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arXiv:quant-ph/0603209

Communication cost of simulating quantum correlations in a class of integer spin singlet states

Ali Ahanj, Pramod S. Joag

Published Mar 23, 2006• 0 citations
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Abstract

We give a classical protocol to exactly simulate quantum correlations implied by a spin $s$ singlet state for the infinite sequence of \textit{integer} spin $s={1,4,13,...}$ satisfying $(2s+1)=3^{n}$,where $n$ is a positive integer.The required amount of communication is found to increase as $log_{3}d^{2}$ where $d=2s+1$ is the dimension of the spin $s$ Hilbert space.

DOI
10.48550/arxiv.quant-ph/0603209

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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.
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.
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.
A quantum system of any fixed dimension can be prepared in a continuum of states, yet it cannot be used to transmit an unlimited amount of classical information. Similarly, the correlations observed between measurement outcomes on separate parts of a shared quantum system can be stronger than classical correlations, but they cannot transmit information. These fundamental limitations suggest that the statistics observed from quantum communication and quantum correlations may admit a simulation using a finite amount of classical communication. This expectation is confirmed in the smallest nontrivial quantum dimension, with two classical bits being necessary and sufficient to exactly simulate qubit communication and all correlations between qubits. Despite significant efforts during the previous decades, this remained the only solved case. Here we resolve both problems for every quantum dimension. The solution reveals an unexpected qualitative transition starting at dimension four: no finite amount of classical communication can exactly simulate ququart communication nor all quantum correlations of two entangled ququarts, even with unlimited shared randomness. One might have expected this transition, if it existed, to appear already for qutrits. Instead, we construct an explicit protocol that exactly simulates qutrit communication using 357357357 classical bits, and consequently, all correlations of two entangled qutrits.
In a recent paper [A. Ahanj et al., Phys. Lett. A368 (2007) 34], 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 = 2n. In the present paper, we have generalized this result by giving a classical protocols 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 ⌈ log 2(s + 1)⌉ in the worst case scenario, where ⌈x⌉ is the least integer greater than or equal to x.
Assume Alice and Bob share some bipartite d-dimensional quantum state. As is well known, by performing two-outcome measurements, Alice and Bob can produce correlations that cannot be obtained classically. We show that by using only two bits of communication, Alice and Bob can classically simulate any such correlations. All previous protocols for exact simulation required the communication to grow to infinity with the dimension d. Our protocol and analysis are based on a power series method, resembling Krivine's bound on Grothendieck's constant, and on the computation of volumes of spherical tetrahedra.