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arXiv:0706.2287

Simulation of two spin-$s$ singlet correlations for all $s$ involving spin measurements

Ali Ahanj, Pramod S. Joag, Sibasish Ghosh A

Published Jun 15, 2007• 0 citations
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Abstract

In a recent paper [A. Ahanj et al., quant-ph/0603053], we gave a classical protocol to simulate quantum correlations corresponding to the spin $s$ singlet state for the infinite sequence of spins satisfying $2s+1 = 2^{n}$. In the present paper, we have generalized this result by giving a classical protocol to exactly simulate quantum correlations implied by the spin-$s$ singlet state corresponding to all integer as well as half-integer spin values $s$. The class of measurements we consider here are only those corresponding to spin observables, as has been done in the above-mentioned paper. The required amount of communication is found to be $\lceil {\rm log}_{2} (s + 1) \rceil$ in the worst case scenario, where $\lceil x \rceil$ is the least integer greater than or equal to $x$.

DOI
10.48550/arxiv.0706.2287

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Simulation of two spin-sss singlet correlations for all sss involving spin measurements
In a recent paper [A. Ahanj et al., quant-ph/0603053], we gave a classical protocol to simulate quantum correlations corresponding to the spin sss singlet state for the infinite sequence of spins satisfying 2s+1=2n2s+1 = 2^{n}2s+1=2n. In the present paper, we have generalized this result by giving a classical protocol to exactly simulate quantum correlations implied by the spin-sss singlet state corresponding to all integer as well as half-integer spin values sss. The class of measurements we consider here are only those corresponding to spin observables, as has been done in the above-mentioned paper. The required amount of communication is found to be ⌈log2(s+1)⌉\lceil {\rm log}_{2} (s + 1) \rceil⌈log2​(s+1)⌉ in the worst case scenario, where ⌈x⌉\lceil x \rceil⌈x⌉ is the least integer greater than or equal to xxx.
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We give a classical protocol to exactly simulate quantum correlations implied by a spin sss singlet state for the infinite sequence of \textit{integer} spin s=1,4,13,...s={1,4,13,...}s=1,4,13,... satisfying (2s+1)=3n(2s+1)=3^{n}(2s+1)=3n,where nnn is a positive integer.The required amount of communication is found to increase as log3d2log_{3}d^{2}log3​d2 where d=2s+1d=2s+1d=2s+1 is the dimension of the spin sss Hilbert space.
We give a classical protocol to simulate quantum correlations implied by a spin s singlet state for the infinite sequence of spins satisfying (2s+1)=2n(2s+1)= 2^{n}(2s+1)=2n, where n>0n>0n>0 is an integer.The required amount of communication is found to increase as log2dlog_{2}dlog2​d where d=2s+1d=2s+1d=2s+1 is the dimension of the spin s Hilbert space.
Analysis of Quantum Correlation in Successive Spin Measurements, Classical Communication of Spin SSS Singlet States and Quantum Nonlocality of two Qubit Entangled States
In this thesis, we study about three subjects 1- Classical simulation of two spin-sss singlet correlations for all sss involving spin measurements, 2- Quantum correlations in successive single spin measurements, 3- Non locality without inequality for almost all two-qubit entangled states based on Cabello's non locality argument.
A long-standing open question is to characterize the classical communication cost of simulating the correlations arising from measurements on finite-dimensional quantum systems. We consider this question in the prepare-and-measure scenario. First we construct a classical simulation model involving a finite average number of bits of one-way communication that simulates exactly the correlations arising from arbitrary ddd-dimensional quantum states and arbitrary measurements. Second, we show that no classical simulation protocol involving a finite average message dimension can exist for d≥4d\geq 4d≥4, hence strengthening a recent result of de Gois, Santos and Vieira. Our first result relies on a more general simulation theorem which can be applied to certain continuous variable (i.e. infinite-dimensional) quantum systems, and even beyond quantum theory, e.g. to generalized probabilistic theories. In particular, we show that multimode Gaussian states of bounded average energy and general Gaussian measurements admit an exact classical simulation with a finite expected number of bits. Overall, our results show that the classical simulation cost of quantum communication is essentially as high as it could be.
Recent work has extended Bell's theorem by quantifying the amount of communication required to simulate entangled quantum systems with classical information. The general scenario is that a bipartite measurement is given from a set of possibilities and the goal is to find a classical scheme that reproduces exactly the correlations that arise when an actual quantum system is measured. Previous results have shown that, using local hidden variables, a finite amount of communication suffices to simulate the correlations for a Bell state. We extend this in a number of ways. First, we show that, when the communication is merely required to be finite on average, Bell states can be simulated without any local hidden variables. More generally, we show that arbitrary positive operator valued measurements on systems of n Bell states can be simulated with O(n2n){O(n2}^{n})O(n2n) bits of communication on average (again, without local hidden variables). On the other hand, when the communication is required to be absolutely bounded, we show that a finite number of bits of local hidden variables is insufficient to simulate a Bell state. This latter result is based on an analysis of the nondeterministic communication complexity of the NOT-EQUAL function, which is constant in the quantum model and logarithmic in the classical model.