arXiv subjects
Publications and source records attributed to Pawel Horodecki.
We claim that the nonlocality without entanglement revealed quite recently by Bennett {\it et al.} [quant-ph/9804053] should be rather interpreted as {\it Einstein-Podolsky-Rosen paradox without entanglement}. It would be true nonlocality without entanglement if one knew that quantum mechnics provides the best possible means for extracting information from physical system i.e. that it is ``operationally complete''.
Bound entanglement is the noisy entanglement which cannot be distilled to a singlet form. Thus it cannot be used alone for quantum communication purposes. Here we show that, nevertheless, the bound entanglement can be, in a sense, pumped into single pair of free entangled particles. It allows for teleportation via the pair with the fidelity impossible to achieve without support of bound entanglement. The result also suggests that the distillable entanglement may be not additive.
We provide some new properties of entanglement of formation. In particular, we obtain an additive lower bound for entanglement of formation. Subsequently we develop the concept of local orthogonality of ensembles which leads to the mixed states with distillable entanglement equal to entanglement of formation. Then we consider thermodynamical analogies within the entanglement processing domain. Especially, we exploit analogy entanglement -- energy. In this scheme the total entanglement i.e. the amount of singlet pairs needed for local preparation of a state corresponds to internal energy while the free entanglement defined as the number of pairs which can be recovered from the state (distillable entanglement) is the counterpart of free energy. In particular, it leads us to the question about ``temperature'' of entanglement. We also propose a scheme of the search of representative state for given entanglement which can be viewed as an analogue of the Jaynes maximum entropy principle.
Suppose that a quantum source is known to have von Neumann entropy less than or equal to S but is otherwise completely unspecified. We describe a method of universal quantum data compression which will faithfully compress the quantum information of any such source to S qubits per signal (in the limit of large block lengths).
We analyse the problem of distillation of entanglement of mixed states in higher dimensional compound systems. Employing the positive maps method [M. Horodecki et al., Phys. Lett. A 223 1 (1996)] we introduce and analyse a criterion of separability which relates the structures of the total density matrix and its reductions. We show that any state violating the criterion can be distilled by suitable generalization of the two-qubit protocol which distills any inseparable two-qubit state. Conversely, all the states which can be distilled by such a protocol must violate the criterion. The proof involves construction of the family of states which are invariant under transformation $\varrho\to U\otimes U^*\varrho U^\dagger\otimes U^{*\dagger}$ where $U$ is a unitary transformation and star denotes complex conjugation. The states are related to the depolarizing channel generalized to non-binary case.
A natural measure in the space of density matrices describing N-dimensional quantum systems is proposed. We study the probability P that a quantum state chosen randomly with respect to the natural measure is not entangled (is separable). We find analytical lower and upper bounds for this quantity. Numerical calculations give P = 0.632 for N=4 and P=0.384 for N=6, and indicate that P decreases exponentially with N. Analysis of a conditional measure of separability under the condition of fixed purity shows a clear dualism between purity and separability: entanglement is typical for pure states, while separability is connected with quantum mixtures. In particular, states of sufficiently low purity are necessarily separable.
We consider the problem of optimal processing of quantum information at incomplete experimental data characterizing the quantum source. In particular, we then prove that for one-qubit quantum source the Jaynes principle offers a simple scheme for optimal compression of quantum information. According to the scheme one should process as if the density matrix of the source were actually equal to the matrix of the Jaynes state.
We show, by explicit examples, that the Jaynes inference scheme based on maximization of entropy can produce inseparable states even if there exists a separable state compatible with the measured data. It can lead to problems with processing of entanglement. The difficulty vanishes when one uses inference scheme based on minimization of entanglement.
It is shown that if a mixed state can be distilled to the singlet form, it must violate partial transposition criterion [A. Peres, Phys. Rev. Lett. 76, 1413 (1996)]. It implies that there are two qualitatively different types of entanglement: ``free'' entanglement which is distillable, and ``bound'' entanglement which cannot be brought to the singlet form useful for quantum communication purposes. Possible physical meaning of the result is discussed.
It is shown that any separable state on Hilbert space ${\cal H}={\cal H}_1\otimes{\cal H}_2$, can be written as a convex combination of N pure product states with $N\leq (dim{\cal H})^2$. Then a new separability criterion for mixed states in terms of range of density matrix is obtained. It is used in construction of inseparable mixed states with positive partial transposition in the case of $3\times 3$ and $2\times 4$ systems. The states represent an entanglement which is hidden in a more subtle way than it has been known so far.
The structure of a local hidden variable model for experiments involving sequences of measurements rigorously is analyzed. Constraints imposed by local realism on the conditional probabilities of the outcomes of such measurement schemes are explicitly derived. The violation of local realism in the case of ``hidden nonlocality'' is illustrated by an operational example.
We apply the inseparability criterion for $2 \times 2$ systems, local filtering and Bennett et al. purification protocol [Phys. Rev. Lett. {\bf 76}, 722 (1996)] to show how to distill {\it any} inseparable $2\times 2$ system. The extended protocol is illustrated geometrically by means of the state parameters in the Hilbert-Schmidt space.
We provide necessary and sufficient conditions for separability of mixed states. As a result we obtain a simple criterion of separability for $2\times2$ and $2\times3$ systems. Here, the positivity of the partial transposition of a state is necessary and sufficient for its separability. However, it is not the case in general. Some examples of mixtures which demonstrate the utility of the criterion are considered.
Relations between teleportation, Bell's inequalities and inseparability are investigated. It is shown that any mixed two spin-$1\over2$ state which violates the Bell-CHSH inequality is useful for teleportation. The result is extended to any Bell's inequalities constructed of the expectation values of products of spin operators. It is also shown that there exist inseparable states which are not useful for teleportation within the standard scheme.