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Andreas Winter

Publications and source records attributed to Andreas Winter.

At least 199 records · Page 11Linked to original sources
arXiv2008

The quantum capacity with symmetric side channels

We present an upper bound for the quantum channel capacity that is both additive and convex. Our bound can be interpreted as the capacity of a channel for high-fidelity quantum communication when assisted by a family of channels that have no capacity on their own. This family of assistance channels, which we call symmetric side channels, consists of all channels mapping symmetrically to their output and environment. The bound seems to be quite tight, and for degradable quantum channels it coincides with the unassisted channel capacity. Using this symmetric side channel capacity, we find new upper bounds on the capacity of the depolarizing channel. We also briefly indicate an analogous notion for distilling entanglement using the same class of (one-way) channels, yielding one of the few entanglement measures that is monotonic under local operations with one-way classical communication (1-LOCC), but not under the more general class of local operations with classical communication (LOCC).

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arXiv2008

Higher entropic uncertainty relations for anti-commuting observables

Uncertainty relations lie at the very core of quantum mechanics, and form the cornerstone of essentially all quantum cryptographic applications. In particular, they play an important role in cryptographic protocols in the bounded-quantum-storage model, where proving the security of all existing protocols ultimately reduces to bounding such relations. Yet, very little is known about such uncertainty relations for more than two measurements. Here, we prove optimal entropic uncertainty relations for anti-commuting binary observables for the Shannon entropy, and nearly optimal relations for the collision entropy. Our results have immediate applications to quantum cryptography.

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arXiv2008

Counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1

For all p > 1, we demonstrate the existence of quantum channels with non-multiplicative maximal output p-norms. Equivalently, for all p >1, the minimum output Renyi entropy of order p of a quantum channel is not additive. The violations found are large; in all cases, the minimum output Renyi entropy of order p for a product channel need not be significantly greater than the minimum output entropy of its individual factors. Since p=1 corresponds to the von Neumann entropy, these counterexamples demonstrate that if the additivity conjecture of quantum information theory is true, it cannot be proved as a consequence of any channel-independent guarantee of maximal p-norm multiplicativity. We also show that a class of channels previously studied in the context of approximate encryption lead to counterexamples for all p > 2.

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arXiv2008

A classical analogue of negative information

Recently, it was discovered that the `quantum partial information' needed to merge one party's state with another party's state is given by the conditional entropy, which can be negative [Horodecki, Oppenheim, and Winter, Nature 436, 673 (2005)]. Here we find a classical analogue of this, based on a long known relationship between entanglement and shared private correlations: namely, we consider a private distribution held between two parties, and correlated to a reference system, and ask how much secret communication is needed for one party to send her distribution to the other. We give optimal protocols for this task, and find that private information can be negative - the sender's distribution can be transferred and the potential to send future distributions in secret is gained through the distillation of a secret key. An analogue of `quantum state exchange' is also discussed and one finds cases where exchanging a distribution costs less than for one party to send it. The results give new classical protocols, and also clarify the various relationships between entanglement and privacy.

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arXiv2008

Entanglement and separability of quantum harmonic oscillator systems at finite temperature

In the present paper we study the entanglement properties of thermal (a.k.a. Gibbs) states of quantum harmonic oscillator systems as functions of the Hamiltonian and the temperature. We prove the physical intuition that at sufficiently high temperatures the thermal state becomes fully separable and we deduce bounds on the critical temperature at which this happens. We show that the bound becomes tight for a wide class of Hamiltonians with sufficient translation symmetry. We find, that at the crossover the thermal energy is of the order of the energy of the strongest normal mode of the system and quantify the degree of entanglement below the critical temperature. Finally, we discuss the example of a ring topology in detail and compare our results with previous work in an entanglement-phase diagram.

