arXiv subjects

Howard Barnum

Publications and source records attributed to Howard Barnum.

48 records · Page 3Linked to original sources

arXiv2002

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 $l_p$ norms for $1 \le p \le \infty$, of separable (unentangled) matrices around the identity matrix. This implies a simple and intutively meaningful geometrical sufficient condition for separability of bipartite density matrices: that their purity $\tr ρ^2$ 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 NMR quantum computation implementations or other experimental situations.

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arXiv2002

Information-disturbance tradeoff in quantum measurement on the uniform ensemble and on the mutually unbiased bases

I consider the tradeoff between the information gained about an initially unknown quantum state, and the disturbance caused to that state by the measurement process. I show that for any distribution of initial states, the information-disturbance frontier is convex, and disturbance is nondecreasing with information gain. I consider the most general model of quantum measurements, and all post-measurement dynamics compatible with a given measurement. For the uniform initial distribution over states, I show that the least-disturbing way of making any measurement is with conditional dynamics satisfying a generalization of the projection postulate, the ``square-root dynamics.'' Thus, procedures for achieving a point on the information-disturbance frontier may be assumed to involve such conditional dynamics. Also, the information-disturbance frontier for the uniform ensemble may be achieved with ``isotropic'' (unitarily covariant) dynamics. This results in a significant simplification of the optimization problem for calculating the tradeoff in this case, giving hope for a closed-form solution. I also show that the discrete ensembles uniform on the d(d+1) vectors of a certain set of d+1 ``mutually unbiased'' or conjugate bases in d dimensions form spherical 2-designs in CP_{d-1} when d is a power of an odd prime. This implies that many of the results of the paper apply also to these discrete ensembles.

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arXiv2002

Quantum information processing and quantum logic: toward mutual illumination

Quantum information and computation may serve as a source of useful axioms and ideas for the quantum logic/quantum structures project of characterizing and classifying types of physical theories, including quantum mechanics and classical mechanics. The axiomatic approach of quantum structures may help isolate what aspects of quantum mechanics are responsible for what aspects of its greater-than-classical information processing power, and whether more general physical theories may escape some common limitations of classical and quantum theories. Also, by by helping us understand how existing quantum algorithms work, quantum structures analyses may suggest new quantum protocols exploiting general features of quantum mechanics. I stress the importance, for these matters, of understanding open and closed-system dynamics, and the structure of composite systems in general frameworks for operational theories, such as effect algebras, convex sets, and related structures.

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arXiv2002

Authentication of Quantum Messages

Authentication is a well-studied area of classical cryptography: a sender S and a receiver R sharing a classical private key want to exchange a classical message with the guarantee that the message has not been modified by any third party with control of the communication line. In this paper we define and investigate the authentication of messages composed of quantum states. Assuming S and R have access to an insecure quantum channel and share a private, classical random key, we provide a non-interactive scheme that enables S both to encrypt and to authenticate (with unconditional security) an m qubit message by encoding it into m+s qubits, where the failure probability decreases exponentially in the security parameter s. The classical private key is 2m+O(s) bits. To achieve this, we give a highly efficient protocol for testing the purity of shared EPR pairs. We also show that any scheme to authenticate quantum messages must also encrypt them. (In contrast, one can authenticate a classical message while leaving it publicly readable.) This has two important consequences: On one hand, it allows us to give a lower bound of 2m key bits for authenticating m qubits, which makes our protocol asymptotically optimal. On the other hand, we use it to show that digitally signing quantum states is impossible, even with only computational security.

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arXiv2002

A lower bound on the quantum query complexity of read-once functions

We establish a lower bound of $Ω{(\sqrt{n})}$ on the bounded-error quantum query complexity of read-once Boolean functions, providing evidence for the conjecture that $Ω(\sqrt{D(f)})$ is a lower bound for all Boolean functions. Our technique extends a result of Ambainis, based on the idea that successful computation of a function requires ``decoherence'' of initially coherently superposed inputs in the query register, having different values of the function. The number of queries is bounded by comparing the required total amount of decoherence of a judiciously selected set of input-output pairs to an upper bound on the amount achievable in a single query step. We use an extension of this result to general weights on input pairs, and general superpositions of inputs.

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arXiv2001

Quantum message authentication codes

This paper has been withdrawn because of an error in the proof of Lemma 1. Without Lemma 1, the result of the main theorem (with the same proof, but omitting the use of Lemma 1 to extend things to general attacks) only holds for attacks in which the attacker applies a tensor product of Pauli matrices (for qubits) or generalized Pauli matrices. The resulting theorem is still relevant for quantum message authentication, but the very strong result claimed in earlier versions of this paper does not hold. These matters will be addressed further in joint work with Claude Crepeau, Daniel Gottesman, Adam Smith, and Alain Tapp. Thanks to them and to Debbie Leung for spotting the error.

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arXiv2000

On quantum coding for ensembles of mixed states

We consider the problem of optimal asymptotically faithful compression for ensembles of mixed quantum states. Although the optimal rate is unknown, we prove upper and lower bounds and describe a series of illustrative examples of compression of mixed states. We also discuss a classical analogue of the problem.

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arXiv1999

A quantum circuit for OR

We give the first quantum circuit for computing $f(0)$ OR $f(1)$ more reliably than is classically possible with a single evaluation of the function. OR therefore joins XOR (i.e. parity, $f(0) \oplus f(1)$) to give the full set of logical connectives (up to relabeling of inputs and outputs) for which there is quantum speedup. The XOR algorithm is of fundamental importance in quantum computation; our OR algorithm (found with the aid of genetic programming), may represent a new quantum computational effect, also useful as a ``subroutine''.

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arXiv1998

Quantum Rate-Distortion Coding

I introduce rate-distortion theory for quantum coding, and derive a lower bound, involving the coherent information, on the rate at which qubits must be used to encode a quantum source with a given maximum level of distortion per source emission. The convexity of the "information rate-distortion function" which defines this bound is also derived.

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arXiv1997

Information-theoretic approach to quantum error correction and reversible measurement

Quantum operations provide a general description of the state changes allowed by quantum mechanics. The reversal of quantum operations is important for quantum error-correcting codes, teleportation, and reversing quantum measurements. We derive information-theoretic conditions and equivalent algebraic conditions that are necessary and sufficient for a general quantum operation to be reversible. We analyze the thermodynamic cost of error correction and show that error correction can be regarded as a kind of ``Maxwell demon,'' for which there is an entropy cost associated with information obtained from measurements performed during error correction. A prescription for thermodynamically efficient error correction is given.

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arXiv1997

Information transmission through a noisy quantum channel

Noisy quantum channels may be used in many information carrying applications. We show that different applications may result in different channel capacities. Upper bounds on several of these capacities are proved. These bounds are based on the coherent information, which plays a role in quantum information theory analogous to that played by the mutual information in classical information theory. Many new properties of the coherent information and entanglement fidelity are proved. Two non-classical features of the coherent information are demonstrated: the failure of subadditivity, and the failure of the pipelining inequality. Both properties arise as a consequence of quantum entanglement, and give quantum information new features not found in classical information theory. The problem of a noisy quantum channel with a classical observer measuring the environment is introduced, and bounds on the corresponding channel capacity proved. These bounds are always greater than for the unobserved channel. We conclude with a summary of open problems.

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arXiv1996

A General Fidelity Limit for Quantum Channels

We derive a general limit on the fidelity of a quantum channel conveying an ensemble of pure states. Unlike previous results, this limit applies to arbitrary coding and decoding schemes, including nonunitary decoding. This establishes the converse of the quantum noiseless coding theorem for all such schemes.

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