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Qubit formula for Relative Entropy of Entanglement |
| contact: |
J. Eisert |
date: |
20 Jun 2003 |
last progress: |
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solved by: |
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The relative entropy of entanglement is an entanglement monotone that
quantifies to what extent a given state can be operationally distinguished (in the
sense of Stein's Lemma) from the closest state which is either separable or has a
positive partial transpose (PPT). For a state \rho it is defined as [1]
ER(\rho) = inf\sigma in D S(\rho||\sigma),
where D stands for the convex sets of separable or PPT states, and S(.||.) is
the quantum relative entropy. The problem is to find a closed formula for this
quantity for systems consisting of two qubits.
The interpretation of the relative entropy of entanglement is a geometrical one:
it is related to the error probability with which a state is mistakenly assumed to
be merely classically correlated or PPT in quantum hypothesis testing. This
entanglement monotone is an upper bound to the distillable entanglement, and in
its asymptotic version conjectured to be identical to the Rains' bound for
distillable entanglement. As most other monotones of entanglement, and all other
known monotones that are provably asymptotically continuous, the actual evaluation
of this quantity amounts to solving an optimization problem. In the case at hand,
it is a convex optimization problem.
The entanglement of formation is a monotone which is also defined as an
optimization problem. If it turned out that the entanglement of formation was in
fact additive (see problem 7), then this quantity could be
interpreted as the entanglement cost, which fleshes out the resource
character of entanglement. Historically, it was very important that for systems
consisting of two qubits, the entanglement of formation can (quite astonishingly)
be evaluated: the Wootters formula [2] is a closed formula for the
entanglement of formation for two-qubit systems. The proof exploits a number of
the particular properties that are available for two-qubit systems [3] - and
only for them. The task is to explicitly solve the convex optimization problem
posed by the relative entropy of entanglement.
So far, there is no published solution to the problem. Ref. [4] presents the
solution to a related problem: for a two-qubit system, given a state on the
boundary of separable states \sigma, it characterizes the states \rho for
which ER(\rho)=S(\rho||\sigma).
| [1] | V. Vedral and M.B. Plenio, Phys. Rev. A 57, 1619 (1998). |
| [2] | W. Wootters, Phys. Rev. Lett. 78, 5022 (1997). |
| [3] | K.G.H. Vollbrecht, and R.F. Werner, J. Math. Phys. 41,
6772 (2000). |
| [4] | S. Ishizaka, Phys. Rev. A 67, 060301(R) (2003). |
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