Table of contents

Guest Editors

During the past two decades, there has emerged the new subject of quantum information and computation which both offers the possibility of powerful new modes of computing and communication and also suggests deep links between the well established disciplines of quantum theory and information theory and computer science.

In recent years, the growth of the subject has been explosive, with significant progress in theory and experiment. The area has a highly interdisciplinary character with contributions from physicists, mathematicians and computer scientists in particular. Developments have occurred in diverse areas including quantum algorithms, quantum communication, quantum cryptography, entanglement and nonlocality.

This progress has been reflected in contributions to Journal of Physics A: Mathematical and General which traditionally provides a natural home for this area of research. Furthermore, the journal's committment to this field has recently been strengthened by the appointments of Sandu Popescu and Nicolas Gisin to the Editorial Board, and in this special issue we take the opportunity to present a snapshot of the present state of the art.

In this paper we analyse the canonical forms into which any pure three-qubit state can be cast. The minimal forms, i.e. the ones with the minimal number of product states built from local bases, are also presented and lead to a complete classification of pure three-qubit states. This classification is related to the values of the polynomial invariants under local unitary transformations by a one-to-one correspondence.

We compare classical and quantum query complexities of total Boolean functions. It is known that for worst-case complexity, the gap between quantum and classical can be at most polynomial. We show that for average-case complexity under the uniform distribution, quantum algorithms can be exponentially faster than classical algorithms. Under non-uniform distributions the gap can even be super-exponential. We also prove some general bounds for average-case complexity and show that the average-case quantum complexity of MAJORITY under the uniform distribution is nearly quadratically better than the classical complexity.

A preliminary version of this paper appeared in the Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science (STACS'2000) (Lecture Notes in Computer Science vol 1770) (Berlin: Springer).

We compare and contrast the error probability and fidelity as measures of the quality of the receiver's measurement strategy for a quantum communications system. The error probability is a measure of the ability to retrieve classical information and the fidelity measures the retrieval of quantum information. We present the optimal measurement strategies for maximizing the fidelity given a source that encodes information on the symmetric qubit-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.

We introduce new entanglement monotones which generalize, to the case of many parties, those which give rise to the majorization-based partial ordering of bipartite states' entanglement. We give some examples of restrictions they impose on deterministic and probabilistic conversion between multipartite states via local actions and classical communication. These include restrictions which do not follow from any bipartite considerations. We derive supermultiplicativity relations between each state's monotones and the monotones for collective processing when the parties share several states. We also investigate polynomial invariants under local unitary transformations, and show that a large class of these are invariant under collective unitary processing and also multiplicative, putting restrictions, for example, on the exact conversion of multiple copies of one state to multiple copies of another.

Entanglement between n particles is a generalization of the entanglement between two particles, and a state is considered entangled if it cannot be written as a mixture of tensor products of the n particles' states. We present the key notion of semi-separability, used to investigate n-particle entanglement by looking at two-party entanglement between its various subsystems. We provide necessary conditions for n-particle separability (that is, sufficient conditions for n-particle entanglement). We also provide necessary and sufficient conditions in the case of pure states. By surprising examples, we show that such conditions are not sufficient for separability in the case of mixed states, suggesting entanglement of a strange type.

We study the optimal cloning transformation for two pairs of orthogonal states of two-dimensional quantum systems, and derive the corresponding optimal fidelities.

We prove a general upper bound on the trade-off between time and space that suffices for the reversible simulation of irreversible computation. Previously, only simulations using exponential time or quadratic space were known. The trade-off shows for the first time that we can simultaneously achieve subexponential time and subquadratic space. The boundary values are the exponential time with hardly any extra space required by the Lange–McKenzie–Tapp method and the (log 3)th power time with square space required by the Bennett method. We also give the first general lower bound on the extra storage space required by general reversible simulation. This lower bound is optimal in that it is achieved by some reversible simulations.

Results obtained in two recent papers, (Kaszlikowski D, Gnacinski P, Zukowski M, Miklaszewski W and Zeilinger A 2000 Phys. Rev. Lett.85 4418 and Durt T, Kaszlikowski D and Zukowski M 2001 Preprint quant-ph/0101084) seem to indicate that the nonlocal character of the correlations between the outcomes of measurements performed on entangled systems separated in space is not robust in the presence of noise. This is surprising, since entanglement itself is robust. Here we revisit this problem and argue that the class of gedanken experiments considered in the two recent papers listed above is too restrictive. By considering a more general class, involving sequences of measurements, we prove that the nonlocal correlations are in fact robust.

