arXiv subjects

Karol Horodecki

Publications and source records attributed to Karol Horodecki.

At least 55 records · Page 3Linked to original sources

arXiv2015

Robust Device-Independent Randomness Amplification with Few Devices

Randomness amplification is the task of transforming a source of somewhat random bits into a source of fully random bits. Although it is impossible to amplify randomness from a single source by classical means, the situation is different considering non-local correlations allowed by quantum mechanics. Here we give the first device-independent protocol for randomness amplification using a constant number of devices. The protocol involves four devices, can amplify any non-deterministic source into a fully random source, tolerates a constant rate of error, and has its correctness based solely on the assumption of no-signaling between the devices. In contrast all previous protocols either required an unbounded number of devices, or could only amplify sources sufficiently close to fully random.

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arXiv2015

Limitations on Quantum Key Repeaters

A major application of quantum communication is the distribution of entangled particles for use in quantum key distribution (QKD). Due to noise in the communication line, QKD is in practice limited to a distance of a few hundred kilometres, and can only be extended to longer distances by use of a quantum repeater, a device which performs entanglement distillation and quantum teleportation. The existence of noisy entangled states that are undistillable but nevertheless useful for QKD raises the question of the feasibility of a quantum key repeater, which would work beyond the limits of entanglement distillation, hence possibly tolerating higher noise levels than existing protocols. Here we exhibit fundamental limits on such a device in the form of bounds on the rate at which it may extract secure key. As a consequence, we give examples of states suitable for QKD but unsuitable for the most general quantum key repeater protocol.

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arXiv2014

When Are Popescu-Rohrlich Boxes and Random Access Codes Equivalent?

We study a problem of interconvertibility of two supra-quantum resources: one is so called PR-box, which violates CHSH inequality up to maximal algebraic bound, and second is so called random access code (RAC). The latter is a functionality that enables Bob (receiver) to choose one of two bits of Alice. It has been known, that PR-box supplemented with one bit of communication can be used to simulate RAC. We ask the converse question: to what extent RAC can simulate PR-box? To this end we introduce racbox: a box such that supplemented with one bit of communication offers RAC. As said, PR-box can simulate racbox. The question we raise, is whether any racbox can simulate PR-box. We show that a non-signaling racbox indeed can simulate PR-box, hence those two resources are equivalent. We also provide an example of signalling racbox which cannot simulate PR-box. We give a resource inequality between racbox es and PR-boxes, and show that it is saturated.

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arXiv2014

On distinguishing of non-signaling boxes via completely locality preserving operations

We consider discriminating between bipartite boxes with 2 binary inputs and 2 binary outputs (2x2) using the class of completely locality preserving operations i.e. those, which transform boxes with local hidden variable model (LHVM) into boxes with LHVM, and have this property even when tensored with identity operation. Following approach developed in entanglement theory we derive linear program which gives an upper bound on the probability of success of discrimination between different isotropic boxes. In particular we provide an upper bound on the probability of success of discrimination between isotropic boxes with the same mixing parameter. As a counterpart of entanglement monotone we use the non-locality cost. Discrimination is restricted by the fact that non-locality cost does not increase under considered class of operations. We also show that with help of allowed class of operations one can distinguish perfectly any two extremal boxes in 2x2 case and any local extremal box from any other extremal box in case of two inputs and two outputs of arbitrary cardinalities.

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arXiv2013

Bound entangled states with extremal properties

Following recent work of Beigi and Shor, we investigate PPT states that are "heavily entangled." We first exploit volumetric methods to show that in a randomly chosen direction, there are PPT states whose distance in trace norm from separable states is (asymptotically) at least 1/4. We then provide explicit examples of PPT states which are nearly as far from separable ones as possible. To obtain a distance of 2-ε from the separable states, we need a dimension of 2^{poly(\log(1/ε))}, as opposed to 2^{poly(1/ε)} given by the construction of Beigi and Shor. We do so by exploiting the so called {\it private states}, introduced earlier in the context of quantum cryptography. We also provide a lower bound for the distance between private states and PPT states and investigate the distance between pure states and the set of PPT states.

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arXiv2013

Robust Device Independent Randomness Amplification

In randomness amplification a slightly random source is used to produce an improved random source. Perhaps surprisingly, a single source of randomness cannot be amplified at all classically. However, the situation is different if one considers correlations allowed by quantum mechanics as an extra resource. Here we present a protocol that amplifies Santha-Vazirani sources arbitrarily close to deterministic into fully random sources. The protocol is device independent, depending only on the observed statistics of the devices and on the validity of the no-signaling principle between different devices. It improves previously-known protocols in two respects. First the protocol is tolerant to noise so that even noisy quantum-mechanical systems give rise to good devices for the protocol. Second it is simpler, being based on the violation of a four-party Bell inequality and on the XOR as a hash function. As a technical tool we prove a new de Finetti theorem where the subsystems are selected from a Santha-Vazirani source.

