In quantum mechanics, separable quantum states are states without quantum entanglement.
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Separable pure states
For simplicity, the following assumes all relevant state spaces are finite-dimensional. First, consider separability for pure states. Separable pure states are also called product states.
Let
with base states
If a pure state
Formally, the embedding of a product of states into the product space is given by the Segre embedding. That is, a quantum-mechanical pure state is separable if and only if it is in the image of the Segre embedding.
The above discussion can be extended to the case of when the state space is infinite-dimensional with virtually nothing changed.
Separability for mixed states
Consider the mixed state case. A mixed state of the composite system is described by a density matrix
where
Otherwise
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 that
In terms of quantum channels, a separable state can be created from any other state using local actions and classical communication while an entangled state cannot.
When 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.
If there is only a single non-zero
Extending to the multipartite case
The above discussion generalizes easily to the case of a quantum system consisting of more than two subsystems. Let a system have n subsystems and have state space
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.
Separability criterion
The problem of deciding whether a state is separable in general is sometimes called the separability problem in quantum information theory. It is considered to be a difficult problem. It has been shown to be NP-hard. 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. We see that the problem quickly becomes intractable, even for low dimensions. Thus more sophisticated formulations are required. The separability problem is a subject of current research.
A separability criterion is a necessary condition a state must satisfy to be separable. 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. Other separability criteria include the range criterion and reduction criterion. See Ref. for a review of separability criteria in discrete variable systems.
In continuous variable systems, the Peres-Horodecki criterion also applies. Specifically, Simon 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 for
Characterization via algebraic geometry
Quantum mechanics may be modelled on a projective Hilbert space, and the categorical product of two such spaces is the Segre embedding. 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" describe the problem and study the geometry of the separable states as a subset of the general state matrices. This subset have some intersection with the subset of states holding Peres-Horodecki criterion. In this paper, Leinaas et al. also give a numerical approach to test for separability in the general case.
Testing for separability
Since separability testing in a general case is an NP-hard. problem, in their paper, Leinaas et al. offer a numerical approach, iteratively refining an estimated separable state towards the target state to be tested, checking if the target state can indeed be reached. An implementation of the algorithm (including a built in Peres-Horodecki criterion testing) is brought in the "StateSeparator" web-app