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Journal of Mathematical Physics

Generalized Einstein-Podolsky-Rosen states

Siendong Huang

Published Nov 1, 2007• 4 citations
Publisher

Abstract

Based on Bohr’s ideas two families of states ωθ and ωϕ sharing the same perfect correlation of the Einstein-Podolsky-Rosen state and maximally violating Bell’s inequalities are constructed. Unlike their finite-dimensional counterpartner, these states are not unitarily equivalent. Hence they cannot be transformed into each other by physical operations of local algebras. This is a new entanglement phenomenon emerging in infinite-dimensional systems. Due to the uncertainty principle the existence of unbounded observables depends on the states ωθ and ωϕ. Therefore the manipulation of unbounded observables in quantum information processes cannot provide information for all states. Especially, the canonical unbounded observables q̂i and p̂i, i=1,2 of individual particles do not exist in the GNS representations associated with ωθ and ωϕ, and hence properties of individual particles cannot be obtained in such states.

DOI
10.1063/1.2809269

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Based on Bohr’s ideas two families of states ωθ and ωϕ sharing the same perfect correlation of the Einstein-Podolsky-Rosen state and maximally violating Bell’s inequalities are constructed. Unlike their finite-dimensional counterpartner, these states are not unitarily equivalent. Hence they cannot be transformed into each other by physical operations of local algebras. This is a new entanglement phenomenon emerging in infinite-dimensional systems. Due to the uncertainty principle the existence of unbounded observables depends on the states ωθ and ωϕ. Therefore the manipulation of unbounded observables in quantum information processes cannot provide information for all states. Especially, the canonical unbounded observables q̂i and p̂i, i=1,2 of individual particles do not exist in the GNS representations associated with ωθ and ωϕ, and hence properties of individual particles cannot be obtained in such states.
The nonlocality of Einstein–Podolsky–Rosen (EPR) states is discussed in the framework of Weyl algebras. To include all possible limits of expectation values, the associated canonical cyclic representations of EPR states are considered. The momentum and position operators of individual particles do not exist in such representations, which explains the uncertainty principle in these cases. However, there are still one-to-one correspondences between the local Weyl operators associated with individual particles. The local observable algebras of individual particles are Type II1 von Neumann algebras and have numerous projection operators. The states obtained by applying local projection operations to the EPR states have perfect correlations between local subalgebras. Bell’s inequality is maximally violated by these states. The lack of atomic projections of Type II1 von Neumann algebras implies that the EPR states cannot be disentangled by local projection operations.
For finite-dimensional systems, states satisfying perfect correlation are just maximally entangled states that play an essential role in quantum information theory. Thus perfect correlation may be a key to entanglement phenomena. For general systems, Einstein–Podolsky–Rosen type states are characterized by perfect correlation and their properties are discussed. Conditions equivalent to perfect correlation are found. As an application of perfect correlation, all Einstein–Podolsky–Rosen type states on a two-particle system are given. Unlike in finite-dimensional systems, they are generally not quasiequivalent.
Expository paper providing a historical survey of the gradual transformation of the "philosophical discussions" between Bohr, Einstein and Schrödinger on foundational issues in quantum mechanics into a quantitative prediction of a new quantum effect, its experimental verification and its proposed (and loudly advertised) applications. The basic idea of the 1935 paper of Einstein-Podolsky-Rosen (EPR) was reformulated by David Bohm for a finite dimensional spin system. This allowed John Bell to derive his inequalities that separate the prediction of quantum entanglement from its possible classical interpretation. We reproduce here their later (1971) version, reviewing on the way the generalization (and mathematical derivation) of Heisenberg's uncertainty relations (due to Weyl and Schrödinger) needed for the passage from EPR to Bell. We also provide an improved derivation of the quantum theoretic violation of Bell's inequalities. Soon after the experimental confirmation of the quantum entanglement (culminating with the work of Alain Aspect) it was Feynman who made public the idea of a quantum computer based on the observed effect.
In this paper the notion of an EPR state for the composite S of two quantum systems S1,S2, relative to S2 and a set 𝒪 of bounded observables of S2, is introduced in the spirit of the classical examples of Einstein–Podolsky–Rosen and Bohm. We restrict ourselves mostly to EPR states of finite norm. The main results are contained in Theorems 3–6 and imply that if EPR states of finite norm relative to (S2,O) exist, then the elements of 𝒪 have discrete probability distributions and the Von Neumann algebra generated by 𝒪 is essentially imbeddable inside S1 by an antiunitary map. The EPR states then correspond to the different imbeddings and certain additional parameters, and are explicitly given by formulas which generalize the famous example of Bohm. If 𝒪 generates all bounded observables, S2 must be of finite dimension and can be imbedded inside S1 by an antiunitary map, and the EPR states relative to S2 are then in canonical bijection with the different imbeddings of S2 inside S1; moreover they are then given by formulas which are exactly those of the generalized Bohm states. The notion of EPR states of infinite norm is also explored and it is shown that the original state of Einstein–Podolsky–Rosen can be realized as a renormalized limit of EPR states of finite quantum systems considered by Weyl, Schwinger, and many others. Finally, a family of states of infinite norm generalizing the Einstein–Podolsky–Rosen example is explicitly given.
We use the uncertainty relation between the operators associated with the total number of particles and with the relative phase of two bosonic modes to construct entanglement and Einstein-Podolsky-Rosen steering criteria. These can be tested experimentally in a variety of systems, such as optical fields, Bose-Einstein condensates, and mechanical oscillators. While known entanglement criteria involving the phase observable typically require us to perform interference measurements by recombining the two systems, our criteria can be tested through local measurements at two spatially distinct positions to investigate the nonlocal nature of quantum correlations. We present simple examples where our criteria are violated and show their robustness to noise. Apart from being useful for state characterization, they might find application in quantum information protocols, for example, based on number-phase teleportation.