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...

We present a complete analysis of variance matrices and quadrature squeezing for arbitrary states of quantum systems with any finite number of degrees of freedom. Basic to our analysis is the recognition of the crucial role played by the real symplectic group Sp(2n,R) of linear canonical transformations on n pairs of canonical variables. We exploit the transformation properties of variance (noise) matrices under symplectic transformations to express the uncertainty-principle restrictions on a general variance matrix in several equivalent forms, each of which is manifestly symplectic invariant. These restrictions go beyond the classically adequate reality, symmetry, and positivity conditions. Towards developing a squeezing criterion for n-mode systems, we distinguish between photon-number-conserving passive linear optical systems and active ones. The former correspond to elements in the maximal compact U ( n ) subgroup of Sp(2n,R), the latter to noncompact elements outside U(n). Based on this distinction, we motivate and state a U(n)-invariant squeezing criterion applicable to any state of an n-mode system, and explore alternative ways of expressing it. The set of all possible quantum-mechanical variance matrices is shown to contain several interesting subsets or subfamilies, whose definitions are related to the fact that a general variance matrix is not diagonalizable within U( n ). Definitions, characterizations, and canonical forms for variance matrices in these subfamilies, as well as general ones, and their squeezing nature, are established. It is shown that all conceivable variance matrices can be generated through squeezed thermal states of the n-mode system and their symplectic transforms. Our formulas are developed in both the real and the complex forms for variance matrices, and ways to pass between them are given.

Gaussian kernels representing operators on the Hilbert space ^H=L 2 (K") are studied. Necessary and sufficient conditions on such a kernel in order that the corresponding operator be positive semidefinite, corresponding to a density matrix (cross-spectral density) in quantum mechanics (optics), are derived. The Wigner distribution method is shown to be a convenient framework for characterizing Gaussian kernels and their unitary evolution under Sp(2«,E) action. The nontrivial role played by a phase term in the kernel is brought out. The entire analysis is presented in a form which is directly applicable to n-dimensional oscillator systems in quantum mechanics and to Gaussian Schell-model partially coherent fields in optics.

Operator-sum or Kraus representations for single-mode Bosonic Gaussian channels are developed, and several of their consequences explored. The fact that the two-mode metaplectic operators acting as unitary purification of these channels do not, in their canonical form, mix the position and momentum variables is exploited to present a procedure which applies uniformly to all families in the Holevo classification. In this procedure the Kraus operators of every quantum-limited Gaussian channel can be simply read off from the matrix elements of a corresponding metaplectic operator. Kraus operators are employed to bring out, in the Fock basis, the manner in which the antilinear, unphysical matrix transposition map when accompanied by injection of a threshold classical noise becomes a physical channel, denoted D(κ) in the Holevo classification. The matrix transposition channels D(κ), D(κ −1 ) turn out to be a dual pair in the sense that their Kraus operators are related by the adjoint operation. The amplifier channel with amplification factor κ and the beamsplitter channel with attenuation factor κ −1 turn out to be mutually dual in the same sense. The action of the quantum-limited attenuator and amplifier channels as simply scaling maps on suitable quasiprobabilities in phase space is examined in the Kraus picture. Consideration of cumulants is used to examine the issue of fixed points. The semigroup property of the amplifier and attenuator families leads in both cases to a Zeno-like effect arising as a consequence of interrupted evolution. In the cases of entanglement-breaking channels a description in terms of rank one Kraus operators is shown to emerge quite simply. In contradistinction, it is shown that there is not even one finite rank operator in the entire linear span of Kraus operators of the quantum-limited amplifier or attenuator families, an assertion far stronger than the statement that these are not entanglement breaking channels. A characterization of extremality in terms of Kraus operators, originally due to Choi, is employed to show that all quantum-limited Gaussian channels are extremal. The fact that every noisy Gaussian channel can be realised as product of a pair of quantum-limited channels is used to construct a discrete set of linearly independent Kraus operators for noisy Gaussian channels, including the classical noise channel, and these Kraus operators have a particularly simple structure.

The issue raised in this Letter is classical, not only in the sense of being nonquantum, but also in the sense of being quite ancient: which subset of 4x4 real matrices should be accepted as physical Mueller matrices in polarization optics? Nonquantum entanglement or inseparability between the polarization and spatial degrees of freedom of an electromagnetic beam whose polarization is not homogeneous is shown to provide the physical basis to resolve this issue in a definitive manner.