Paweł Horodecki

Bio: Paweł Horodecki is an academic researcher from Gdańsk University of Technology. The author has contributed to research in topics: Quantum entanglement & Quantum information. The author has an hindex of 46, co-authored 171 publications receiving 16704 citations. Previous affiliations of Paweł Horodecki include University of Gdańsk.

Quantum entanglement

Abstract: All our former experience with application of quantum theory seems to say: {\it what is predicted by quantum formalism must occur in laboratory} But the essence of quantum formalism - entanglement, recognized by Einstein, Podolsky, Rosen and Schr\"odinger - waited over 70 years to enter to laboratories as a new resource as real as energy This holistic property of compound quantum systems, which involves nonclassical correlations between subsystems, is a potential for many quantum processes, including ``canonical'' ones: quantum cryptography, quantum teleportation and dense coding However, it appeared that this new resource is very complex and difficult to detect Being usually fragile to environment, it is robust against conceptual and mathematical tools, the task of which is to decipher its rich structure This article reviews basic aspects of entanglement including its characterization, detection, distillation and quantifying In particular, the authors discuss various manifestations of entanglement via Bell inequalities, entropic inequalities, entanglement witnesses, quantum cryptography and point out some interrelations They also discuss a basic role of entanglement in quantum communication within distant labs paradigm and stress some peculiarities such as irreversibility of entanglement manipulations including its extremal form - bound entanglement phenomenon A basic role of entanglement witnesses in detection of entanglement is emphasized

Volume of the set of separable states

Abstract: The question of how many entangled or, respectively, separable states there are in the set of all quantum states is considered. We propose a natural measure in the space of density matrices $\ensuremath{\varrho}$ describing $N$-dimensional quantum systems. We prove that, under this measure, the set of separable states possesses a nonzero volume. Analytical lower and upper bounds of this volume are also derived for $N=2\ifmmode\times\else\texttimes\fi{}2$ and $N=2\ifmmode\times\else\texttimes\fi{}3$ cases. Finally, numerical Monte Carlo calculations allow us to estimate the volume of separable states, providing numerical evidence that it decreases exponentially with the dimension of the composite system. We have also analyzed a conditional measure of separability under the condition of fixed purity. Our results display a clear dualism between purity and separability: entanglement is typical of pure states, while separability is connected with quantum mixtures. In particular, states of sufficiently low purity are necessarily separable.

Mixed-State Entanglement and Distillation: Is there a “Bound” Entanglement in Nature?

Abstract: 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. A possible physical meaning of the result is discussed.

General teleportation channel, singlet fraction, and quasidistillation

Abstract: We prove a theorem on direct relation between the optimal fidelity ${f}_{\mathrm{max}}$ of teleportation and the maximal singlet fraction ${F}_{\mathrm{max}}$ attainable by means of trace-preserving local quantum and classical communication (LQCC) action. For a given bipartite state acting on ${C}^{d}\ensuremath{\bigotimes}{C}^{d}$ we have ${f}_{\mathrm{max}}{=(F}_{\mathrm{max}}d+1)/(d+1)$. 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 $U\ensuremath{\bigotimes}{U}^{*}$ 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 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.

Reduction criterion of separability and limits for a class of distillation protocols

Abstract: We analyze 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 analyze a criterion of separability that 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 that distills any inseparable two-qubit state. In particular, this means that any state $\ensuremath{\varrho}$ of two N-level systems with $〈{\ensuremath{\psi}}_{+}|\ensuremath{\varrho}|{\ensuremath{\psi}}_{+}〉g1/N$ can be distilled $({\ensuremath{\psi}}_{+}$ is the singlet state generalized to higher dimension). The criterion also singles out all the states that can be distilled by a class of protocols. The proof involves construction of the family of states that are invariant under transformation $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\varrho}}U\ensuremath{\bigotimes}{U}^{*}\ensuremath{\varrho}{U}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\bigotimes}{U}^{*\ifmmode\dagger\else\textdagger\fi{}},$ where U is a unitary transformation and the asterisk denotes complex conjugation. The states are related to the depolarizing channel generalized to the nonbinary case.

