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Physical Review A

Violation of Bell’s inequalities for a two-mode squeezed vacuum state in lossy transmission lines

Radim Filip, Ladislav Mišta

Published Oct 30, 2002• 30 citations
Publisher

Abstract

We investigate the transfer of a two-mode squeezed vacuum state through lossy transmission lines. It is shown that the state can violate Bell's inequalities, and consequently, excludes a local hidden variable model at pronouncedly larger transmission lengths in comparison with previous claims. This can be important for security of continuous-variable quantum cryptography employing entangled states.

DOI
10.1103/physreva.66.044309

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We investigate the transfer of a two-mode squeezed vacuum state through lossy transmission lines. It is shown that the state can violate Bell's inequalities, and consequently, excludes a local hidden variable model at pronouncedly larger transmission lengths in comparison with previous claims. This can be important for security of continuous-variable quantum cryptography employing entangled states.
A Bell inequality is a fundamental test to rule out local hidden variable model descriptions of correlations between two physically separated systems. There have been a number of experiments in which a Bell inequality has been violated using discrete-variable systems. We demonstrate a violation of Bell's inequality using continuous variable quadrature measurements. By creating a four-mode entangled state with homodyne detection, we recorded a clear violation with a Bell value of B=2.31±0.02. This opens new possibilities for using continuous variable states for device independent quantum protocols.
We generalize Bell's inequalities to biparticle systems with continuous quantum variables. This is achieved by introducing the Bell operator in perfect analogy to the usual spin-1/2 systems. It is then demonstrated that two-mode squeezed vacuum states display quantum nonlocality by using the generalized Bell operator. In particular, the original Einstein-Podolsky-Rosen entangled states, which are the limiting case of the two-mode squeezed vacuum states, can maximally violate Bell's inequality due to Clauser, Horne, Shimony and Holt. The experimental aspect of our scheme and nonlocality of arbitrary biparticle entangled pure states of continuous variables are briefly considered.
The ultimate limits of continuous-variable single-mode quantum teleportation due to absorption are studied, with special emphasis on (quasi-) monochromatic optical fields propagating through fibers. It is shown that even if an infinitely squeezed two-mode squeezed vacuum were used, the amount of information that would be transferred quantum mechanically over a finite distance is limited and effectively approaches zero on a length scale that is much shorter than the (classical) absorption length. The state-dependent teleportation fidelity can be close to unity only for short distances. To realize the largest possibly fidelity, asymmetrical equipment must be used, where the source of the two-mode squeezed vacuum is nearer to Alice than to Bob and as a consequence the coherent displacement performed by Bob cannot be chosen independently of the transmission lengths.
Quantum entanglement and nonlocality properties of a family of two-mode Gaussian pure states have been investigated. The results show that the entanglement of these states is determined by both the two-mode squeezing parameter and the difference of the two single-mode squeezing parameters. For the same two-mode squeezing parameter, these states show larger entanglement than the usual two-mode squeezed vacuum state. The violation of Bell inequality depends strongly on all the squeezing parameters of these states and disappears completely in the limit of large squeezing. In particular, these states can exhibit much stronger violation of local realism than two-mode squeezed vacuum state in the range of experimentally available squeezing values.
It is a long-standing belief, as pointed out by Bell in 1986, that it is impossible to use a two-mode Gaussian state possessing a positive-definite Wigner function to demonstrate nonlocality as the Wigner function itself provides a local hidden-variable model. In particular, when one performs continuous-variable (CV) quadrature measurements upon a routinely generated CV entanglement, namely, the two-mode squeezed vacuum (TMSV) state, the resulting Wigner function is positive-definite and as such, the TMSV state cannot violate any Bell inequality using CV quadrature measurements. We show here, however, that a Bell inequality for CV states in terms of entropies can be quantum mechanically violated by the TMSV state with two coarse-grained quadrature measurements per site within experimentally accessible parameter regime. The proposed CV entropic Bell inequality is advantageous for an experimental test, especially for a possible loophole-free test of nonlocality, as the quadrature measurements can be implemented with homodyne detections of nearly 100100%100 detection efficiency under current technology.