We propose an interpretation of quantum separability based on a physical principle: local time reversal. It immediately leads to a simple characterization of separable quantum states that reproduces results known to hold for binary composite systems and which thereby is complete for low dimensions. We then describe a constructive algorithm for finding the canonical decomposition of separable and non separable mixed states of dimensions 2x2 and 2x3.
By using the "subtracting projectors" method in proving the separability of PPT states on multiple quantum spaces, we derive a canonical form of PPT states in ${\Cb}^{K_1} \otimes {\Cb}^{K_2} \otimes ... \otimes {\Cb}^{K_m} \otimes {\Cb}^N$ composite quantum systems with rank $N$, from which a sufficient separability condition for these states is presented.
We investigate the canonical forms of positive partial transposition (PPT) density matrices in ${\cal C}^2 \otimes {\cal C}^M \otimes {\cal C}^N$ composite quantum systems with rank $N$. A general expression for these PPT states are explicitly obtained. From this canonical form a sufficient separability condition is presented.
We give a canonical form of PPT states in ${\cal C}^2 \otimes {\cal C}^2\otimes {\cal C}^2 \otimes {\cal C}^N$ with rank=$N$. From this canonical form a necessary separable condition for these states is presented.
The problem of constructing a necessary and sufficient condition for establishing the separability of continuous variable systems is revisited. Simon [R. Simon, Phys. Rev. Lett. 84, 2726 (2000)] pointed out that such a criterion may be constructed by drawing a parallel between the Peres' partial transpose criterion for finite dimensional systems and partial time reversal transformation for continuous variable systems. We generalize the partial time reversal transformation to a partial scaling transformation and re-examine the problem using a tomographic description of the continuous variable q…
We analyze the properties of the conditional amplitude operator, the quantum analog of the conditional probability which has been introduced in [quant-ph/9512022]. The spectrum of the conditional operator characterizing a quantum bipartite system is invariant under local unitary transformations and reflects its inseparability. More specifically, it is shown that the conditional amplitude operator of a separable state cannot have an eigenvalue exceeding 1, which results in a necessary condition for separability. This leads us to consider a related separability criterion based on the positive ma…
Time reversal symmetry is studied in a space with noncommutativity of coordinates and noncommutativity of momenta of canonical type. The circular motion is examined as an apparent example of time reversal symmetry breaking in the space. On the basis of exact solution of the problem we show that because of noncommutativity the period of the circular motion depends on its direction. We propose the way to recover the time reversal and rotational symmetries in noncommutative phase space of canonical type. Namely, on the basis of idea of generalization of parameters of noncommutativity to tensors w…
For n an even number of qubits and v a unitary evolution, a matrix decomposition v=k1 a k2 of the unitary group is explicitly computable and allows for study of the dynamics of the concurrence entanglement monotone. The side factors k1 and k2 of this Concurrence Canonical Decomposition (CCD) are concurrence symmetries, so the dynamics reduce to consideration of the a factor. In this work, we provide an explicit numerical algorithm computing v=k1 a k2 for n odd. Further, in the odd case we lift the monotone to a two-argument function, allowing for a theory of concurrence dynamics in odd qubits.…
The structure of the state spaces of bipartite (N tensor N) quantum systems which are invariant under product representations of the group SO(3) of three-dimensional proper rotations is analyzed. The subsystems represent particles of arbitrary spin j which transform according to an irreducible representation of the rotation group. A positive map theta is introduced which describes the time reversal symmetry of the local states and which is unitarily equivalent to the transposition of matrices. It is shown that the partial time reversal transformation theta_2 = (I tensor theta) acting on the co…
Noncommutative algebra which is rotationally invariant, time reversal invariant and equivalent to noncommutative algebra of canonical type is considered. Perihelion shift of orbit of a particle in Coulomb potential in the rotationally-invariant noncommutative phase space is found up to the second order in the parameters of noncommutativity. Applying the result to the case of Mercury planet and using observable results for precession of its orbit we find upper bounds on the parameters of noncommutativity in the rotationally-invariant noncommutative phase space. The obtained upper bound for the …
A hierarchy of multimode uncertainty relations on the second moments of n pairs of canonical operators is derived in terms of quantities invariant under linear canonical (i.e. symplectic) transformations. Conditions for the separability of multimode continuous variable states are derived from the uncertainty relations, generalizing the inequalities obtained in [Phys. Rev. Lett. 96, 110402 (2006)] to states with some transposed symplectic eigenvalues equal to 1. Finally, to illustrate the methodology proposed for the detection of continuous variable entanglement, the separability of multimode n…
Internal symmetries can be used to classify multiple solutions to the time energy canonical commutation relation (TE-CCR). The dynamical behavior of solutions to the TE-CCR posessing particular internal symmetries involving time reversal differ significantly from solutions to the TE-CCR without those particular symmetries, implying a connection between the internal symmetries of a quantum system, its internal unitary dynamics, and the TE-CCR.
General physical background of Peres-Horodecki positive partial transpose (ppt-) separability criterion is revealed. Especially, the physical sense of partial transpose operation is shown to be equivalent to the "local causality reversal" (LCR-) procedure for all separable quantum systems or to the uncertainty in a global time arrow direction in all entangled cases. Using these universal causal considerations the heuristic causal separability criterion has been proposed for arbitrary $ D^{N} \times D^{N}$ density matrices acting in $ \mathcal{H}_{D}^{\otimes N} $ Hilbert spaces which describe …
Quantized space described by time reversal invariant and rotationally invariant noncommutative algebra of canonical type is studied. A particle in uniform field is considered. We find exactly the energy of a particle in uniform field in the quantized space and its wavefunctions. It is shown that the motion of the particle in the field direction in time reversal invariant and rotationally invariant noncommutative space is the same as in the ordinary space (space with the ordinary commutation relations for operators of coordinates and operators of momenta). Noncommutativity of coordinates has in…
We develop rigorous notions of causality and causal separability in the process framework introduced in [Oreshkov, Costa, Brukner, Nat. Commun. 3, 1092 (2012)], which describes correlations between separate local experiments without a prior assumption of causal order between them. We consider the general multipartite case and take into account the possibility for dynamical causal order, where the order of a set of events can depend on other events in the past. Starting from a general definition of causality, we derive an iteratively formulated canonical decomposition of multipartite causal pro…
By constructing a hydrodynamic canonical formalism, we show that the occurrence of an arbitrary density-dependent gauge potential in the meanfield Hamiltonian of a Bose-condensed fluid invariably leads to nonlinear flow-dependent terms in the wave equation for the phase, where such terms arise due to the explicit dependence of the mechanical flow on the fluid density. In addition, we derive a canonical momentum transport equation for this class of nonlinear fluid and obtain an expression for the stress tensor. Further, we study the hydrodynamic equations in a particular nonlinear fluid, where …
This work concerns the statistics of the Two-Time Measurement definition of heat variation in each reservoir of a thermodynamic quantum system. We study the cumulant generating function of the heat flows in the thermodynamic and large-time limits. It is well-known that, if the system is time-reversal invariant, this cumulant generating function satisfies the celebrated Evans--Searles symmetry. We show in addition that, under appropriate ultraviolet regularity assumptions on the local interaction between the reservoirs, it satisfies a translation-invariance property, as proposed in [Andrieux et…