I and i

Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality. Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

Quantum Computation and Quantum Information

Abstract: Part I. Fundamental Concepts: 1. Introduction and overview 2. Introduction to quantum mechanics 3. Introduction to computer science Part II. Quantum Computation: 4. Quantum circuits 5. The quantum Fourier transform and its application 6. Quantum search algorithms 7. Quantum computers: physical realization Part III. Quantum Information: 8. Quantum noise and quantum operations 9. Distance measures for quantum information 10. Quantum error-correction 11. Entropy and information 12. Quantum information theory Appendices References Index.

Quantum entanglement

Abstract: All our former experience with application of quantum theory seems to say: {\it what is predicted by quantum formalism must occur in laboratory} But the essence of quantum formalism - entanglement, recognized by Einstein, Podolsky, Rosen and Schr\"odinger - waited over 70 years to enter to laboratories as a new resource as real as energy This holistic property of compound quantum systems, which involves nonclassical correlations between subsystems, is a potential for many quantum processes, including ``canonical'' ones: quantum cryptography, quantum teleportation and dense coding However, it appeared that this new resource is very complex and difficult to detect Being usually fragile to environment, it is robust against conceptual and mathematical tools, the task of which is to decipher its rich structure This article reviews basic aspects of entanglement including its characterization, detection, distillation and quantifying In particular, the authors discuss various manifestations of entanglement via Bell inequalities, entropic inequalities, entanglement witnesses, quantum cryptography and point out some interrelations They also discuss a basic role of entanglement in quantum communication within distant labs paradigm and stress some peculiarities such as irreversibility of entanglement manipulations including its extremal form - bound entanglement phenomenon A basic role of entanglement witnesses in detection of entanglement is emphasized

Mixed State Entanglement and Quantum Error Correction

Abstract: Entanglement purification protocols (EPPs) and quantum error-correcting codes (QECCs) provide two ways of protecting quantum states from interaction with the environment. In an EPP, perfectly entangled pure states are extracted, with some yield D, from a mixed state M shared by two parties; with a QECC, an arbitrary quantum state |\ensuremath{\xi}〉 can be transmitted at some rate Q through a noisy channel \ensuremath{\chi} without degradation. We prove that an EPP involving one-way classical communication and acting on mixed state M^(\ensuremath{\chi}) (obtained by sharing halves of Einstein-Podolsky-Rosen pairs through a channel \ensuremath{\chi}) yields a QECC on \ensuremath{\chi} with rate Q=D, and vice versa. We compare the amount of entanglement E(M) required to prepare a mixed state M by local actions with the amounts ${\mathit{D}}_{1}$(M) and ${\mathit{D}}_{2}$(M) that can be locally distilled from it by EPPs using one- and two-way classical communication, respectively, and give an exact expression for E(M) when M is Bell diagonal. While EPPs require classical communication, QECCs do not, and we prove Q is not increased by adding one-way classical communication. However, both D and Q can be increased by adding two-way communication. We show that certain noisy quantum channels, for example a 50% depolarizing channel, can be used for reliable transmission of quantum states if two-way communication is available, but cannot be used if only one-way communication is available. We exhibit a family of codes based on universal hashing able to achieve an asymptotic Q (or D) of 1-S for simple noise models, where S is the error entropy. We also obtain a specific, simple 5-bit single-error-correcting quantum block code. We prove that iff a QECC results in high fidelity for the case of no error then the QECC can be recast into a form where the encoder is the matrix inverse of the decoder. \textcopyright{} 1996 The American Physical Society.

Entanglement in many-body systems

Abstract: Recent interest in aspects common to quantum information and condensed matter has prompted a flurry of activity at the border of these disciplines that were far distant until a few years ago. Numerous interesting questions have been addressed so far. Here an important part of this field, the properties of the entanglement in many-body systems, are reviewed. The zero and finite temperature properties of entanglement in interacting spin, fermion, and boson model systems are discussed. Both bipartite and multipartite entanglement will be considered. In equilibrium entanglement is shown tightly connected to the characteristics of the phase diagram. The behavior of entanglement can be related, via certain witnesses, to thermodynamic quantities thus offering interesting possibilities for an experimental test. Out of equilibrium entangled states are generated and manipulated by means of many-body Hamiltonians.