Volume 43, Issue 9, September 2002
 ENTANGLEMENT


Characterizing entanglement
View Description Hide DescriptionQuantum entanglement is at the heart of many tasks in quantum information. Apart from simple cases (low dimensions, few particles, pure states), however, the mathematical structure of entanglement is not yet fully understood. This tutorial is an introduction to our present knowledge about how to decide whether a given state is separable or entangled, how to characterize entanglement via witness operators, how to classify entangled states according to their usefulness (i.e., distillability), and how to quantify entanglement with appropriate measures.

The uniqueness theorem for entanglement measures
View Description Hide DescriptionWe explore and develop the mathematics of the theory of entanglement measures. After a careful review and analysis of definitions, of preliminary results, and of connections between conditions on entanglement measures, we prove a sharpened version of a uniqueness theorem which gives necessary and sufficient conditions for an entanglement measure to coincide with the reduced von Neumann entropy on pure states. We also prove several versions of a theorem on extreme entanglement measures in the case of mixed states. We analyze properties of the asymptotic regularization of entanglement measures proving, for example, convexity for the entanglement cost and for the regularized relative entropy of entangle ment.

Global entanglement in multiparticle systems
View Description Hide DescriptionWe define a polynomialmeasure of multiparticle entanglement which is scalable, i.e., which applies to any number of spin particles. By evaluating it for three particle states, for eigenstates of the one dimensional Heisenberg antiferromagnet and on quantum error correcting codesubspaces, we illustrate the extent to which it quantifies global entanglement. We also apply it to track the evolution of entanglement during a quantum computation.

Entanglement and perfect quantum error correction
View Description Hide DescriptionThe entanglement of formation gives a necessary and sufficient condition for the existence of a perfect quantum error correction procedure.

The entanglement of purification
View Description Hide DescriptionWe introduce a measure of both quantum as well as classical correlations in a quantum state, the entanglement of purification. We show that the (regularized) entanglement of purification is equal to the entanglement cost of creating a state ρ asymptotically from maximally entangled states, with negligible communication. We prove that the classical mutual information and the quantum mutual information divided by two are lower bounds for the regularized entanglement of purification. We present numerical results of the entanglement of purification for Werner states in

Conditional entropies and their relation to entanglement criteria
View Description Hide DescriptionWe discuss conditional Rényi and Tsallis entropies for bipartite quantum systems of finite dimension. We investigate the relation between the positivity of conditional entropies and entanglement properties. It is in particular shown that any state having a negative conditional entropy with respect to any value of the entropic parameter is distillable since it violates the reduction criterion. Moreover, we show that the entanglement of Werner states in odd dimensions can neither be detected by entropic criteria nor by any other spectral criterion.

Parallel transport in an entangled ring
View Description Hide DescriptionThis article defines a notion of parallel transport in a lattice of quantum particles, such that the transformation associated with each link of the lattice is determined by the quantum state of the two particles joined by that link. We focus particularly on a onedimensional lattice—a ring—of entangled rebits, which are binary quantum objects confined to a real state space. We consider states of the ring that maximize the correlation between nearest neighbors, and show that some correlation must be sacrificed in order to have nontrivial parallel transport around the ring. An analogy is made with lattice gauge theory, in which nontrivial parallel transport around closed loops is associated with a reduction in the probability of the field configuration. We discuss the possibility of extending our result to qubits and to higher dimensional lattices.
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 NOISE, ENTROPY AND CHANNEL CAPACITY


On entanglementassisted classical capacity
View Description Hide DescriptionWe give a modified proof of the recent result of C. H. Bennett, P. W. Shor, J. A. Smolin, and A. V. Thapliyal concerning entanglementassisted classical capacity of a quantum channel and discuss the relation between entanglementassisted and unassisted classical capacities.

Additivity of the classical capacity of entanglementbreaking quantum channels
View Description Hide DescriptionWe show that for the tensor product of an entanglementbreaking quantum channel with an arbitrary quantum channel, both the minimum entropy of an output of the channel and the Holevo–Schumacher–Westmoreland capacity are additive. In addition, for the tensor product of two arbitrary quantum channels, we give a bound involving entanglement of formation for the amount of subadditivity (for minimum entropy output) or superadditivity (for classical capacity) that can occur.

