Phys. A: Math. Theor.

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The squashed entanglement quantifies the amount of entanglement in a bipartite quantum state, and it satisfies all of the axioms desired for an entanglement measure. The quantum discord is a measure of quantum correlations that are different from those due to entanglement. What these two measures have in common is that they are both based upon the conditional quantum mutual information. In [Berta et al., J. Math. Phys. 56, 022205 (2015)], we recently proposed Rényi generalizations of the conditional quantum mutual information of a tripartite state on ABC (with C being the conditioning system), which were shown to satisfy some properties that hold for the original quantity, such as non-negativity, duality, and monotonicity with respect to local operations on the system B (with it being left open to show that the Rényi quantity is monotone with respect to local operations on system A). Here we define a Rényi squashed entanglement and a Rényi quantum discord based on a Rényi conditional quantum mutual information and investigate these quantities in detail. Taking as a conjecture that the Rényi conditional quantum mutual information is monotone with respect to local operations on both systems A and B, we prove that the Rényi squashed entanglement and the Rényi quantum discord satisfy many of the properties of the respective original von Neumann entropy based quantities. In our prior work [Berta et al., Phys. Rev. A 91, 022333 (2015)], we also detailed a procedure to obtain Rényi generalizations of any quantum information measure that is equal to a linear combination of von Neumann entropies with coefficients chosen from the set {−1, 0, 1}. Here, we extend this procedure to include differences of relative entropies. Using the extended procedure and a conjectured monotonicity of the Rényi generalizations in the Rényi parameter, we discuss potential remainder terms for well known inequalities such as monotonicity of the relative entropy, joint convexity of the relative entropy, and the Holevo bound.

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Section: Rényi Entropies and Information Measuresmentioning

confidence: 99%

Rényi squashed entanglement, discord, and relative entropy differences

Seshadreesan¹,

Berta²,

Wilde³

2015

J. Phys. A: Math. Theor.

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“…Note that this state in general depends on the output of the previous measurement and their application therefore constitutes an adaptive strategy. In general the input state to the j-th application of the measurement can be chosen as theory [29][30][31][32][33], particularly in quantum channel discrimination [34,35]. It follows from the structure of E k n in Eq.…”

mentioning

confidence: 99%

“…theory [29][30][31][32][33], particularly in quantum channel discrimination [34,35]. It follows from the structure of E k n in Eq.…”

mentioning

confidence: 99%

Discrimination Power of a Quantum Detector

et al. 2017

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We investigate the ability of a quantum measurement device to discriminate two states or, generically, two hypothesis. In full generality, the measurement can be performed a number n of times, and arbitrary pre-processing of the states and post-processing of the obtained data is allowed. Even if the two hypothesis correspond to orthogonal states, perfect discrimination is not always possible. There is thus an intrinsic error associated to the measurement device, which we aim to quantify, that limits its discrimination power. We minimize various error probabilities (averaged or constrained) over all pairs of n-partite input states. These probabilities, or their exponential rates of decrease in the case of large n, give measures of the discrimination power of the device. For the asymptotic rate of the averaged error probability, we obtain a Chernoff-type bound, dual to the standard Chernoff bound for which the state pair is fixed and the optimization is over all measurements. The key point in the derivation is that i.i.d. states become optimal in asymptotic settings. Minimum asymptotic rates are also obtained for constrained error probabilities, dual to Stein's Lemma and Hoeffding's bound. We further show that adaptive protocols where the state preparer gets feedback from the measurer do not improve the asymptotic rates. These rates thus quantify the ultimate discrimination power of a measurement device.Quantum-enabled technologies exploit the laws that govern the microscopic world to outperform their classical counterparts. Detectors, or measurement devices, are a key ingredient in quantum protocols. They are the interface that connects the microscopic world of quantum phenomena and the world of classical, macroscopically distinct, events that we observe. It is only through measurements that we can access the information residing in quantum systems and ultimately make use of any quantum advantage.We often encounter experimental situations where measurement devices (e.g., Stern Gerlach apparatus, heterodyne detectors, photon counters, fluorescence spectrometers) are a given. A natural question is then to ask about the ability or power of those devices to perform certain quantum informationprocessing tasks. The informational power of a measurement has been addressed in several ways [1], e.g., via the "intrinsic data" it provides [2] or the capacity of the quantum-classical channel it defines [1, 3-6], or via some associated entropic quantities [7][8][9].In this letter we focus on what is arguably the most fundamental primitive in quantum information processing: state discrimination, or generically, quantum hypothesis testing. Our aim is to explore how well a quantum measurement device can discriminate two hypotheses. This problem is dual to that of exploring how well two given quantum states can be discriminated [10]. This is of practical interest since preparing states is often easier than tailoring optimal measurements for a given state pair.In a generic discrimination protocol the measurement is performed not ju...

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“…A direct proof of this can be found in [26] (see also [27]). Furthermore, Bowen [10] (alternatively again the Quantum Reverse Shannon Theorem) showed that feedback does not increase the entanglement-assisted capacity.…”

Section: Proof (Of Theorem 15)mentioning

confidence: 77%

On Zero-Error Communication via Quantum Channels in the Presence of Noiseless Feedback

Duan

Severini

Winter

2016

IEEE Trans. Inform. Theory

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We initiate the study of zero-error communication via quantum channels when the receiver and sender have at their disposal a noiseless feedback channel of unlimited quantum capacity, generalizing Shannon's zero-error communication theory with instantaneous feedback.We first show that this capacity is a function only of the linear span of Choi-Kraus operators of the channel, which generalizes the bipartite equivocation graph of a classical channel, and which we dub "non-commutative bipartite graph". Then we go on to show that the feedback-assisted capacity is non-zero (allowing for a constant amount of activating noiseless communication) if and only if the non-commutative bipartite graph is non-trivial, and give a number of equivalent characterizations. This result involves a far-reaching extension of the "conclusive exclusion" of quantum states [Pusey/Barrett/Rudolph, Nature Phys. 8(6):475-478, 2012].We then present an upper bound on the feedback-assisted zero-error capacity, motivated by a conjecture originally made by Shannon and proved later by Ahlswede. We demonstrate this bound to have many good properties, including being additive and given by a minimax formula. We also prove a coding theorem showing that this quantity is the entanglement-assisted capacity against an adversarially chosen channel from the set of all channels with the same Choi-Kraus span, which can also be interpreted as the feedbackassisted unambiguous capacity. The proof relies on a generalization of the "Postselection Lemma" (de Finetti reduction) [Christandl/König/Renner, Phys. Rev. Lett. 102:020504, 2009] that allows to reflect additional constraints, and which we believe to be of independent interest. This capacity is a relaxation of the feedback-assisted zero-error capacity; however, we have to leave open the question of whether they coincide in general.We illustrate our ideas with a number of examples, including classical-quantum channels and Weyl diagonal channels, and close with an extensive discussion of open questions. a A preliminary version of this paper was presented as a poster at QIP 2012, 12-16 December 2011

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Strong Converse Exponents for a Quantum Channel Discrimination Problem and Quantum-Feedback-Assisted Communication

Cited by 142 publications

References 69 publications

Rényi squashed entanglement, discord, and relative entropy differences

Rényi squashed entanglement, discord, and relative entropy differences

Discrimination Power of a Quantum Detector

On Zero-Error Communication via Quantum Channels in the Presence of Noiseless Feedback

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