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Pawel Horodecki

Publications and source records attributed to Pawel Horodecki.

At least 91 records · Page 5Linked to original sources
arXiv2001

Rank Two Bipartite Bound Entangled States Do Not Exist

We explore the relation between the rank of a bipartite density matrix and the existence of bound entanglement. We show a relation between the rank, marginal ranks, and distillability of a mixed state and use this to prove that any rank n bound entangled state must have support on no more than an n \times n Hilbert space. A direct consequence of this result is that there are no bipartite bound entangled states of rank two. We also show that a separability condition in terms of a quantum entropy inequality is associated with the above results. We explore the idea of how many pure states are needed in a mixture to cancel the distillable entanglement of a Schmidt rank n pure state and provide a lower bound of n-1. We also prove that a mixture of a non-zero amount of any pure entangled state with a pure product state is distillable.

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arXiv2001

Classical capacity of a noiseless quantum channel assisted by noisy entanglement

We derive the general formula for the capacity of a noiseless quantum channel assisted by an arbitrary amount of noisy entanglement. In this capacity formula, the ratio of the quantum mutual information and the von Neumann entropy of the sender's share of the noisy entanglement plays the role of mutual information in the completely classical case. A consequence of our results is that bound entangled states cannot increase the capacity of a noiseless quantum channel.

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arXiv2001

Concurrence in arbitrary dimensions

We argue that a complete characterisation of quantum correlations in bipartite systems of many dimensions may require a quantity which, even for pure states, does not reduce to a single number. Subsequently, we introduce multi-dimensional generalizations of concurrence and find evidence that they may provide useful tools for the analysis of quantum correlations in mixed bipartite states. We also introudce {\it biconcurrence} that leads to a necessary and sufficient condition for separability.

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arXiv2001

Dynamical description of quantum computing: generic nonlocality of quantum noise

We develop dynamical non-Markovian description of quantum computing in weak coupling limit, in lowest order approximation. We show that long range memory of quantum reservoir produces strong interrelation between structure of noise and quantum algorithm, implying nonlocal attacks of noise. We then argue that the quantum error correction method fails to protect quantum computation against electromagnetic or phonon vacuum which exhibit $1/t^4$ memory. This shows that the implicit assumption of quantum error correction theory -- independence of noise and self-dynamics -- fails in long time regimes. We also use our approach to present {\it pure} decoherence and decoherence accompanied by dissipation in terms of spectral density of reservoir. The so-called {\it dynamical decoupling} method is discussed in this context. Finally, we propose {\it minimal decoherence model}, in which the only source of decoherence is vacuum. We optimize fidelity of quantum information processing under the trade-off between speed of gate and strength of decoherence.

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arXiv2000

Relativistic J-matrix method

The relativistic version of the J-matrix method for a scattering problem on the potential vanishing faster than the Coulomb one is formulated. As in the non-relativistic case it leads to a finite algebraic eigenvalue problem. The derived expression for the tangent of phase shift is simply related to the non-relativistic case formula and gives the latter as a limit case. It is due to the fact that the used basis set satisfies the ``kinetic balance condition''.

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arXiv2000

Bound entanglement and continuous variables

We introduce the definition of generic bound entanglement for the case of continuous variables. We provide some examples of bound entangled states for that case, and discuss their physical sense in the context of quantum optics. We rise the question of whether the entanglement of these states is generic. As a byproduct we obtain a new many parameter family of bound entangled states with positive partial transpose. We also point out that the ``entanglement witnesses'' and positive maps revealing the corresponding bound entanglement can be easily constructed.

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arXiv2000

On balance of information in bipartite quantum communication systems: entanglement-energy analogy

We adopt the view according to which information is the primary physical entity that posseses objective meaning. Basing on two postulates that (i) entanglement is a form of quantum information corresponding to internal energy (ii) sending qubits corresponds to work, we show that in the closed bipartite quantum communication systems the information is conserved. We also discuss entanglement-energy analogy in context of the Gibbs-Hemholtz-like equation connecting the entanglement of formation, distillable entanglement and bound entanglement. Then we show that in the deterministic protocols of distillation the information is conserved. We also discuss the objectivity of quantum information in context of information interpretation of quantum states and alghoritmic complexity.

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arXiv2000

Separability of n-particle mixed states: necessary and sufficient conditions in terms of linear maps

We provide necessary and sufficient conditions for separability of mixed states of n-particle systems. The conditions are formulated in terms of maps which are positive on product states of $n-1$ particles. The method of providing of the maps on the basis of unextendable product bases is provided. The three qubit state problem is reformulated in the form suggesting possibility of explicite characterisation of all maps needed for separability condition.

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arXiv2000

Unified approach to quantum capacities: towards quantum noisy coding theorem

Basing on unified approach to {\it all} kinds of quantum capacities we show that the rate of quantum information transmission is bounded by the maximal attainable rate of coherent information. Moreover, we show that, if for any bipartite state the one-way distillable entanglement is no less than coherent information, then one obtains Shannon-like formulas for all the capacities. The inequality also implies that the decrease of distillable entanglement due to mixing process does not exceed of corresponding loss information about a system.