Scalable programmable quantum gates and a new aspect of the additivity problem for the classical capacity of quantum channels
View Description Hide DescriptionWe consider two apparently separated problems: in the first part of the article we study the concept of a scalable (approximate) programmable quantum gate (SPQG). These are special (approximate) programmable quantum gates, with nice properties that could have implications on the theory of universal computation. Unfortunately, as we prove, such objects do not exist in the domain of usual quantum theory. In the second part the problem of noisy dense coding (and generalizations) is addressed. We observe that the additivity problem for the classical capacity obtained is of apparently greater generality than for the usual quantum channel (completely positive maps): i.e., the latter occurs as a special case of the former, but, as we shall argue with the help of the nonexistence result of the first part, the former cannot be reduced to an instance of the latter. We conclude by suggesting that the additivity problem for the classical capacity of quantum channels, as posed until now, may conceptually not be in its appropriate generality.

Counterexample to an additivity conjecture for output purity of quantum channels
View Description Hide DescriptionA conjecture arising naturally in the investigation of additivity of classical information capacity of quantum channels states that the maximal purity of outputs from a quantum channel, as measured by the norm, should be multiplicative with respect to the tensor product of channels. We disprove this conjecture for The same example (with also disproves a conjecture for the multiplicativity of the injective norm of Hilbert spacetensor products.

Inequalities for quantum entropy: A review with conditions for equality
View Description Hide DescriptionThis article presents selfcontained proofs of the strong subadditivity inequality for von Neumann’s quantum entropy, and some related inequalities for the quantum relative entropy, most notably its convexity and its monotonicity under stochastic maps. Moreover, the approach presented here, which is based on Klein’s inequality and Lieb’s theorem that the function is concave, allows one to obtain conditions for equality. In the case of strong subadditivity, which states that where the subscripts denote subsystems of a composite system, equality holds if and only if Using the fact that the Holevo bound on the accessible information in a quantum ensemble can be obtained as a consequence of the monotonicity of relative entropy, we show that equality can be attained for that bound only when the states in the ensemble commute. The article concludes with an Appendix giving a short description of Epstein’s elegant proof of Lieb’s theorem.

Quantum operations that cannot be implemented using a small mixed environment
View Description Hide DescriptionTo implement any quantum operation (a.k.a. “superoperator” or “CP map”) on a dimensional quantum system, it is enough to apply a suitable overall unitary transformation to the system and a dimensional environment which is initialized in a fixed pure state. It has been suggested that a dimensional environment might be enough if we could initialize the environment in a mixed state of our choosing. In this note we show with elementary means that certain explicit quantum operations cannot be realized in this way. Our counterexamples map some pure states to pure states, giving strong and easily manageable conditions on the overall unitary transformation. Everything works in the more general setting of quantum operations from dimensional to dimensional spaces, so we place our counterexamples within this more general framework.

A lower bound on the quantum capacity of channels with correlated errors
View Description Hide DescriptionThe highest fidelity of quantum errorcorrecting codes of length and rate is proven to be lower bounded by for some function on noisy quantum channels that are subject to not necessarily independent errors. The is positive below some threshold which implies is a lower bound on the quantum capacity. This work is an extension of the author’s previous works [M. Hamada, Phys. Rev. A, 65, 052305 (2002); quantph/0109114, LANL (2001); M. Hamada, quantph/0112103, LANL (2001)], which presented the bound for channels subject to independent errors, or channels modeled as tensor products of copies of a completely positive linear map. The relation of the channel class treated in this article to those in the previous works is similar to that of Markov chains to sequences of independent identically distributed random variables.

Lower bound for the quantum capacity of a discrete memoryless quantum channel
View Description Hide DescriptionWe generalize the random coding argument of stabilizer codes and derive a lower bound on the quantum capacity of an arbitrary discrete memoryless quantum channel. For the depolarizing channel, our lower bound coincides with that obtained by Bennett et al. We also slightly improve the quantum Gilbert–Varshamov bound for general stabilizer codes, and establish an analog of the quantum Gilbert–Varshamov bound for linear stabilizer codes. Our proof is restricted to the binary quantum channels, but its extension of to adic channels is straight forward.

