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arXiv:2012.14897

$\mathcal{PT}$-Symmetric Quantum Discrimination of Three States

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

Published Dec 29, 2020• 2 citations
arXivHTMLPDF

Abstract

If the system is known to be in one of two non-orthogonal quantum states, $|ψ_1\rangle$ or $|ψ_2\rangle$, 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 $p < 1$, while in $\mathcal{PT}$-symmetric quantum mechanics a \textit{simulated single-measurement} quantum state discrimination with the success rate $p$ can be done. We extend the $\mathcal{PT}$-symmetric quantum state discrimination approach for the case of three pure quantum states, $|ψ_1\rangle$, $|ψ_2\rangle$ and $|ψ_3\rangle$ 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 $\mathcal{PT}$ symmetry on the IBM quantum processor.

DOI
10.48550/arxiv.2012.14897

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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.
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.
As a fundamental and challenging problem in quantum information processing, quantum state discrimination (QSD) has recently been revisited from the perspective of non-Hermitian (NH) physics. The existing work about QSD in NH systems is mostly limited to PT-symmetric or pseudo-Hermitian systems with real spectra, while generic NH Hamiltonians possess complex spectra. To explore the underlying physics of quantum state discrimination in NH systems, we first demonstrate that the unambiguous discrimination of non-orthogonal quantum states can be realized with P-pseudo-Hermitian and PT-symmetric Hamiltonians in the broken phase and the required evolution time can be arbitrarily small. We further extend the regime to generic non-Hermitian Hamiltonians and demonstrate the feasibility of unambiguous discrimination. Under the same energy constraint, we establish the criteria for constructing P-pseudo-Hermitian Hamiltonians that enable unambiguous discrimination of quantum states with smaller angular separation or within shorter evolution time than any fixed PT-symmetric system. Furthermore, we also discuss the potential impact of exceptional points (EPs) on QSD. Our results suggest that the non-orthogonal eigenstates of NH Hamiltonians, rather than PT symmetry or pseudo-Hermitian symmetry, are fundamental to unambiguous discrimination and information flow in NH systems.
Abstract We present a novel PT\mathcal{PT}PT-symmetric approach for discriminating three pure qubit states and its implementation by the dilation method. Our approach based on the dilation method manipulating the Hilbert space of the system by two-staged PT\mathcal{PT}PT-symmetric evolution expedites the three-state discrimination process at the cost of introducing an inconclusive outcome. This modification of the Hilbert space of the qubit makes it possible to eliminate one of three states in the limit α→±π2\alpha\rightarrow\pm\frac{\pi}{2}α→±2π​ in the vicinity of the exceptional point corresponding to coalescing eigenvectors and eigenvalues, effectively reducing the problem to two-state discrimination, or reducing the geometry to the mirror-symmetric states for which an efficient discrimination strategy is developed. To demonstrate the effectiveness of our approach, we implement it on a superconducting quantum processor provided by IBM Quantum Experience employing an ancilla qubit. We conclude our paper by identifying potential applications, including those in quantum communication and cryptography, to facilitate further research and development.
PT -symmetric systems have garnered significant attention due to their unconventional properties. Despite the growing interest, there remains an ongoing debate about whether these systems outperform their Hermitian counterparts in practical applications, and if so, by what metrics this performance should be measured. We developed PT -symmetric approach for mapping N=3 pure qubit states to address this, implemented it using the dilation method, and demonstrated it on a superconducting quantum processor from the IBM Quantum Experience. For the first time, we derived exact expressions for the population of the post-selected PT -symmetric subspace for both N=2 and N=3 states. When applied to the discrimination of N=2 pure states, our algorithm provides an equivalent result to the conventional unambiguous quantum state discrimination. For N=3 states, our approach introduces novel capabilities not available in traditional Hermitian systems, enabling the transformation of an arbitrary set of three pure quantum states into another, at the cost of introducing an inconclusive outcome. Our algorithm has the same error rate for the attack on the three-state QKD protocol as the conventional minimum error, maximum confidence, and maximum mutual information strategies. For post-selected quantum metrology, our results provide precise conditions where PT -symmetric quantum sensors outperform their Hermitian counterparts in terms of information-cost rate. Combined with punctuated unstructured quantum database search, our method significantly reduces the qubit readout requirements at the cost of adding an ancilla, while maintaining the same average number of oracle calls as the original punctuated Grover's algorithm. Our work opens new pathways for applying PT symmetry in quantum communications, computing, and cryptography.