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arXiv2013

Free randomness amplification using bipartite chain correlations

A direct analysis of the protocol of randomness amplification using Bell inequality violation is performed in terms of the convex combination of no-signaling boxes required to simulate quantum violation of the inequality. The probability distributions of bits generated by a Santha-Vazirani source are shown to be mixtures of permutations of Bernoulli distributions with parameter defined by the source. An intuitive proof is provided for the range of partial randomness from which perfect randomness can be extracted using quantum correlations violating the chain inequalities. Exact values are derived in the asymptotic limit of a large number of measurement settings.

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arXiv2011

Quantum privacy witness

While it is usually known that the mean value of a single observable is enough to detect entanglement or its distillability, the counterpart of such an approach in the case of quatum privacy has been missing. Here we develop the concept of a privacy witness, i.e. a single observable that may detect presence of the secure key even in the case of bound entanglement. Then we develop the notion of secret key estimation based on few observables and discuss the witness decomposition into local measurements. The surprising property of the witness is that with the help of a low number of product mesurements involved it may still report the key values that are {\it strictly above} distillable entanglement of the state. For an exemplary four-qubit state studied in a recent experiment [K. Dobek {\em et al.}, Phys. Rev. Lett. {\bf 106}, 030501 (2011)] this means 6 Pauli operator product measurements versus 81 needed to carry out the complete quantum state tomography. The present approach may be viewed as a paradigm for the general program of experimentally friendly detection and estimation of task-dedicated quantum entanglement.

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arXiv2010

Contextuality offers security

The discovery of quantum key distribution by Bennett and Brassard (BB84) bases on the fundamental quantum feature: incompatibility of measurements of quantum non-commuting observables. In 1991 Ekert showed that cryptographic key can be generated at a distance with help of entangled (correlated) quantum particles. Recently Barrett, Hardy and Kent showed that the non-locality used by Ekert is itself a good resource of cryptographic key even beyond quantum mechanics. Their result paved the way to new generation of quantum cryptographic protocols - secure even if the devices are built by the very eavesdropper. However, there is a question, which is fundamental from both practical and philosophical point of view: does Nature offer security on operational level based on the original concept behind quantum cryptography - that information gain about one bservables must cause disturbance to another, incompatible one? Here we resolve this problem by using in place of non-locality another striking feature of quantum world - contextuality. It is a strong version of incompatibility manifested in the famous Kochen-Specker paradox. The crucial concept is the use of a new class of families of bipartite probability distributions which locally exhibit the Kochen-Specker paradox conditions and, in addition, exhibit perfect correlations. We show that if two persons share systems described by such a family then they can extract secure key. This is the first operational protocol that directly implements the fundamental feature of Nature: the information gain vs. disturbance trade-off. At the same time it provides natural device-independent cryptographic scheme within quantum mechanics suitable for current technology.

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arXiv2010

Contextuality offers device-independent security

The discovery of quantum key distribution by Bennett and Brassard (BB84) bases on the fundamental quantum feature: incompatibility of measurements of quantum non-commuting observables. In 1991 Ekert showed that cryptographic key can be generated at a distance with help of entangled (correlated) quantum particles. Recently Barrett, Hardy and Kent showed that the non-locality used by Ekert is itself a good resource of cryptographic key even beyond quantum mechanics. Their result paved the way to new generation of quantum cryptographic protocols - secure even if the devices are built by the very eavesdropper. However, there is a question, which is fundamental from both practical and philosophical point of view: does Nature offer security on operational level based on the original concept behind quantum cryptography - that information gain about one observable must cause disturbance to another, incompatible one? Here we resolve this problem by using another striking feature of quantum world - contextuality. It is a strong version of incompatibility manifested in the famous Kochen-Specker paradox. The crucial concept is the use of a new class of families of bipartite probability distributions . We show that if two persons share systems which locally exhibit the Kochen-Specker paradox and, in addition, exhibit perfect correlations, then they can extract secure key, even if they do not trust the quantum devices. This is the first operational protocol that directly implements the fundamental feature of Nature: the information gain vs. disturbance trade-off. For sake of proof we exhibit a new Bell's inequality which is interesting in itself. The security is proved not by exploiting strong violation of the inequality by quantum mechanics (as one usually proceeds), but rather by arguing, that quantum mechanics cannot violate it too much.