Trading quantum for classical resources in quantum data compression
View Description Hide DescriptionWe study the visible compression of a source of pure quantum signal states or, more formally, the minimal resources per signal required to represent arbitrarily long strings of signals with arbitrarily high fidelity, when the compressor is given the identity of the input state sequence as classical information. According to the quantum source coding theorem, the optimal quantum rate is the von Neumann entropyqubits per signal. We develop a refinement of this theorem in order to analyze the situation in which the states are coded into classical and quantum bits that are quantified separately. This leads to a tradeoff curve where qubits per signal is the optimal quantum rate for a given classical rate of bits per signal. Our main result is an explicit characterization of this tradeoff function by a simple formula in terms of only singlesignal, perfect fidelity encodings of the source. We give a thorough discussion of many further mathematical properties of our formula, including an analysis of its behavior for group covariant sources and a generalization to sources with continuously parametrized states. We also show that our result leads to a number of corollaries characterizing the tradeoff between information gain and state disturbance for quantum sources. In addition, we indicate how our techniques also provide a solution to the socalled remote state preparation problem. Finally, we develop a probabilityfree version of our main result which may be interpreted as an answer to the question: “How many classical bits does a qubit cost?” This theorem provides a type of dual to Holevo’s theorem, insofar as the latter characterizes the cost of coding classical bits into qubits.
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 QUANTUM COMPUTING ARCHITECTURE


Efficient discrete approximations of quantum gates
View Description Hide DescriptionQuantum compiling addresses the problem of approximating an arbitrary quantum gate with a string of gates drawn from a particular finite set. It has been shown that this is possible for almost all choices of base sets and, furthermore, that the number of gates required for precision ε is only polynomial in Here we prove that using certain sets of base gates quantum compiling requires a string length that is linear in a result which matches the lower bound from counting volume up to constant factor.

Topological quantum memory
View Description Hide DescriptionWe analyzesurface codes, the topological quantum errorcorrecting codes introduced by Kitaev. In these codes, qubits are arranged in a twodimensional array on a surface of nontrivial topology, and encoded quantum operations are associated with nontrivial homology cycles of the surface. We formulate protocols for error recovery, and study the efficacy of these protocols. An orderdisorder phase transition occurs in this system at a nonzero critical value of the error rate; if the error rate is below the critical value (the accuracy threshold), encoded information can be protected arbitrarily well in the limit of a large code block. This phase transition can be accurately modeled by a threedimensional lattice gauge theory with quenched disorder. We estimate the accuracy threshold, assuming that all quantum gates are local, that qubits can be measured rapidly, and that polynomialsize classical computations can be executed instantaneously. We also devise a robust recovery procedure that does not require measurement or fast classical processing; however, for this procedure the quantum gates are local only if the qubits are arranged in four or more spatial dimensions. We discuss procedures for encoding, measurement, and performing faulttolerant universal quantum computation with surface codes, and argue that these codes provide a promising framework for quantum computing architectures.

Qubits as parafermions
View Description Hide DescriptionQubits are neither fermions nor bosons. A Fock space description of qubits leads to a mapping from qubits to parafermions: particles with a hybrid bosonfermion quantum statistics. We study this mapping in detail, and use it to provide a classification of the algebras of operators acting on qubits. These algebras in turn classify the universality of different classes of physically relevant qubitqubit interaction Hamiltonians. The mapping is further used to elucidate the connections between qubits, bosons, and fermions. These connections allow us to share universality results between the different particle types. Finally, we use the mapping to study the quantum computational power of certain anisotropic exchange Hamiltonians. In particular, we prove that the XY model with nearestneighbor interactions only is not computationally universal. We also generalize previous results about universal quantum computation with encoded qubits to codes with higher rates.
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 QUANTUM CRYPTOGRAPHY


Secrecy capacity in the fourstate protocol of quantum key distribution
View Description Hide DescriptionThe process of key distillation from the quantum transmission in quantum key distribution is reviewed, with the objective of calculating the secrecy capacity of the fourstate protocol in the presence of an individual attack in which the eavesdropping probe is entangled with the signal states, and states of the probe become correlated with the states measured by the legitimate receiver. Emphasis is placed on information leakage to the eavesdropping probe. The calculation generalizes earlier work to include an arbitrary angle between the signal bases.
