Minimum-error discrimination of quantum states: New bounds and comparison

Abstract

The minimum-error probability of ambiguous discrimination for two quantum states is the well-known {\it Helstrom limit} presented in 1976. Since then, it has been thought of as an intractable problem to obtain the minimum-error probability for ambiguously discriminating arbitrary $m$ quantum states. In this paper, we obtain a new lower bound on the minimum-error probability for ambiguous discrimination and compare this bound with six other bounds in the literature. Moreover, we show that the bound between ambiguous and unambiguous discrimination does not extend to ensembles of more than two states. Specifically, the main technical contributions are described as follows: (1) We derive a new lower bound on the minimum-error probability for ambiguous discrimination among arbitrary $m$ mixed quantum states with given prior probabilities, and we present a necessary and sufficient condition to show that this lower bound is attainable. (2) We compare this new lower bound with six other bounds in the literature in detail, and, in some cases, this bound is optimal. (3) It is known that if $m=2$, the optimal inconclusive probability of unambiguous discrimination $Q_U$ and the minimum-error probability of ambiguous discrimination $Q_E$ between arbitrary given $m$ mixed quantum states have the relationship $Q_U\ge 2Q_E$. In this paper, we show that, however, if $m>2$, the relationship $Q_U\ge 2Q_E$ may not hold again in general, and there may be no supremum of $Q_U/Q_E$ for more than two states, which may also reflect an essential difference between discrimination for two-states and multi-states. (4) A number of examples are constructed.

25 pages; some new results have been added while some proofs have been shortened; comments are welcome

References in corpus (19)

• Quantum Cryptography
• Experimental Quantum State Tomography of Optical Fields and Ultrafast Statistical Sampling
• Unknown Quantum States and Operations, a Bayesian View
• Unambiguous Discrimination Between Linearly-Independent Quantum States
• Reversing quantum dynamics with near-optimal quantum and classical fidelity
• Optimum Unambiguous Discrimination Between Linearly Independent Symmetric States
• Designing Optimal Quantum Detectors Via Semidefinite Programming
• Unambiguous discrimination of mixed states
• Optimum unambiguous discrimination of two mixed quantum states
• Optimal discrimination of mixed quantum states involving inconclusive results
• Minimum-error discrimination between three mirror-symmetric states
• Experimental realization of optimal detection strategies for overcomplete states
• Unambiguous discrimination between quantum mixed states
• Minimum-error discrimination between symmetric mixed quantum states
• Reduction Theorems for Optimal Unambiguous State Discrimination of Density Matrices
• Minimum-error discrimination between mixed quantum states
• Distinguishing mixed quantum states: Minimum-error discrimination versus optimum unambiguous discrimination
• Multiparty data hiding of quantum information
• A lower bound on the probability of error in quantum state discrimination

Cited by in corpus (1)

• Error rates of Belavkin weighted quantum measurements and a converse to Holevo's asymptotic optimality theorem