Operational meaning of quantum measures of recovery

Cooney, Tom; Hirche, Christoph; Morgan, Ciara; Olson, Jonathan P.; Seshadreesan, Kaushik P.; Watrous, John; Wilde, Mark M.

doi:10.1103/physreva.94.022310

Cited by 17 publications

(19 citation statements)

References 66 publications

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“…With the above arguments we can now show in an intuitive way that channels of this form achieve equality for the quantum side of our conjecture. where the first inequality follows from the chain rule for mutual information, the second one from Equation 44, the third from the classical Mrs. Gerbers Lemma and the final one from Equation 43. Note that the equality holds because in the classical Mrs. Gerbers Lemma binary symmetric channels achieve equality.…”

Section: A Quantum States That Achieve Equalitymentioning

confidence: 99%

Bounds on Information Combining With Quantum Side Information

Hirche

Reeb

2018

IEEE Trans. Inform. Theory

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Bounds on information combining" are entropic inequalities that determine how the information (entropy) of a set of random variables can change when these are combined in certain prescribed ways. Such bounds play an important role in classical information theory, particularly in coding and Shannon theory; entropy power inequalities are special instances of them. The arguably most elementary kind of information combining is the addition of two binary random variables (a CNOT gate), and the resulting quantities play an important role in Belief propagation and Polar coding. We investigate this problem in the setting where quantum side information is available, which has been recognized as a hard setting for entropy power inequalities.Our main technical result is a non-trivial, and close to optimal, lower bound on the combined entropy, which can be seen as an almost optimal "quantum Mrs. Gerber's Lemma". Our proof uses three main ingredients: (1) a new bound on the concavity of von Neumann entropy, which is tight in the regime of low pairwise state fidelities; (2) the quantitative improvement of strong subadditivity due to Fawzi-Renner, in which we manage to handle the minimization over recovery maps; (3) recent duality results on classical-quantum-channels due to Renes et al. We furthermore present conjectures on the optimal lower and upper bounds under quantum side information, supported by interesting analytical observations and strong numerical evidence.We finally apply our bounds to Polar coding for binary-input classical-quantum channels, and show the following three results: (A) Even non-stationary channels polarize under the polar transform. (B) The blocklength required to approach the symmetric capacity scales at most sub-exponentially in the gap to capacity. (C) Under the aforementioned lower bound conjecture, a blocklength polynomial in the gap suffices.C. Hirche is with the

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Section: A Quantum States That Achieve Equalitymentioning

confidence: 99%

Bounds on Information Combining With Quantum Side Information

Hirche

Reeb

2018

IEEE Trans. Inform. Theory

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“…Similarly, the exponential rate of β is given by Stein's Lemma [14,15] if an upperbound is set on α, or by Hoeffding's bound [16][17][18] if instead a lowerbound is set on the exponential rate of α [19]. These asymptotic bounds have found many useful applications in quantum information theory, such as providing an alternative proof of the classical capacity of a quantum channel [20,21], giving operational meaning to abstract quantities [22], quantum reading [23], or in quantum illumination [24,25].Coming back to our original problem, we wish to assess the discrimination power of a device given by a specific POVM, E = {E 1 , . .…”

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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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“…By Uhlmann's theorem, there exists an extension σ ABXaX b of σ AB such that P ρ ′ ABXaX b , σ ABXaX b ≤ ε with the X a -and X b -registers classical in the same basis as in Eq. (14). Additionally, the extension can be cho-…”

Section: Entanglement Measuresmentioning

confidence: 99%

“…and its regularized version D ∞ (A; B|C) ρ [7,30]. The latter quantity has an operational interpretation in composite asymmetric quantum hypothesis testing as the asymptotic exponential rate of mistakenly identifying ρ ABC instead of a corresponding locally recovered state (I B ⊗ R C→BC )(ρ AC ) [14]. Moreover, it was recently shown that [18]…”

Section: Entanglement Measuresmentioning

confidence: 99%

Disentanglement Cost of Quantum States

Berta

Majenz

2018

Phys. Rev. Lett.

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We show that the minimal rate of noise needed to catalytically erase the entanglement in a bipartite quantum state is given by the regularized relative entropy of entanglement. This offers a solution to the central open question raised in [Groisman et al., PRA 72, 032317 (2005)] and complements their main result that the minimal rate of noise needed to erase all correlations is given by the quantum mutual information. We extend our discussion to the tripartite setting where we show that an asymptotic rate of noise given by the regularized relative entropy of recovery is sufficient to catalytically transform the state to a locally recoverable version of the state.

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Operational meaning of quantum measures of recovery

Cited by 17 publications

References 66 publications

Bounds on Information Combining With Quantum Side Information

Bounds on Information Combining With Quantum Side Information

Discrimination Power of a Quantum Detector

Disentanglement Cost of Quantum States

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