Quantum Chernoff bound as a measure of distinguishability between density matrices: Application to qubit and Gaussian states

Abstract

Hypothesis testing is a fundamental issue in statistical inference and has been a crucial element in the development of information sciences. The Chernoff bound gives the minimal Bayesian error probability when discriminating two hypotheses given a large number of observations. Recently the combined work of Audenaert et al. ͓Phys. Rev. Lett. 98, 160501 ͑2007͔͒ and Nussbaum and Szkola ͓e-print arXiv:quant-ph/0607216͔ has proved the quantum analog of this bound, which applies when the hypotheses correspond to two quantum states. Based on this quantum Chernoff bound, we define a physically meaningful distinguishability measure and its corresponding metric in the space of states; the latter is shown to coincide with the Wigner-Yanase metric. Along the same lines, we define a second, more easily implementable, distinguishability measure based on the error probability of discrimination when the same local measurement is performed on every copy. We study some general properties of these measures, including the probability distribution of density matrices, defined via the volume element induced by the metric. It is shown that the Bures and the local-measurement based metrics are always proportional. Finally, we illustrate their use in the paradigmatic cases of qubits and Gaussian infinite-dimensional states.

Quantum probes for fractional Gaussian processes

“…We thus need to analyze the quantity P e,n = 1 2 1 − Tr 1 2 (|ρ ⊗n γ 2 − ρ ⊗n γ 1 )| . The evaluation of the trace distance for increasing n may be difficult and for this reason, one usually resort to the quantum Chernoff bound, which gives an upper bound to the probability of error [56,57,58,59,60,61] P e,n ≤ 1 2 Q n where…”

Monotone Riemannian metrics and dynamic structure factor in condensed matter physics

“…The quantum Chernoff metric also enjoys increasing popularity in the information approach in physics [39][40][41]. It appears under several different names, among which are (up to a factor of one forth) the Hellinger metric [4] or the Wigner-Yanase metric [21][22][23].…”

“…Much as for quantification of the asymptotic behavior of the error in the quantum state discrimination problem, one may define the quantum Chernoff metric d 2 QC , which is expressed in terms of the non-logarithmic variety of the quantum Chernoff bound [21][22][23]. The quantum Chernoff metric also enjoys increasing popularity in the information approach in physics [39][40][41].…”

“…The notion of distance has found applications in different fields. For example: thermodynamic of small systems [5,16], geometrical description of phase transitions and quantum criticality [6][7][8][9][10][11][12][13][14][15], quantum estimation of Hamiltonian parameters [18,19] and hypothesis testing and discrimination of states [20][21][22][23], are only a part of them. If we consider the distance between two states obtained by an infinitesimal change in the values of the parameters that specifie the quantum state, we come to the notion of a metric tensor, i.e.…”

Adaptive Quantum State Tomography Improves Accuracy Quadratically

“…Unlike alternative metrics, 1 − F (ρ, ρ) quantifies an important operational quantity: how many copies are required to reliably distinguishρ from ρ?. Without doing justice to the rich body of research behind this simple statement (e.g., [8][9][10][11][12][13]. .…”