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arXiv2008

Uncommon information (the cost of exchanging a quantum state)

If two parties share an unknown quantum state, one can ask how much quantum communication is needed for party A to send her share to party B. Recently, it was found that the number of qubits which should be sent is given by the conditional entropy. This quantifies the notion of partial information, and it can even be negative. Here, we not only demand that A send her state to B, but additionally, B should send his state to A. Paradoxically, we find that requiring that the parties perform this additional task can lower the amount of quantum communication required. This primitive, which we call quantum state exchange, can be used to quantify the notion of uncommon information, since the two parties only need to send each other the parts of their state they don't hold in common. In the classical case, the concept of uncommon information follows trivially from the concept of partial information. We find that for quantum states, this is not so. We prove upper and lower bounds for the uncommon information and find optimal protocols for several classes of states.

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arXiv2008

Counterexamples to additivity of minimum output p-Renyi entropy for p close to 0

Complementing recent progress on the additivity conjecture of quantum information theory, showing that the minimum output p-Renyi entropies of channels are not generally additive for p>1, we demonstrate here by a careful random selection argument that also at p=0, and consequently for sufficiently small p, there exist counterexamples. An explicit construction of two channels from 4 to 3 dimensions is given, which have non-multiplicative minimum output rank; for this pair of channels, numerics strongly suggest that the p-Renyi entropy is non-additive for all p < 0.11. We conjecture however that violations of additivity exist for all p<1.

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arXiv2008

Quantum coding theorem from privacy and distinguishability

We prove direct quantum coding theorem for random quantum codes. The problem is separated into two parts: proof of distinguishability of codewords by receiver, and that of indistinguishability of codewords by environment (privacy). For a large class of codes, only privacy has to be checked.

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arXiv2007

Random quantum codes from Gaussian ensembles and an uncertainty relation

Using random Gaussian vectors and an information-uncertainty relation, we give a proof that the coherent information is an achievable rate for entanglement transmission through a noisy quantum channel. The codes are random subspaces selected according to the Haar measure, but distorted as a function of the sender's input density operator. Using large deviations techniques, we show that classical data transmitted in either of two Fourier-conjugate bases for the coding subspace can be decoded with low probability of error. A recently discovered information-uncertainty relation then implies that the quantum mutual information for entanglement encoded into the subspace and transmitted through the channel will be high. The monogamy of quantum correlations finally implies that the environment of the channel cannot be significantly coupled to the entanglement, and concluding, which ensures the existence of a decoding by the receiver.

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arXiv2006

Robustness of quantum Markov chains

If the conditional information of a classical probability distribution of three random variables is zero, then it obeys a Markov chain condition. If the conditional information is close to zero, then it is known that the distance (minimum relative entropy) of the distribution to the nearest Markov chain distribution is precisely the conditional information. We prove here that this simple situation does not obtain for quantum conditional information. We show that for tri-partite quantum states the quantum conditional information is always a lower bound for the minimum relative entropy distance to a quantum Markov chain state, but the distance can be much greater; indeed the two quantities can be of different asymptotic order and may even differ by a dimensional factor.

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arXiv2006

A lower bound on entanglement-assisted quantum communication complexity

We prove a general lower bound on the bounded-error entanglement-assisted quantum communication complexity of Boolean functions. The bound is based on the concept that any classical or quantum protocol to evaluate a function on distributed inputs can be turned into a quantum communication protocol. As an application of this bound, we give a very simple proof of the statement that almost all Boolean functions on n+n bits have linear communication complexity, even in the presence of unlimited entanglement.

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arXiv2006

The foundations of statistical mechanics from entanglement: Individual states vs. averages

We consider an alternative approach to the foundations of statistical mechanics, in which subjective randomness, ensemble-averaging or time-averaging are not required. Instead, the universe (i.e. the system together with a sufficiently large environment) is in a quantum pure state subject to a global constraint, and thermalisation results from entanglement between system and environment. We formulate and prove a "General Canonical Principle", which states that the system will be thermalised for almost all pure states of the universe, and provide rigorous quantitative bounds using Levy's Lemma.

