Optimization of Quantum States for Signaling Across an Arbitrary Qubit Noise Channel With Minimum-Error Detection

Abstract

For discrimination between two signaling states of a qubit, the optimal detector minimizing the probability of error is applied to the situation where detection has to be performed from a noisy qubit affected by an arbitrary quantum noise separately characterized. With no noise, any pair of orthogonal pure quantum states is optimal for signaling as it enables error-free detection. In the presence of noise, detection errors are in general inevitable, and the pairs of signaling states best resistant to such noise are investigated. With an arbitrary quantum noise, modeled as a channel affecting the qubit, and when minimum-error detection is performed from the output, a characterization of the optimal input signaling pairs and of their best detection performance is obtained through a simple maximization of a quadratic scalar criterion in three constrained real variables. This general characterization enables to establish that such optimal signaling pairs are always made of two orthogonal pure quantum states, but that they must be specifically selected to match the noise properties and prior probabilities. The maximization is explicitly solved for several generic quantum noise processes relevant to the qubit, such as the squeezed generalized amplitude damping noise which describes interaction with a thermal bath representing a decohering environment and which includes as special cases both the generalized and the regular amplitude damping noise processes, and such as general Pauli noise processes which include for instance the bit-flip noise and the depolarizing noise. Also, examined is the situation of one imposed (pure or mixed) signaling state, for which the other associated signaling state optimal for noisy detection is determined as a pure state, yet not necessarily orthogonal.Index Terms-Quantum state discrimination, quantum detection, quantum noise, noisy qubit, decoherence.

Stochastic Resonance with Unital Quantum Noise

“…The action of a quantum noise is generally representable by a completely positive trace-preserving superoperator N ðÁÞ producing the noisy quantum state ¼ N ð 1 ðÞÞ. This is equivalent to a Bloch vector r specifying supplied by the a±ne transformation [28,30]…”

Optimizing qubit phase estimation

“…On a qubit state ρ 1 (ξ ) with Bloch vector r 1 (ξ ), the action of the quantum noise expressed by Eq. (32) can always be described as an affine transformation on the Bloch vectors reading [20,24,26]…”

Qubit state detection and enhancement by quantum thermal noise

“…Qubit detection with noise: We address the common problem in quantum communication or binary coding consisting of detecting or discriminating between two noisy quantum states [1,2]. We apply the approach to the qubit, which stands as a fundamental reference for quantum information, representing for instance situations with photons with their two states of polarisation or electrons with their two states of spin, individually serving as information carrier.…”

“…The qubit is then altered by decoherence as a quantum noise producing the noisy state r r ′ = N (r). The noise is generally represented by the completely positive trace-preserving superoperator N (•) acting on the Bloch vectors according to the affine map [2,3] r…”

“…We note that our general condition P 0 ≤ P 1 implies 0 ≤ P 1 − P 0 ≤ 1, and that t = P 1 r 1 − P 0 r 0 is a vector of R 3 which can be anywhere in the Bloch ball satisfying t ≤ 1. We now want to study the impact of a specific quantum noise of great relevance to the qubit, which is the generalised amplitude damping noise or quantum thermal noise [2,3]. It is defined in (2) by…”