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The reduction of the closest disentangled states

Ishizaka, S

Published Mar 18, 2002• 5 citations
OpenAlex

Abstract

We show that, for a given entangled state in any system (multi-party with any dimension), the closest disentangled state always has the same reduction as the corresponding reduction of the given entangled state. This implies that any party cannot distinguish these two state without any classical communications. Furthermore, we determined 12 parameters in the Hilbert-Schmidt representation of the closest disentangled state in two qubits, and the minimization problem of the relative entropy of entanglement is reduced to the problem minimizing three parameters.

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We show that, for a given entangled state in any system (multi-party with any dimension), the closest disentangled state always has the same reduction as the corresponding reduction of the given entangled state. This implies that any party cannot distinguish these two state without any classical communications. Furthermore, we determined 12 parameters in the Hilbert-Schmidt representation of the closest disentangled state in two qubits, and the minimization problem of the relative entropy of entanglement is reduced to the problem minimizing three parameters.
We study the closest disentangled state to a given entangled state in any system (multi-party with any dimension). We obtain the set of equations the closest disentangled state must satisfy, and show that its reduction is strongly related to the extremal condition of the local filtering on each party. Although the equations we obtain are not still tractable, we find some sufficient conditions for which the closest disentangled state has the same reduction as the given entangled state. Further, we suggest a prescription to obtain a tight upper bound of the relative entropy of entanglement in two-qubit systems.
The separable state closest to a given entangled state in the relative entropy measure is called the closest disentangled state. We provide an analytical formula connecting the entangled states and the closest disentangled state in two qubits. Using this formula, when any disentangled state $\ensuremath{\sigma}$ located at the entangle-disentangle boundary is given, entangled states to which $\ensuremath{\sigma}$ is closest can be obtained analytically. Further, this formula naturally defines the direction normal to the boundary surface. The direction is uniquely determined by $\ensuremath{\sigma}$ in almost all cases.
The basic question that is addressed in this paper is finding the closest separable state for a given entangled state, measured with the Hilbert-Schmidt distance. While this problem is in general very difficult, we show that the following strongly related problem can be solved: find the Hilbert-Schmidt distance of an entangled state to the set of all partially transposed states. We prove that this latter distance can be expressed as a function of the negative eigenvalues of the partial transpose of the entangled state, and show how it is related to the distance of a state to the set of positive, partially transposed (PPT) states. We illustrate this by calculating the closest biseparable state to the W state and give a simple and very general proof for the fact that the set of W-type states is not of measure zero. Next we show that all surfaces with states whose partial transposes have constant minimal negative eigenvalue are similar to the boundary of PPT states. We illustrate this with some examples on bipartite qubit states, where contours of constant negativity are plotted on two-dimensional intersections of the complete state space.
I consider the deterministic distinguishability of a set of orthogonal, bipartite states when only a single copy is available and the parties are restricted to local operations and classical communication, but with the additional requirement that entanglement must be preserved in the process. Several general theorems aimed at characterizing sets of states with which the parties can succeed in such a task are proven. These include (i) a maximum for the number of states when the Schmidt rank of every outcome must be at least a given minimum, (ii) an upper bound (equal to the dimension of Hilbert space if entanglement need not be preserved) for the sum over Schmidt ranks of the initial states when only one-way classical communication is allowed, and (iii) separately, a necessary and a sufficient condition on the states such that their original Schmidt ranks can always be preserved. Two additional theorems explicitly demonstrate a trade-off between the extent to which the set of states fill Hilbert space, as measured by their Schmidt ranks, and how refined the parties must make their measurements, an important factor in determining the Schmidt rank the state can retain after it has been identified. It is shown that our bound on the sum of Schmidt ranks can be exceeded if two-way communication is permitted, and this includes the case that entanglement need not be preserved, so that this sum can exceed the dimension of Hilbert space. Such questions, concerning how the various results are affected by the resources used by the parties (amount of classical communication and types of local operations), are addressed for each theorem. This subject is closely related to the problem of locally purifying an entangled state from a mixed state, which is of direct relevance to teleportation and dense coding using a mixed-state resource. In an appendix, I give an extremely simple and transparent proof of ``nonlocality without entanglement,'' a phenomenon originally discussed by Bennett and co-workers several years ago.
The long-standing problem of finding a closed formula for the relative entropy of entanglement (REE) for two qubits is addressed. A compact-form solution to the inverse problem, which characterizes an entangled state for a given closest separable state, is obtained. Analysis of the formula for a large class of entangled states strongly suggests that a compact analytical solution of the original problem, which corresponds to finding the closest separable state for a given entangled state, can be given only in some special cases. A few applications of the compact-form formula are given to show additivity of the REE, to relate the REE with the Rains upper bound for distillable entanglement, and to show that a Bell state does not have a unique closest separable state.
We consider a variation of the multi-party communication complexity scenario where the parties are supplied with an extra resource: particles in an entangled quantum state. We show that, although a prior quantum entanglement cannot be used to simulate a communication channel, it can reduce the communication complexity of functions in some cases. Specifically, we show that, for a particular function among three parties (each of which possesses part of the function's input), a prior quantum entanglement enables them to learn the value of the function with only three bits of communication occurring among the parties, whereas, without quantum entanglement, four bits of communication are necessary. We also show that, for a particular two-party probabilistic communication complexity problem, quantum entanglement results in less communication than is required with only classical random correlations (instead of quantum entanglement). These results are a noteworthy contrast to the well-known fact that quantum entanglement cannot be used to actually simulate communication among remote parties.