Discrimination Power of a Quantum Detector

Abstract

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...

Robustness of Measurement, Discrimination Games, and Accessible Information

“…Moreover, we also show that the robustness of measurement is naturally related to a single-shot generalisation of the accessible information of a quantum-classical channel. Thus although our starting point was different to previous work, we indeed find a close connection to many ideas already explored [1][2][3][4][5][6][7][8][9][10][11][12][13], as one might expect.…”

“…where the equality in the first line follows from the operational significance of the RoM as the optimal advantage for any discrimination game specified by an ensemble E , (13), and where we have defined the optimal ensemble E * for the measurement M with o M outcomes. The second inequality follows by the fact that the advantage for the optimal ensemble is not smaller than the advantage for the ensemble E * (with o M outcomes) which is the optimal ensemble for the measurement M .…”

“…The third inequality uses (5), i.e. that simulability implies that the guessing probability cannot be larger for any ensemble, and then the final equality uses again (13).…”

“…The study of how informative a quantum measurement is it not new. There has been a series of papers studying this question, from an information-theoretic perspective [1][2][3][4][5][6][7][8][9][10][11][12][13]. The novel approach we adopt here comes from taking a 'resourcetheoretic' point of view.…”

Quantum Network Discrimination

“…Hypothesis testing not only allows us to investigate the usually unavoidable error occurring when discriminating between two possible quantum states or channels, the framework has also proven useful in giving bounds, determining properties and proof operational interpretations of quantities such as the capacity of a quantum channel [13,46], the entanglement in a quantum state [5] and much more [10,25]. In the case of discriminating two quantum states, a wide body of literature is available determining the optimal single-copy and asymptotic errors in several different scenarios, see e.g.…”