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arXiv:1109.1584 · Physical Review A

Distinguishability of hyperentangled Bell states by linear evolution and local projective measurement

Neal C. Pisenti, Carl Philipp Gaebler, T. W. Lynn

Published Aug 26, 2011• 39 citations
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Abstract

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 $n$ two-state variables. We show that at most ${2}^{n+1}\ensuremath{-}1$ classes out of ${4}^{n}$ 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.

DOI
10.1103/physreva.84.022340

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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 2n+1\ensuremath−1{2}^{n+1}\ensuremath{-}12n+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.
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.
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.
It is known that it is impossible to unambiguously distinguish the four Bell states encoded in pairs of photon polarizations using only linear optics. However, hyperentanglement, the simultaneous entanglement in more than one degree of freedom, has been shown to assist the complete Bell analysis of the four Bell states (given a fixed state of the other degrees of freedom). Yet introducing other degrees of freedom also enlarges the total number of Bell-like states. We investigate the limits for unambiguously distinguishing these Bell-like states. In particular, when the additional degree of freedom is qubitlike, we find that the optimal one-shot discrimination schemes are to group the 16 states into seven distinguishable classes, and that an unambiguous discrimination is possible with two identical copies.
Distinguishing limit of Bell states for any nnn-photon DDD-dimensional hyperentanglement
Bell state measurement is crucial to quantum information protocols, but it is impossible to unambiguously distinguish all the Bell states encoded in multi-photon using only linear optics. There is a maximum number of distinguished Bell states, i.e. distinguising limit which is very important for increasing the channel capacity of quantum communications. In this paper, we separate nnn-photon DDD-dimensional hyperentanglement into two groups. For the first group of UUU (G1G_1G1​), we obtain the limit N1=nD−(n−1){N_1} = nD - (n - 1)N1​=nD−(n−1), which can be applied for both bosons' and fermions' cases. We further discuss the limit NNN for any nDnDnD system with the second group of UUU (G2G_2G2​), inferring that at least Dn−1{D^{n - 1}}Dn−1 Bell states can be distinguished due to the symmetry of Bell states. Obviously, N1≤N2{N_1} \le {N_2}N1​≤N2​ for those systems with n>2n>2n>2. Finally, we theoretically design an optical setup for Bell state measurement of two-photon eight-dimensional hyperentanglement of spin, path and orbital angular momentum (OAM) and distinguish 15 classes of 64 Bell states. Our results provide a theoretical basis and practical reference to increase the channel capacity of the quantum communication.
It has been proven that, with a single copy provided, the four Bell states cannot be distinguished by local operations and classical communications (LOCC). Traditionally, a Bell basis projective measurement is needed to distinguish the four Bell states, which is usually carried out with a local interference between two particles. This paper presents an interesting protocol that allows two remote parties to distinguish four Bell states deterministically. We prove that our protocol of distinguishing remote Bell states is beyond LOCC.