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Reduction criterion

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In quantum information theory, the reduction criterion is a necessary condition a mixed state must satisfy in order for it to be separable. In other words, the reduction criterion is a separability criterion. It was first proved[1] and independently formulated in 1999.[2] Violation of the reduction criterion is closely related to the distillability of the state in question.[1]

Details

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Let H1 and H2 be Hilbert spaces of finite dimensions n and m respectively. L(Hi) will denote the space of linear operators acting on Hi. Consider a bipartite quantum system whose state space is the tensor product

An (un-normalized) mixed state ρ is a positive linear operator (density matrix) acting on H.

A linear map Φ: L(H2) → L(H1) is said to be positive if it preserves the cone of positive elements, i.e. A is positive implied Φ(A) is also.

From the one-to-one correspondence between positive maps and entanglement witnesses, we have that a state ρ is entangled if and only if there exists a positive map Φ such that

is not positive. Therefore, if ρ is separable, then for all positive map Φ,

Thus every positive, but not completely positive, map Φ gives rise to a necessary condition for separability in this way. The reduction criterion is a particular example of this.

Suppose H1 = H2. Define the positive map Φ: L(H2) → L(H1) by

It is known that Φ is positive but not completely positive. So a mixed state ρ being separable implies

Direct calculation shows that the above expression is the same as

where ρ1 is the partial trace of ρ with respect to the second system. The dual relation

is obtained in the analogous fashion. The reduction criterion consists of the above two inequalities.

Connection with Fréchet bounds

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The above last two inequalities together with lower bounds for ρ can be seen as quantum Fréchet inequalities, that is as the quantum analogous of the classical Fréchet probabilistic bounds, that hold for separable quantum states. The upper bounds are the previous ones , , and the lower bounds are the obvious constraint together with , where are identity matrices of suitable dimensions. The lower bounds have been obtained in.[3]: Theorem A.16  These bounds are satisfied by separable density matrices, while entangled states can violate them. Entangled states exhibit a form of stochastic dependence stronger than the strongest classical dependence and in fact they violate Fréchet like bounds. It is also worth mentioning that is possible to give a Bayesian interpretation of these bounds.[3]

References

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  1. ^ a b M. Horodecki and P. Horodecki (1999). "Reduction criterion of separability and limits for a class of distillation protocols". Phys. Rev. A. 59 (6): 4206–4216. arXiv:quant-ph/9708015. Bibcode:1999PhRvA..59.4206H. doi:10.1103/PhysRevA.59.4206.
  2. ^ N. Cerf; et al. (1999). "Reduction criterion for separability". Phys. Rev. A. 60 (2): 898–909. arXiv:quant-ph/9710001. Bibcode:1999PhRvA..60..898C. doi:10.1103/PhysRevA.60.898.
  3. ^ a b Benavoli, A.; Facchini, A.; Zaffalon, M. (10 October 2016). "Quantum mechanics: The Bayesian theory generalized to the space of Hermitian matrices". Physical Review A. 94 (4): 042106. arXiv:1605.08177. Bibcode:2016PhRvA..94d2106B. doi:10.1103/PhysRevA.94.042106. S2CID 19791149.