Michal Horodecki

Separability of mixed states: necessary and sufficient conditions

A state acting on Hilbert space H1 ⊗ H2 is called separable if it can be approximated in trace norm by convex combinations of product states. We provide necessary and sufficient conditions for separability of mixed states in terms of functionals and positive maps. As a result we obtain a complete characterization of separable states for 2 × 2 and 2 × 3 systems. Here, the positivity of the partial transposition of a state is necessary and sufficient for its separability. Typeset using REVTEX ∗e-mail: fizrh@univ.gda.pl 1 Quantum inseparability, first recognized in 1935 by Einstein, Podolsky and Rosen [1] and Schrödinger [2], is one of the most astonishing features of quantum formalism. After over sixty years it is still a fascinating object from both theoretical and experimental points of view. Recently, together with a dynamical development of experimental methods, a number of possible practical applications of quantum inseparable states has been proposed including quantum computation [3] and quantum teleportation [4]. The above ideas are based on the fact that the quantum inseparability implies, in particular, the existence of the pure entangled states which produce nonclassical phenomena. However, in laboratory one deals with mixed states rather than pure ones. This is due to the uncontrolled interaction with the environment. Then it is very important to know which mixed states can produce quantum effects. The problem is much more complicated than in the pure states case. It may be due to the fact that mixed states apparently possess the ability to behave classically in some respects but quantum mechanically in others. In accordance with the so-called generalized inseparability principle [5] we will call a mixed state of compound quantum system inseparable if it cannot be written as convex combination of product states. The problem of inseparability of mixed states was first raised by Werner [6], who constructed a family of inseparable states which admit the local hidden variable model. It has been pointed pointed out [7] that, nevertheless, some of them are nonlocal and this “hidden” nonlocality can be revealed by subjecting them to more complicated experiments than single von Neumann measurements considered by Werner. This shows that it is hard to divide the mixed states into definitely quantum and classical ones. Recently the separable states have been investigated within the information-theoretic approach [5,8–10]. It has been shown that they satisfy a series of the so-called quantum α-entropy inequalities (for α = 1, 2 [8,9] and α = ∞ [10]). Moreover, the separable two spin2 states with maximal entropies of subsystems have been completely characterized in terms of α-entropy inequalities [5]. It is remarkable that there exist inseparable states which do not reveal nonclassical features under the entropy criterion [9]. 2 Then the fundamental problem of an “operational” characterization of the separable states arises. So far only some necessary conditions of separability have been found [5,6,8,9,11]. An important step is due to Peres [12], who has provided a very strong condition. Namely, he noticed that the separable states remain positive if subjected to partial transposition. Then he conjectured that this is also sufficient condition. In this paper we present two necessary and sufficient conditions for separability of mixed states. It provides a complete, operational characterization of separable states for 2× 2 and 2 × 3 systems. It appears that Peres’ conjecture is valid for those cases. However, as we show in the Appendix, the conjecture is not valid in general. To make our considerations more clear, we start from the following notation and definitions. We will deal with the states on the finite dimensional Hilbert space H = H1 ⊗H2. An operator % acting on H is a state if Tr% = 1 and if it is a positive operator i.e.Abstract We provide necessary and sufficient conditions for the separability of mixed states. As a result we obtain a simple criterion of the separability for 2 × 2 and 2 × 3 systems. Here, the positivity of the partial transposition of a state is necessary and sufficient for its separability. However, this is not the case in general. Some examples of mixtures which demonstrate the utility of the criterion are considered.

Violating Bell inequality by mixed spin-12 states: necessary and sufficient condition

Abstract The necessary and sufficient condition for violating the Clauser-Horne-Shimony-Holt (CHSH) inequality by an arbitrary mixed spin- 1 2 state is presented. Some examples of mixtures which demonstrate the utility of the condition are considered. In particular, it is shown that the local hidden variable (LHV) model for mixed states introduced by Werner [Phys. Rev. A 40 (1989) 4277] is forbidden in some region.

Information-theoretic aspects of inseparability of mixed states.

