Separable state
The physical intuition behind these definitions is that product states have no correlation between the different degrees of freedom, while separable states might have correlations, but all such correlations can be explained as due to a classical random variable, as opposed as being due to entanglement.In general, determining if a state is separable is not straightforward and the problem is classed as NP-hard.By a postulate of quantum mechanics these can be described as vectors in the tensor product spaceIn this discussion we will focus on the case of the Hilbert spacesFrom the very definition of the tensor product, any vector of norm 1, i.e. a pure state of the composite system, can be written as wherePhysically, this means that it is not possible to assign a definite (pure) state to the subsystems, which instead ought to be described as statistical ensembles of pure states, that is, as density matrices.is thus entangled if and only if the von Neumann entropy of the partial state[1] That is, a quantum-mechanical pure state is separable if and only if it is in the image of the Segre embedding.A mixed state of the composite system is described by a density matrixare all rank-1 projections, that is, they represent pure ensembles of the appropriate subsystems.It is clear from the definition that the family of separable states is a convex set.Notice that, again from the definition of the tensor product, any density matrix, indeed any matrix acting on the composite state space, can be trivially written in the desired form, if we drop the requirement thatWhen the state spaces are infinite-dimensional, density matrices are replaced by positive trace class operators with trace 1, and a state is separable if it can be approximated, in trace norm, by states of the above form.and is called simply separable or product state.One property of the product state is that in terms of entropy, The above discussion generalizes easily to the case of a quantum system consisting of more than two subsystems.is separable if it takes the form Similarly, a mixed state ρ acting on H is separable if it is a convex sum Or, in the infinite-dimensional case, ρ is separable if it can be approximated in the trace norm by states of the above form.It has been shown to be NP-hard in many cases [2][3] and is believed to be so in general.Some appreciation for this difficulty can be obtained if one attempts to solve the problem by employing the direct brute force approach, for a fixed dimension.The problem quickly becomes intractable, even for low dimensions.The separability problem is a subject of current research.In the low-dimensional (2 X 2 and 2 X 3) cases, the Peres-Horodecki criterion is actually a necessary and sufficient condition for separability.[8] for a review of separability criteria in discrete variable systems.In continuous variable systems, the Peres-Horodecki criterion also applies.Specifically, Simon [9] formulated a particular version of the Peres-Horodecki criterion in terms of the second-order moments of canonical operators and showed that it is necessary and sufficient forSimon's condition can be generalized by taking into account the higher order moments of canonical operators [12][13] or by using entropic measures.In the bipartite case, a quantum state is separable if and only if it lies in the image of the Segre embedding.Jon Magne Leinaas, Jan Myrheim and Eirik Ovrum in their paper "Geometrical aspects of entanglement"[16] describe the problem and study the geometry of the separable states as a subset of the general state matrices.In this paper, Leinaas et al. also give a numerical approach to test for separability in the general case.Testing for separability in the general case is an NP-hard problem.[2][3] Leinaas et al.[16] formulated an iterative, probabilistic algorithm for testing if a given state is separable.