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arXiv2000

A Schmidt number for density matrices

We introduce the notion of a Schmidt number of a bipartite density matrix, characterizing the minimum Schmidt rank of the pure states that are needed to construct the density matrix. We prove that Schmidt number is nonincreasing under local quantum operations and classical communication. We show that $k$-positive maps witness Schmidt number, in the same way that positive maps witness entanglement. We show that the family of states which is made from mixing the completely mixed state and a maximally entangled state have increasing Schmidt number depending on the amount of maximally entangled state that is mixed in. We show that Schmidt number {\it does not necessarily increase} when taking tensor copies of a density matrix $ρ$; we give an example of a density matrix for which the Schmidt numbers of $ρ$ and $ρ\otimes ρ$ are both 2.

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arXiv2000

Limits for entanglement measures

We show that {\it any} entanglement measure $E$ suitable for the regime of high number of entangled pairs satisfies $E_D\leq E\leq E_F$ where $E_D$ and $E_F$ are entanglement of distillation and formation respectively. We also exhibit a general theorem on bounds for distillable entanglement. The results are obtained by use of a very transparent reasoning based on the fundamental principle of entanglement theory saying that entanglement cannot increase under local operations and classical communication.

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arXiv2000

Operational criterion and constructive checks for the separability of low rank density matrices

We consider low rank density operators $\varrho$ supported on a $M\times N$ Hilbert space for arbitrary $M$ and $N$ ($M\leq N$) and with a positive partial transpose (PPT) $\varrho^{T_A}\ge 0$. For rank $r(\varrho) \leq N$ we prove that having a PPT is necessary and sufficient for $\varrho$ to be separable; in this case we also provide its minimal decomposition in terms of pure product states. It follows from this result that there is no rank 3 bound entangled states having a PPT. We also present a necessary and sufficient condition for the separability of generic density matrices for which the sum of the ranks of $\varrho$ and $\varrho^{T_A}$ satisfies $r(\varrho)+r(\varrho^{T_A}) \le 2MN-M-N+2$. This separability condition has the form of a constructive check, providing thus also a pure product state decomposition for separable states, and it works in those cases where a system of couple polynomial equations has a finite number of solutions, as expected in most cases.

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arXiv1999

Local environment can enhance fidelity of quantum teleportation

We show how an interaction with the environment can enhance fidelity of quantum teleportation. To this end, we present examples of states which cannot be made useful for teleportation by any local unitary transformations; nevertheless, after being subjected to a dissipative interaction with the local environment, the states allow for teleportation with genuinely quantum fidelity. The surprising fact here is that the necessary interaction does not require any intelligent action from the parties sharing the states. In passing, we produce some general results regarding optimization of teleportation fidelity by local action. We show that bistochastic processes cannot improve fidelity of two-qubit states. We also show that in order to have their fidelity improvable by a local process, the bipartite states must violate the so-called reduction criterion of separability.

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arXiv1999

MR spectroscopy with parabolic magnetic field: spin-oscillator coupling effect

The spectrum of the spin particle in oscillatory potential subjected to external parabolic magnetic field ${\bf B}=(B_0+Gx+\tilde G x^2){\bf \hat z}$ is obtained. The structure of energy levels of the considered system allows to identify the frequency of the oscillator via the spectrum of spin sublevels coming only from {\it one} oscillatory level. The effect is due to the gradient terms in the form of the field.

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arXiv1999

Teleportation via generalized measurements, and conclusive teleportation

In this work we show that teleportation is a special case of a generalized Einstein, Podolsky, Rosen (EPR) non-locality. Based on the connection between teleportation and generalized measurements we define conclusive teleportation. We show that perfect conclusive teleportation can be obtained with any pure entangled state, and it can be arbitrarily approached with a particular mixed state.

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arXiv1999

Binding entanglement channels

We define the binding entanglement channel as the quantum channel through which quantum information cannot be reliably transmitted, but which can be used to share bound entanglement. We provide a characterization of such class of channels. We also show that any bound entangled state can be used to construction of the map corresponding the binding entanglement channel.

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arXiv1999

General teleportation channel, singlet fraction and quasi-distillation

We prove a theorem on direct relation between the optimal fidelity $f_{max}$ of teleportation and the maximal singlet fraction $F_{max}$ attainable by means of trace-preserving LQCC action (local quantum and classical communication). For a given bipartite state acting on $C^d\otimes C^d$ we have $f_{max}= {F_{max}d+1\over d+1}$. We assume completely general teleportation scheme (trace preserving LQCC action over the pair and the third particle in unknown state). The proof involves the isomorphism between quantum channels and a class of bipartite states. We also exploit the technique of $U\otimes U^*$ twirling states (random application of unitary transformation of the above form) and the introduced analogous twirling of channels. We illustrate the power of the theorem by showing that {\it any} bound entangled state does not provide better fidelity of teleportation than for the purely classical channel. Subsequently, we apply our tools to the problem of the so-called conclusive teleportation, then reduced to the question of optimal conclusive increasing of singlet fraction. We provide an example of state for which Alice and Bob have no chance to obtain perfect singlet by LQCC action, but still singlet fraction arbitrarily close to unity can be obtained with nonzero probability. We show that a slight modification of the state has a threshold for singlet fraction which cannot be exceeded anymore.

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