We discuss several aspects of multiparticle mixed-state entanglement and its experimental detection. First we consider entanglement between two particles which is robust against disposals of other particles. To completely detect these kinds of entanglement, full knowledge of the multiparticle density matrix (or of all reduced density matrices) is required. Then we review the relation of the separability properties of l-partite splittings of a state ρ to its multipartite entanglement properties. We show that it suffices to determine the diagonal matrix elements of ρ in a certain basis in order to detect multiparticle entanglement properties of ρ. We apply these observations to analyse two recent experiments, where multiparticle entangled states of 3 (4) particles were produced. Finally, we focus on bound entangled states (non-separable, non-distillable states) and show that they can be activated by joint actions of the parties. We also provide several examples which show the activation of bound entanglement with bound entanglement.

In a previous paper, we adopted the method using quantum mutual entropy to measure the degree of entanglement in the time development of the Jaynes–Cummings model. In this paper, we formulate the entanglement in the time development of the Jaynes–Cummings model with squeezed states, and then show that the entanglement can be controlled by means of squeezing.

We extend algorithmic information theory to quantum mechanics, taking a universal semicomputable density matrix (`universal probability') as a starting point, and define complexity (an operator) as its negative logarithm.

A number of properties of Kolmogorov complexity extend naturally to the new domain. Approximately, a quantum state is simple if it is within a small distance from a low-dimensional subspace of low Kolmogorov complexity. The von Neumann entropy of a computable density matrix is within an additive constant from the average complexity. Some of the theory of randomness translates to the new domain.

We explore the relations of the new quantity to the quantum Kolmogorov complexity defined by Vitányi (we show that the latter is sometimes as large as 2n − 2 log n) and the qubit complexity defined by Berthiaume, Dam and Laplante. The `cloning' properties of our complexity measure are similar to those of qubit complexity.

An alternative proof for existence of `quantum nonlocality without entanglement', i.e. existence of variables with product-state eigenstates which cannot be measured locally, is presented. A simple `nonlocal' variable for the case of one-way communication is given and the limit for its approximate measurability is found.

We give a detailed proof of the conjecture that the asymptotic entanglement cost of preparing a state ρ is equal to lim _n→∞E_f (ρ^⊗n)/n where E_f is the entanglement of formation.

We discuss the problem of separating consistently the total correlations in a bipartite quantum state into a quantum and a purely classical part. A measure of classical correlations is proposed and its properties are explored.

I investigate dense coding with a general mixed state on the Hilbert space C^d ⊗ C^d shared between a sender and receiver. The following result is proved. When the sender prepares the signal states by mutually orthogonal unitary transformations with equal a priori probabilities, the capacity of dense coding is maximized. It is also proved that the optimal capacity of dense coding χ* satisfies E_R(ρ) ≤ χ* ≤ E_R(ρ) + log₂d, where E_R(ρ) is the relative entropy of entanglement of the shared entangled state.

We investigate a new strategy for incoherent eavesdropping in Ekert's entanglement-based quantum key distribution protocol. We show that under certain assumptions of symmetry the effectiveness of this strategy reduces to that of the original single-qubit protocol of Bennett and Brassard.

We investigate optimal separable approximations (decompositions) of states ϱ of bipartite quantum systems A and B of arbitrary dimensions M × N following the lines of Lewenstein and Sanpera. Such approximations allow to represent in an optimal way any density operator as a sum of a separable state and an entangled state of a certain form. For two-qubit systems (M = N = 2) the best separable approximation has the form of a mixture of a separable state and a projector onto a pure entangled state. We formulate a necessary condition that the pure state in the best separable approximation is not maximally entangled. We demonstrate that the weight of the entangled state in the best separable approximation in arbitrary dimensions provides a good entanglement measure. We prove for arbitrary M and N that the best separable approximation corresponds to a mixture of separable and entangled states, both of which are unique. We develop also a theory of optimal separable approximations for states with positive partial transpose (PPT states). Such approximations allow to decompose any density operator with positive partial transpose as a sum of a separable state and an entangled PPT state. We discuss procedures for constructing such decompositions.

If two parties, Alice and Bob, share some number, n, of partially entangled pairs of qubits, then it is possible for them to concentrate these pairs into some smaller number of maximally entangled states. We present a simplified version of the algorithm for such entanglement concentration, and we describe efficient networks for implementing these operations.

