Recently, there.have lots of work in quantum hypothesis testing that lies in the core of fascinating applications in quantum information theory.They mainly deal with a simple hypothesis because their main concern is the dis- tinguishability of two quantum states.However, in a practical application, composite hypothesis testing is much important as in classical statistics.In the present paper, we focus on the unbiased test in the quantum composite hypothesis testing.We derive necessary and sufficient conditions on exis- tence of a nontrivial unbiased test in the finite-dimensional Hilbert space.As a practical application, we consider the statistical hypothesis testing of entanglements.It is shown that the uniformly most powerful unbiased test is the random guess for the hypothesis testing.It implies that quan- tum fluctuation conflicts with the concept of hypothesis testing in classical statistics.
Previous studies in quantum information have recognized that specific types of noise can encode information in certain applications. However, the role of noise in quantum hypothesis testing, traditionally assumed to undermine performance, has not been thoroughly explored. Our study provides sufficient conditions for general noisy dynamics to surpass noiseless (unitary) dynamics within certain time interval. We then design and experimentally implement a noise-assisted quantum hypothesis testing protocol on ultralow-field nuclear magnetic resonance systems, which demonstrates that the success probability under certain noisy dynamics can indeed surpass the ceiling set by unitary evolution. Moreover, we show that in cases where noise initially hampers performance, strategic application of coherent controls on the system can transform those previously detrimental noises into advantageous ones. Our results, both theoretical and experimental, demonstrates the potential to leveraging noise in quantum hypothesis testing, which pushes the boundaries of quantum hypothesis testing and general quantum information processing. In quantum science, quantum hypothesis testing (QHT) is used to determine the model of a given quantum system, but normally the presence of inherent quantum noise hampers perfect hypothesis testing. The authors theoretically and experimentally explore the potential of leveraging noise in quantum hypothesis testing (QHT) to surpass the success probabilities achievable under noiseless dynamics.
The task of binary quantum hypothesis testing is to determine the state of a quantum system via measurements on it, given the side information that it is in one of two possible states, say
ρ and
σ. This task is generally studied in either the symmetric setting, in which the two possible errors incurred in the task (the so-called type I and type II errors) are treated on an equal footing, or the asymmetric setting in which one minimizes the type II error probability under the constraint that the corresponding type I error probability is below a given threshold. Here we define a one-parameter family of binary quantum hypothesis testing tasks, which we call
s-hypothesis testing, and in which the relative significance of the two errors are weighted by a parameter
s. In particular,
s-hypothesis testing interpolates continuously between the regimes of symmetric and asymmetric hypothesis testing. Moreover, if arbitrarily many identical copies of the system are assumed to be available, then the minimal error probability of
s-hypothesis testing is shown to decay exponentially in the number of copies, with a decay rate given by a quantum divergence which we denote as
ξs(ρ∥σ), and which satisfies a host of interesting properties. Moreover, this one-parameter family of divergences interpolates continuously between the corresponding decay rates for symmetric hypothesis testing (the quantum Chernoff divergence) for
s=1, and asymmetric hypothesis testing (the Umegaki relative entropy) for
s=0.