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arXiv2009

Squashed entanglement for multipartite states and entanglement measures based on the mixed convex roof

New measures of multipartite entanglement are constructed based on two definitions of multipartite information and different methods of optimizing over extensions of the states. One is a generalization of the squashed entanglement where one takes the mutual information of parties conditioned on the state's extension and takes the infimum over such extensions. Additivity of the multipartite squashed entanglement is proved for both versions of the multipartite information which turn out to be related. The second one is based on taking classical extensions. This scheme is generalized, which enables to construct measures of entanglement based on the {\it mixed convex roof} of a quantity, which in contrast to the standard convex roof method involves optimization over all decompositions of a density matrix rather than just the decompositions into pure states. As one of the possible applications of these results we prove that any multipartite monotone is an upper bound on the amount of multipartite distillable key. The findings are finally related to analogous results in classical key agreement.

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arXiv2009

When non i.i.d. information sources can be communicationally useful?

Information sources used for communication purposes usually are assumed to be i.i.d. type, especially as far as entanglement or nonlocality properties are concerned. Here we proposed simple scheme for detection of nonlocality of non i.i.d. sources based on the program idea which can be useful in communication tasks including quantum cryptography and communication complexity reduction. The principle of the scheme is rather general and can be applied to other problems like detection of entanglement coming from non-i.i.d sources.

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arXiv2008

General paradigm for distilling classical key from quantum states

We develop a formalism for distilling a classical key from a quantum state in a systematic way, expanding on our previous work on secure key from bound entanglement [K. Horodecki et. al., Phys. Rev. Lett. 94 (2005)]. More detailed proofs, discussion and examples are provided of the main results. Namely, we demonstrate that all quantum cryptographic protocols can be recast in a way which looks like entanglement theory, with the only change being that instead of distilling EPR pairs, the parties distill private states. The form of these general private states are given, and we show that there are a number of useful ways of expressing them. Some of the private states can be approximated by certain states which are bound entangled. Thus distillable entanglement is not a requirement for a private key. We find that such bound entangled states are useful for a cryptographic primitive we call a controlled private quantum channel. We also find a general class of states which have negative partial transpose (are NPT), but which appear to be bound entangled. The relative entropy distance is shown to be an upper bound on the rate of key. This allows us to compute the exact value of distillable key for a certain class of private states.

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arXiv2007

Quantum key distribution based on private states: unconditional security over untrusted channels with zero quantum capacity

We prove unconditional security for a quantum key distribution (QKD) protocol based on distilling pbits (twisted ebits) [quant-ph/0309110] from an arbitrary untrusted state that is claimed to contain distillable key. Our main result is that we can verify security using only public communication -- via parameter estimation of the given untrusted state. The technique applies even to bound entangled states, thus extending QKD to the regime where the available quantum channel has zero quantum capacity. We also show how to convert our purification-based QKD schemes to prepare-measure schemes.

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arXiv2007

Quantum entanglement

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 Schrödinger - 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.

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arXiv2007

Unconditional privacy over channels which cannot convey quantum information

By sending systems in specially prepared quantum states, two parties can communicate without an eavesdropper being able to listen. The technique, called quantum cryptography, enables one to verify that the state of the quantum system has not been tampered with, and thus one can obtain privacy regardless of the power of the eavesdropper. All previous protocols relied on the ability to faithfully send quantum states. In fact, until recently, they could all be reduced to a single protocol where security is ensured though sharing maximally entangled states. Here we show this need not be the case -- one can obtain verifiable privacy even through some channels which cannot be used to reliably send quantum states.

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arXiv2005

Are quantum correlations symmetric ?

To our knowledge, all known bipartite entanglement measures are symmetric under exchange of subsystems. We ask if an entanglement measure that is not symmetric can exist. A related question is if there is a state that cannot be swapped by means of LOCC. We show, that in general one cannot swap states by LOCC. This allows to construct nonsymmetric measure of entanglement, and a parameter that reports asymmetry of entanglement contents of quantum state. We propose asymptotic measure of asymmetry of entanglement, and show that states for which it is nonzero, contain necessarily bound entanglement.

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arXiv2005

Quantum key distribution based on arbitrarily-weak distillable entangled states

States with private correlations but little or no distillable entanglement were recently reported. Here, we consider the secure distribution of such states, i.e., the situation when an adversary gives two parties such states and they have to verify privacy. We present a protocol which enables the parties to extract from such untrusted states an arbitrarily long and secure key, even though the amount of distillable entanglement of the untrusted states can be arbitrarily small.

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