Information-theoretic aspects of quantum inseparability of mixed states are investigated in terms of the {alpha} entropy inequalities and teleportation fidelity. Inseparability of mixed states is defined and a complete characterization of the inseparable 2{times}2 systems with maximally disordered subsystems is presented within the Hilbert-Schmidt space formalism. A connection between teleportation and negative conditional {alpha} entropy is also emphasized. {copyright} {ital 1996 The American Physical Society.}

Partial quantum information

Information—be it classical or quantum—is measured by the amount of communication needed to convey it. In the classical case, if the receiver has some prior information about the messages being conveyed, less communication is needed. Here we explore the concept of prior quantum information: given an unknown quantum state distributed over two systems, we determine how much quantum communication is needed to transfer the full state to one system. This communication measures the partial information one system needs, conditioned on its prior information. We find that it is given by the conditional entropy—a quantity that was known previously, but lacked an operational meaning. In the classical case, partial information must always be positive, but we find that in the quantum world this physical quantity can be negative. If the partial information is positive, its sender needs to communicate this number of quantum bits to the receiver; if it is negative, then sender and receiver instead gain the corresponding potential for future quantum communication. We introduce a protocol that we term ‘quantum state merging’ which optimally transfers partial information. We show how it enables a systematic understanding of quantum network theory, and discuss several important applications including distributed compression, noiseless coding with side information, multiple access channels and assisted entanglement distillation.

The second laws of quantum thermodynamics.

Significance In ordinary thermodynamics, transitions are governed by a single quantity–the free energy. Its monotonicity is a formulation of the second law. Here, we find that the second law for microscopic or highly correlated systems takes on a very different form than it does at the macroscopic scale, imposing not just one constraint on state transformations, but many. We find a family of quantum free energies which generalize the standard free energy, and can never increase. The ordinary second law corresponds to the nonincreasing of one of these free energies, with the remainder imposing additional constraints on thermodynamic transitions. In the thermodynamic limit, these additional second laws become equivalent to the standard one. We also prove a strengthened version of the zeroth law of thermodynamics, allowing a definition of temperature. The second law of thermodynamics places constraints on state transformations. It applies to systems composed of many particles, however, we are seeing that one can formulate laws of thermodynamics when only a small number of particles are interacting with a heat bath. Is there a second law of thermodynamics in this regime? Here, we find that for processes which are approximately cyclic, the second law for microscopic systems takes on a different form compared to the macroscopic scale, imposing not just one constraint on state transformations, but an entire family of constraints. We find a family of free energies which generalize the traditional one, and show that they can never increase. The ordinary second law relates to one of these, with the remainder imposing additional constraints on thermodynamic transitions. We find three regimes which determine which family of second laws govern state transitions, depending on how cyclic the process is. In one regime one can cause an apparent violation of the usual second law, through a process of embezzling work from a large system which remains arbitrarily close to its original state. These second laws are relevant for small systems, and also apply to individual macroscopic systems interacting via long-range interactions. By making precise the definition of thermal operations, the laws of thermodynamics are unified in this framework, with the first law defining the class of operations, the zeroth law emerging as an equivalence relation between thermal states, and the remaining laws being monotonicity of our generalized free energies.

Quantum state merging and negative information

We consider a quantum state shared between many distant locations, and define a quantum information processing primitive, state merging, that optimally merges the state into one location. As announced in [Horodecki, Oppenheim, Winter, Nature 436, 673 (2005)], the optimal entanglement cost of this task is the conditional entropy if classical communication is free. Since this quantity can be negative, and the state merging rate measures partial quantum information, we find that quantum information can be negative. The classical communication rate also has a minimum rate: a certain quantum mutual information. State merging enabled one to solve a number of open problems: distributed quantum data compression, quantum coding with side information at the decoder and sender, multi-party entanglement of assistance, and the capacity of the quantum multiple access channel. It also provides an operational proof of strong subadditivity. Here, we give precise definitions and prove these results rigorously.

Limits for Entanglement Measures

The basic principle of entanglement processing says that entanglement cannot increase under local operations and classical communication. Based on this principle, we show that any entanglement measure E suitable for the regime of a high number of identically prepared entangled pairs satisfies ED < or = E < or = EF, where ED and EF are the entanglement of distillation and formation, respectively. Moreover, we exhibit a theorem establishing a very general form of bounds for distillable entanglement.

Secure Key from Bound Entanglement

We characterize the set of shared quantum states which contain a cryptographically private key. This allows us to recast the theory of privacy as a paradigm closely related to that used in entanglement manipulation. It is shown that one can distill an arbitrarily secure key from bound entangled states. There are also states that have less distillable private keys than the entanglement cost of the state. In general, the amount of distillable key is bounded from above by the relative entropy of entanglement. Relationships between distillability and distinguishability are found for a class of states which have Bell states correlated to separable hiding states. We also describe a technique for finding states exhibiting irreversibility in entanglement distillation.

Bound entanglement can be activated

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.

The asymptotic entanglement cost of preparing a quantum state

We give a detailed proof of the conjecture that the asymptotic entanglement cost of preparing a state ρ is equal to limn→∞ Ef (ρ ⊗n )/n where Ef is the entanglement of formation.