Dirk Schlingemann

Quantum error-correcting codes associated with graphs

We present a construction for quantum error correcting codes. The basic ingredients are a graph and a finite Abelian group, from which the code can explicitly be obtained. We prove necessary and sufficient conditions for the graph such that the resulting code corrects a certain number of errors. This allows a simple verification of the one-error correcting property of codes of length 5 in any dimension. As examples, we construct a large class of maximum distance separable codes, i.e. codes saturating the Singleton bound, as well as a code of length 10 detecting three errors.

The Information-Disturbance Tradeoff and the Continuity of Stinespring's Representation

Stinesprings dilation theorem is the basic structure theorem for quantum channels: it states that any quantum channel arises from a unitary evolution on a larger system. Here we prove a continuity theorem for Stinesprings dilation: if two quantum channels are close in cb-norm, then it is always possible to find unitary implementations which are close in operator norm, with dimension-independent bounds. This result generalizes Uhlmanns theorem from states to channels and allows to derive a formulation of the information-disturbance tradeoff in terms of quantum channels, as well as a continuity estimate for the no-broadcasting theorem. We briefly discuss further implications for quantum cryptography, thermalization processes, and the black hole information loss puzzle.

Semicausal operations are semilocalizable

We prove a conjecture by DiVincenzo, which in the terminology of Preskill et al. states that semicausal operations are semilocalizable. That is, we show that any operation on the combined system of Alice and Bob, which does not allow Bob to send messages to Alice, can be represented as an operation by Alice, transmitting a quantum particle to Bob, and a local operation by Bob. The proof is based on the uniqueness of the Stinespring representation for a completely positive map. We sketch some of the problems in transferring these concepts to the context of relativistic quantum field theory.

A short impossibility proof of quantum bit commitment

Bit commitment protocols, whose security is based on the laws of quantum mechanics alone, are generally held to be impossible on the basis of a concealment-bindingness tradeoff (1, 2). A strengthened and explicit impossibility proof has been given in Ref. (3) in the Heisenberg picture and in a C � -algebraic framework, considering all conceivable protocols in which both classical and quantum information are exchanged. In the present paper we provide a new impossibility proof in the Schrodinger picture, greatly simplifying the classification of protocols and strategies using the mathematical formulation in terms of quantum combs (4), with each single-party strategy represented by a conditional comb. We prove that assuming a stronger notion of concealment—worst-case over the classical information histories—allows Alices cheat to pass also the worst-case Bobs test. The present approach allows us to restate the concealment-bindingness tradeoff in terms of the continuity of dilations of probabilistic quantum combs with respect to the comb-discriminability distance. PACS numbers: 03.67.Dd Bit commitment involves two mistrustful parties— Alice and Bob—of which Alice submits to Bob a piece of evidence that he will use to confirm a bit value that she will later reveal, whereas Bob cannot determine the bit value from the evidence alone. A good bit commit- ment protocol should be simultaneously concealing and binding, namely the evidence should be submitted to Bob in such a way that he has (almost) no chance to identify the committed bit value before Alice later decodes it for him, whereas Alice has (almost) no way of changing the value of the committed bit once she has submitted the evidence. In the easiest example to illustrate bit com- mitment, Alice writes the bit down on a piece of paper, which is then locked in a safe and sent to Bob, whereas Alice keeps the key. At a later time, she will unveil the bit by handing over the key to Bob. However, Bob may be able to open the safe in the meantime, and this scheme is in principle insecure. Yet all bit commitment schemes currently used rely on strongboxes and keys made of com- putations that are (supposedly) hard to perform (see Ref. (3) for a list of references), and cryptographers have long known that bit commitment (like any other interesting two-party cryptographic primitive) cannot be securely implemented with classical information (5).

Entanglement, haag-duality and type properties of infinite quantum spin chains

We consider an infinite spin chain as a bipartite system consisting of the left and right half-chains and analyze entanglement properties of pure states with respect to this splitting. In this context, we show that the amount of entanglement contained in a given state is deeply related to the von Neumann type of the observable algebras associated to the half-chains. Only the type I case belongs to the usual entanglement theory which deals with density operators on tensor product Hilbert spaces, and only in this situation separable normal states exist. In all other cases, the corresponding state is infinitely entangled in the sense that one copy of the system in such a state is sufficient to distill an infinite amount of maximally entangled qubit pairs. We apply this results to the critical XY model and show that its unique ground state φS provides a particular example for this type of entanglement.

How continuous quantum measurements in finite dimensions are actually discrete.

We show that in finite dimensions a quantum measurement with a continuous set of outcomes can be always realized as a continuous random choice of measurements with a finite number of outcomes.

Barycentric decomposition of quantum measurements in finite dimensions

We analyze the convex structure of the set of positive operator valued measures (POVMs) representing quantum measurements on a given finite dimensional quantum system, with outcomes in a given locally compact Hausdorff space. The extreme points of the convex set are operator valued measures concentrated on a finite set of k \le d^2 points of the outcome space, d< \infty being the dimension of the Hilbert space. We prove that for second countable outcome spaces any POVM admits a Choquet representation as the barycenter of the set of extreme points with respect to a suitable probability measure. In the general case, Krein-Milman theorem is invoked to represent POVMs as barycenters of a certain set of POVMs concentrated on k \le d^2 points of the outcome space.

ON HAAG DUALITY FOR PURE STATES OF QUANTUM SPIN CHAINS

In this note, we consider quantum spin chains and their translationally invariant pure states. We prove Haag duality for quasilocal observables localized in semi-infinite intervals (-∞ , 0] and [1, ∞) when the von Neumann algebra generated by observables localized in [0, ∞) is non-type I.