Peres-Horodecki Separability Criterion for Continuous Variable Systems

Abstract

The Peres-Horodecki criterion of positivity under partial transpose is studied in the context of separability of bipartite continuous variable states. The partial transpose operation admits, in the continuous case, a geometric interpretation as mirror reflection in phase space. This recognition leads to uncertainty principles, stronger than the traditional ones, to be obeyed by all separable states. For all bipartite Gaussian states, the Peres-Horodecki criterion turns out to be necessary and sufficient condition for separability.PACS numbers: 42.50.Dv, 89.70.+c Entanglement or inseparability is central to all branches of the emerging field of quantum information and quantum computation [1]. A particularly elegant criterion for checking if a given state is separable or not was proposed by Peres [2]. This condition is necessary and sufficient for separability in the 2 × 2 and 2 × 3 dimensional cases, but ceases to be so in higher dimensions as shown by Horodecki [3].While a major part of the effort in quantum information theory has been in the context of systems with finite number of Hilbert space dimensions, more specifically the qubits, recently there has been much interest in the canonical continuous case [4][5][6][7][8][9]. We may mention in particular the experimental realization of quantum teleportation of coherent states [10]. It is therefore important to be able to know if a given state of a bipartite canonical continuous system is entangled or separable.With increasing Hilbert space dimension, any test for separability will be expected to become more and more difficult to implement in practice. In this paper we show that in the limit of infinite dimension, corresponding to continuous variable bipartite states, the Peres-Horodecki criterion leads to a test that is extremely easy to implement. Central to our work is the recognition that the partial transpose operation acquires, in the continuous case, a beautiful geometric interpretation as mirror reflection in the Wigner phase space. Separability forces on the second moments (uncertainties) a restriction that is stronger than the traditional uncertainty principle; even commuting variables need to obey an uncertainty relation. This restriction is used to prove that the Peres-Horodecki criterion is necessary and sufficient separability condition for all bipartite Gaussian states. Consider a single mode described by annihilation operatorâ = (q + ip)/ √ 2, obeying the standard commutation relation [q,p] = i, which is equivalent to [â,â † ] = 1. There is a one-to-one correpondence between density operators and c-number Wigner distribution functions W (q, p) [11]. The latter are real functions over the phase space and satisfy an additional property coding the nonnegativity of the density operator. It follows from the definition of Wigner distribution that the transpose operation T , which takes everyρ to its transposeρ T , is equivalent to a mirror reflection in phase space:ρMirror reflection is not a canonical transformation in phase space, and cannot be i...

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