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

Ali Ahanj, Pramod S. Joag

Published Mar 7, 2006• 0 citations

Abstract

We give a classical protocol to simulate quantum correlations implied by a spin s singlet state for the infinite sequence of spins satisfying $(2s+1)= 2^{n}$, where $n>0$ is an integer.The required amount of communication is found to increase as $log_{2}d$ where $d=2s+1$ is the dimension of the spin s Hilbert space.

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

Ali Ahanj, Pramod S. Joag

Published Mar 7, 2006• 0 citations

OpenAlex

Abstract

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)= 2^{n}

, where

n>0

is an integer.The required amount of communication is found to increase as

log_{2}d

where

d=2s+1

is the dimension of the spin s Hilbert space.

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

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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.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.

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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 $x$ .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 $x$.

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International Journal of Quantum InformationJun 1, 20100 citations

The cost of classically simulating quantum communication is as high as it could be

Riccardo Castellano, Pavel Sekatski, Nicolas Brunner

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 $d$ -dimensional quantum states and arbitrary measurements. Second, we show that no classical simulation protocol involving a finite average message dimension can exist for $d\geq 4$ , 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.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 $d$-dimensional quantum states and arbitrary measurements. Second, we show that no classical simulation protocol involving a finite average message dimension can exist for $d\geq 4$, 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.

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Ali Ahanj, Pramod S. Joag

This paper has been withdrawn due to a fatal error in the argument of the simulation scheme.

arXiv:quant-ph/0604015Apr 4, 20060 citations

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)= 2^{n}

, where

n>0

is an integer.The required amount of communication is found to increase as

log_{2}d

where

d=2s+1

is the dimension of the spin s Hilbert space.

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.

Simulation of two spin-

s

singlet correlations for all

s

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

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

357

classical bits, and consequently, all correlations of two entangled qutrits.

d

-dimensional quantum states and arbitrary measurements. Second, we show that no classical simulation protocol involving a finite average message dimension can exist for

d\geq 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.