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arXiv:1903.02655 · Bulletin of the American Physical Society

Maximal LELM Distinguishability of Qubit and Qutrit Bell States Using Projective and Non-Projective Measurements

Nathaniel Leslie, Julien Devin, T. W. Lynn

Published Mar 6, 2019• 1 citation
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Abstract

Numerous quantum information protocols make use of maximally entangled two-particle states, or Bell states, in which information is stored in the correlations between the two particles rather than their individual properties. Retrieving information stored in this way means distinguishing between different Bell states, yet the well known no-go theorem establishes that projective linear evolution and local measurement (LELM) detection schemes can only reliably distinguish three of the four qubit Bell states. We establish maximum distinguishability of the qutrit Bell states of bosons via projective LELM measurements; only three of the nine Bell states can be distinguished. Next, we extend to the case of non-projective measurements. We strengthen the no-go theorem by showing that general LELM measurements cannot reliably distinguish all four qubit Bell states. We also establish that at most five qutrit Bell states can be distinguished with generalized LELM measurements.

DOI
10.48550/arxiv.1903.02655

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Numerous quantum information protocols make use of maximally entangled two-particle states, or Bell states, in which information is stored in the correlations between the two particles rather than their individual properties. Retrieving information stored in this way means distinguishing between different Bell states, yet the well known no-go theorem establishes that projective linear evolution and local measurement (LELM) detection schemes can only reliably distinguish three of the four qubit Bell states. We establish maximum distinguishability of the qutrit Bell states of bosons via projective LELM measurements; only three of the nine Bell states can be distinguished. Next, we extend to the case of non-projective measurements. We strengthen the no-go theorem by showing that general LELM measurements cannot reliably distinguish all four qubit Bell states. We also establish that at most five qutrit Bell states can be distinguished with generalized LELM measurements.
Many quantum information protocols rely on the ability to distinguish between entangled quantum states known as Bell states. However, theoretical limits exist on the maximal distinguishability of these entangled states using linear evolution and local measurement (LELM) devices. In the case of two particles entangled in multiple qubit variables, the maximum number of distinguishable Bell states is known. However, in the more general case of two particles entangled in multiple qudit variables, only an upper bound is known under additional assumptions. I have written software in Matlab and Mathematica to explore computationally the maximum number of Bell states that can be distinguished in the case of two particles entangled in a qutrit variable, and the case of two particles entangled in both a qutrit and qubit variable. Using code I have written in Mathematica, I have reduced the number of cases to check for sets of 9 qubit x qutrit Bell states from 94,143,280 to 10,365. Further work needs to be done to computationally check these cases for distinguishability by an LELM apparatus.
Measuring an entangled state of two particles is crucial to many quantum communication protocols. Yet Bell-state distinguishability using a finite apparatus obeying linear evolution and local measurement is theoretically limited. We extend known bounds for Bell-state distinguishability in one and two variables to the general case of entanglement in nnn two-state variables. We show that at most ${2}^{n+1}\ensuremath{-}1$ classes out of 4n{4}^{n}4n hyper-Bell states can be distinguished with one copy of the input state. With two copies, complete distinguishability is possible. We present optimal schemes in each case.
Bell-state measurements are essential ingredients in many protocols for quantum information processing, ranging from quantum teleportation and dense coding to entanglement distribution in quantum networks. Their power relies on the fact that they are measurements in an entangled basis of a two-particle system and that the used Bell-state basis can be generated from a single Bell state by local unitary transformations. How can these measurements be generalized to more particles? We develop a general framework for this state-to-measurement problem: We introduce a hierarchy of classes of measurement bases, distinguished by the local transformations the parties may use for their generation from a single state. This leads to a generalization of the concept of maximally entangleable (or weighted hypergraph) states and the identification of a novel maximally entangled basis of four qubits, being a candidate for data-hiding tasks or distillation protocols. Finally, we prove that not all forms of entanglement can be encoded in an entire measurement basis.
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Entanglement of qudit pairs, with single-particle Hilbert space dimension d , has important potential for quantum information processing, with applications in cryptography, algorithms, and error correction. For a pair of qudits of arbitrary even dimension d , we introduce a generalized Bell basis with definite symmetry under exchange of internal states between the two particles. We show that no complete exchange-symmetrized basis can exist for odd d . This framework extends prior work on exchange-symmetrized hyperentangled qubit bases, where d is a power of 2. For our exchange-symmetrized basis, we show that measurement devices restricted to linear evolution and local measurement (LELM) can unambiguously distinguish 2 d − 1 qudit Bell states for any even d . This achieves the upper bound in general for reliable Bell-state distinguishability via LELM and augments previously known limits for d = 2 n and d = 3 . This result is relevant to near-term realizations of quantum communication protocols.