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arXiv:2008.06503

Discriminating an Arbitrary Number of Pure Quantum States by the Combined $\mathcal{CPT}$ and Hermitian Measurements

Yaroslav Balytskyi, Sang‐Yoon Chang, Anatoliy O. Pinchuk, Manohar Raavi

Published Aug 16, 2020• 0 citations
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Abstract

If the system is known to be in one of two non-orthogonal quantum states, $|ψ_1\rangle$ or $|ψ_2\rangle$, $\mathcal{PT}$-symmetric quantum mechanics can discriminate them, \textit{in principle}, by a single measurement. We extend this approach by combining $\mathcal{PT}$-symmetric and Hermitian measurements and show that it's possible to distinguish an arbitrary number of pure quantum states by an appropriate choice of the parameters of $\mathcal{PT}$-symmetric Hamiltonian.

DOI
10.48550/arxiv.2008.06503

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Discriminating an Arbitrary Number of Pure Quantum States by the Combined CPT\mathcal{CPT}CPT and Hermitian Measurements
If the system is known to be in one of two non-orthogonal quantum states, ∣ψ1⟩|ψ_1\rangle∣ψ1​⟩ or ∣ψ2⟩|ψ_2\rangle∣ψ2​⟩, PT\mathcal{PT}PT-symmetric quantum mechanics can discriminate them, \textit{in principle}, by a single measurement. We extend this approach by combining PT\mathcal{PT}PT-symmetric and Hermitian measurements and show that it's possible to distinguish an arbitrary number of pure quantum states by an appropriate choice of the parameters of PT\mathcal{PT}PT-symmetric Hamiltonian.
If the system is known to be in one of two non-orthogonal quantum states, ∣ψ1⟩|\psi_1\rangle∣ψ1​⟩ or ∣ψ2⟩|\psi_2\rangle∣ψ2​⟩, PT\mathcal{PT}PT-symmetric quantum mechanics can discriminate them, \textit{in principle}, by a single measurement. We extend this approach by combining PT\mathcal{PT}PT-symmetric and Hermitian measurements and show that it's possible to distinguish an arbitrary number of pure quantum states by an appropriate choice of the parameters of PT\mathcal{PT}PT-symmetric Hamiltonian.
PT\mathcal{PT}PT-Symmetric Quantum Discrimination of Three States
If the system is known to be in one of two non-orthogonal quantum states, ∣ψ1⟩|ψ_1\rangle∣ψ1​⟩ or ∣ψ2⟩|ψ_2\rangle∣ψ2​⟩, it is not possible to discriminate them by a single measurement due to the unitarity constraint. In a regular Hermitian quantum mechanics, the successful discrimination is possible to perform with the probability p<1p < 1p<1, while in PT\mathcal{PT}PT-symmetric quantum mechanics a \textit{simulated single-measurement} quantum state discrimination with the success rate ppp can be done. We extend the PT\mathcal{PT}PT-symmetric quantum state discrimination approach for the case of three pure quantum states, ∣ψ1⟩|ψ_1\rangle∣ψ1​⟩, ∣ψ2⟩|ψ_2\rangle∣ψ2​⟩ and ∣ψ3⟩|ψ_3\rangle∣ψ3​⟩ without any additional restrictions on the geometry and symmetry possession of these states. We discuss the relation of our approach with the recent implementation of PT\mathcal{PT}PT symmetry on the IBM quantum processor.
The objective of this paper is to explain and elucidate the formalism of PT quantum mechanics by applying it to a well-known problem in conventional Hermitian quantum mechanics, namely the problem of state discrimination. Suppose that a system is known to be in one of two quantum states, |ψ(1)> or |ψ(2)>. If these states are not orthogonal, then the requirement of unitarity forbids the possibility of discriminating between these two states with one measurement; that is, determining with one measurement what state the system is in. In conventional quantum mechanics, there is a strategy in which successful state discrimination can be achieved with a single measurement but only with a success probability p that is less than unity. In this paper, the state-discrimination problem is examined in the context of PT quantum mechanics and the approach is based on the fact that a non-Hermitian PT-symmetric Hamiltonian determines the inner product that is appropriate for the Hilbert space of physical states. It is shown that it is always possible to choose this inner product so that the two states |ψ(1)> and |ψ(2)> are orthogonal. Using PT quantum mechanics, one cannot achieve a better state discrimination than in ordinary quantum mechanics, but one can instead perform a simulated quantum state discrimination, in which with a single measurement a perfect state discrimination is realized with probability p.
Quantum state discrimination (QSD) is a fundamental task in quantum information processing, improving the computation efficiency and communication security. Non-Hermitian (NH) PT-symmetric systems were found to be able to discriminate two quantum states better than the Hermitian strategy. In this work, we propose a QSD approach based on P-pseudo-Hermitian systems with real spectra. We theoretically prove the feasibility of realizing QSD in the real-spectrum phase of a P-pseudo-Hermitian system, i.e., two arbitrary non-orthogonal quantum states can be discriminated by a suitable P-pseudo-Hermitian Hamiltonian. In detail, we decide the minimal angular separation between two non-orthogonal quantum states for a fixed P-pseudo-Hermitian Hamiltonian, and we find the orthogonal evolution time is able to approach zero under suitable conditions, while both the trace distance and the quantum relative entropy are employed to judge their orthogonality. We give a criterion to choose the parameters of a P-pseudo-Hermitian Hamiltonian that evolves the two initial orthogonal states faster than a fixed arbitrary PT-symmetric one with an identical energy difference. Our work expands the NH family for QSD, and can be used to explore real quantum systems in the future.
Abstract Distinguishing physical processes is one of the fundamental problems in quantum physics. Although distinguishability of quantum preparations and quantum channels have been studied considerably, distinguishability of quantum measurements remains largely unexplored. We investigate the problem of single-shot discrimination of quantum measurements using two strategies, one based on single quantum systems and the other one based on entangled quantum systems. First, we formally define both scenarios. We then construct sets of measurements (including non-projective) in arbitrary finite dimensions that are perfectly distinguishable within the second scenario using quantum entanglement, while not in the one based on single quantum systems. Furthermore, we show that any advantage in measurement discrimination tasks over single systems is a demonstration of Einstein–Podolsky–Rosen ‘quantum steering’. Alongside, we prove that all pure two-qubit entangled states provide an advantage in a measurement discrimination task over one-qubit systems.
Abstract We address perfect discrimination of two separable states. When available states are restricted to separable states, we can theoretically consider a larger class of measurements than the class of measurements allowed in quantum theory. The framework composed of the class of separable states and the above extended class of measurements is a typical example of general probabilistic theories. In this framework, we give a necessary and sufficient condition to discriminate two separable pure states perfectly. In particular, we derive measurements explicitly to discriminate two separable pure states perfectly, and find that some non-orthogonal states are perfectly distinguishable. However, the above framework does not improve the capacity , namely, the maximum number of states that are simultaneously and perfectly distinguishable.