Howard Barnum

Authentication of quantum messages

Authentication is a well-studied area of classical cryptography: a sender A and a receiver B sharing a classical secret key want to exchange a classical message with the guarantee that the message has not been modified or replaced by a dishonest party with control of the communication line. In this paper we study the authentication of messages composed of quantum states. We give a formal definition of authentication in the quantum setting. Assuming A and B have access to an insecure quantum channel and share a secret, classical random key, we provide a non-interactive scheme that enables A to both encrypt and authenticate an m qubit message by encoding it into m+s qubits, where the error probability decreases exponentially in the security parameter s. The scheme requires a secret key of size 2m+O(s). To achieve this, we give a highly efficient protocol for testing the purity of shared EPR pairs. It has long been known that learning information about a general quantum state will necessarily disturb it. We refine this result to show that such a disturbance can be done with few side effects, allowing it to circumvent cryptographic protections. Consequently, any scheme to authenticate quantum messages must also encrypt them. In contrast, no such constraint exists classically. This reasoning has two important consequences: It allows us to give a lower bound of 2m key bits for authenticating m qubits, which makes our protocol asymptotically optimal. Moreover, we use it to show that digitally signing quantum states is impossible.

A subsystem-independent generalization of entanglement.

We present a generalization of entanglement based on the idea that entanglement is relative to a distinguished subspace of observables rather than a distinguished subsystem decomposition. A pure quantum state is entangled relative to such a subspace if its expectations are a proper mixture of those of other states. Many information-theoretic aspects of entanglement can be extended to this observable-based setting, suggesting new ways of measuring and classifying multipartite entanglement. By going beyond the distinguishable-subsystem framework, generalized entanglement also provides novel tools for probing quantum correlations in interacting many-body systems.

Largest separable balls around the maximally mixed bipartite quantum state

For finite-dimensional bipartite quantum systems, we find the exact size of the largest balls, in spectral

Information Processing in Convex Operational Theories

{l}_{p}

Separable balls around the maximally mixed multipartite quantum states

norms for

Generalizations of entanglement based on coherent states and convex sets

1l~pl~ensuremath{infty},

Quantum query complexity and semi-definite programming

of separable (unentangled) matrices around the identity matrix. This implies a simple and intuitively meaningful geometrical sufficient condition for separability of bipartite density matrices: that their purity

Better bound on the exponent of the radius of the multipartite separable ball

mathrm{tr}{ensuremath{rho}}^{2}

Nonclassicality without entanglement enables bit commitment

not be too large. Theoretical and experimental applications of these results include algorithmic problems such as computing whether or not a state is entangled, and practical ones such as obtaining information about the existence or nature of entanglement in states reached by nuclear magnetic resonance quantum computation implementations or other experimental situations.

A Generalization of Entanglement to Convex Operational Theories: Entanglement Relative to a Subspace of Observables

In order to understand the source and extent of the greater-than-classical information processing power of quantum systems, one wants to characterize both classical and quantum mechanics as points in a broader space of possible theories. One approach to doing this, pioneered by Abramsky and Coecke, is to abstract the essential categorical features of classical and quantum mechanics that support various information-theoretic constraints and possibilities, e.g., the impossibility of cloning in the latter, and the possibility of teleportation in both. Another approach, pursued by the authors and various collaborators, is to begin with a very conservative, and in a sense very concrete, generalization of classical probability theory - which is still sufficient to encompass quantum theory - and to ask which quantum informational phenomena can be reproduced in this much looser setting. In this paper, we review the progress to date in this second programme, and offer some suggestions as to how to link it with the categorical semantics for quantum processes offered by Abramsky and Coecke.