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arXiv2006

No quantum advantage for nonlocal computation

We investigate the problem of "nonlocal" computation, in which separated parties must compute a function with nonlocally encoded inputs and output, such that each party individually learns nothing, yet together they compute the correct function output. We show that the best that can be done classically is a trivial linear approximation. Surprisingly, we also show that quantum entanglement provides no advantage over the classical case. On the other hand, generalized (i.e. super-quantum) nonlocal correlations allow perfect nonlocal computation. This gives new insights into the nature of quantum nonlocality and its relationship to generalised nonlocal correlations.

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arXiv2006

On the quantum chromatic number of a graph

We investigate the notion of quantum chromatic number of a graph, which is the minimal number of colours necessary in a protocol in which two separated provers can convince an interrogator with certainty that they have a colouring of the graph. After discussing this notion from first principles, we go on to establish relations with the clique number and orthogonal representations of the graph. We also prove several general facts about this graph parameter and find large separations between the clique number and the quantum chromatic number by looking at random graphs. Finally, we show that there can be no separation between classical and quantum chromatic number if the latter is 2, nor if it is 3 in a restricted quantum model; on the other hand, we exhibit a graph on 18 vertices and 44 edges with chromatic number 5 and quantum chromatic number 4.

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arXiv2006

State Discrimination with Post-Measurement Information

We introduce a new state discrimination problem in which we are given additional information about the state after the measurement, or more generally, after a quantum memory bound applies. In particular, the following special case plays an important role in quantum cryptographic protocols in the bounded storage model: Given a string x encoded in an unknown basis chosen from a set of mutually unbiased bases, you may perform any measurement, but then store at most q qubits of quantum information. Later on, you learn which basis was used. How well can you compute a function f(x) of x, given the initial measurement outcome, the q qubits and the additional basis information? We first show a lower bound on the success probability for any balanced function, and any number of mutually unbiased bases, beating the naive strategy of simply guessing the basis. We then show that for two bases, any Boolean function f(x) can be computed perfectly if you are allowed to store just a single qubit, independent of the number of possible input strings x. However, we show how to construct three bases, such that you need to store all qubits in order to compute f(x) perfectly. We then investigate how much advantage the additional basis information can give for a Boolean function. To this end, we prove optimal bounds for the success probability for the AND and the XOR function for up to three mutually unbiased bases. Our result shows that the gap in success probability can be maximal: without the basis information, you can never do better than guessing the basis, but with this information, you can compute f(x) perfectly. We also exhibit an example where the extra information does not give any advantage at all.

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arXiv2006

Optimal superdense coding of entangled states

We present a one-shot method for preparing pure entangled states between a sender and a receiver at a minimal cost of entanglement and quantum communication. In the case of preparing unentangled states, an earlier paper showed that a 2n-qubit quantum state could be communicated to a receiver by physically transmitting only n+o(n) qubits in addition to consuming n ebits of entanglement and some shared randomness. When the states to be prepared are entangled, we find that there is a reduction in the number of qubits that need to be transmitted, interpolating between no communication at all for maximally entangled states and the earlier two-for-one result of the unentangled case, all without the use of any shared randomness. We also present two applications of our result: a direct proof of the achievability of the optimal superdense coding protocol for entangled states produced by a memoryless source, and a demonstration that the quantum identification capacity of an ebit is two qubits.

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arXiv2006

Implications of Superstrong Nonlocality for Cryptography

Non-local boxes are hypothetical ``machines'' that give rise to superstrong non-local correlations, leading to a stronger violation of Bell/CHSH inequalities than is possible within the framework of quantum mechanics. We show how non-local boxes can be used to perform any two-party secure computation. We first construct a protocol for bit commitment and then show how to achieve oblivious transfer using non-local boxes. Both have been shown to be impossible using quantum mechanics alone.

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