The role of the off-diagonal density-matrix elements of the entangled pair is investigated in the quantum teleportation of a qubit. The dependence between these elements and the off-diagonal elements of the teleported density matrix is shown to be linear. In this way ideal quantum teleportation is related to a completely classical communication protocol: the one-time pad cypher. The latter can be regarded as the classical counterpart of Bennett's quantum-teleportation scheme. The quantum-to-classical transition is demonstrated by a gedankenexperiment.

Quantum key distribution is the best known application of quantum cryptography. Previously proposed proofs of security of quantum key distribution contain various technical subtleties. Here, a conceptually simpler proof of security of quantum key distribution is presented. The new insight is the invariance of the error rate of a teleportation channel: we show that the error rate of a teleportation channel is independent of the signals being transmitted. This is because the non-trivial error patterns are permuted under teleportation. This new insight is combined with the recently proposed quantum-to-classical reduction theorem. Our result shows that assuming that Alice and Bob have fault-tolerant quantum computers, quantum key distribution can be made unconditionally secure over arbitrarily long distances even against the most general type of eavesdropping attacks and in the presence of all types of noises.

We study the entanglement properties of two-mode Gaussian light emerging from a generic SU(1,1) interferometer. Our tool is the two-mode characteristic function which is determined by the 4×4 covariance matrix. For an initial product of two mixed single-mode Gaussian states we investigate the output two-mode covariance matrix. Its structure displays the noise properties of the reduced states as well as the correlations between modes. Classicality of the output two-mode state is characterized by the existence of the Glauber-Sudarshan P representation. For testing separability we apply the Peres-Simon criterion requiring preservation of the positivity of the density matrix under partial transposition. Since inseparability entails nonclassicality, the threshold gain above which nonclassicality of the output state becomes manifest is lower than that allowing for its inseparability. We find that only for a thermal input do nonclassicality and inseparability of the output have the same threshold gain.

The evolution of a quantum lattice gas automaton (QLGA) for a single charged particle is invariant under multiplication of the wave function by a global phase. Requiring invariance under the corresponding local gauge transformations determines the rule for minimal coupling to an arbitrary external electromagnetic field. We develop the Aharonov–Bohm effect in the resulting model into a constant time algorithm to distinguish a one-dimensional periodic lattice from one with boundaries; any classical deterministic lattice gas automaton (LGA) algorithm distinguishing these two spatial topologies would have expected running time on the order of the cardinality of the lattice.

In a good physical theory dimensionless quantities, such as the ratio m_p/m_e of the mass of the proton to the mass of the electron, do not depend on the system of units being used. This paper demonstrates that one widely used method for defining measures of entanglement violates this principle. Specifically, in this approach dimensionless ratios E(ρ)/E(σ) of entanglement measures may depend on what state is chosen as the basic unit of entanglement. This observation leads us to suggest three novel approaches to the quantification of entanglement. These approaches lead to unit-free definitions for the entanglement of formation and the distillable entanglement, and suggest natural measures of entanglement for multipartite systems. We also show that the behaviour of one of these novel measures, the entanglement of computation, is related to some open problems in computational complexity.

We present upper and lower bounds to the relative entropy of entanglement of multi-party systems in terms of the bi-partite entanglements of formation and distillation and entropies of various subsystems. We point out implications of our results to the local reversible convertibility of multi-party pure states and discuss their physical basis in terms of deleting information.

We propose a protocol, based on entanglement procedures recently suggested by Jaksch et al, which allows the teleportation of an unknown state of a neutral atom in an optical lattice to another atom in another site of the lattice without any irreversible detection.

We present an optimal strategy having finite outcomes for estimating a single parameter of the displacement operator on an arbitrary finite-dimensional system using a finite number of identical samples. Assuming the uniform a priori distribution for the displacement parameter, an optimal strategy can be constructed by making the square root measurement based on uniformly distributed sample points. This type of measurement automatically ensures the global maximality of the figure of merit, that is, the so-called average score or fidelity. Quantum circuit implementations for the optimal strategies are provided in the case of a two-dimensional system.

We address the `major open problem' of evaluating how much increased efficiency in estimation is possible using non-separable—as opposed to separable—measurements of N copies of m-level quantum systems. First, we study the six cases m = 2, N = 2, ..., 7 by computing the 3 × 3 Fisher information matrices for the corresponding optimal measurements recently devised by Vidal et al. We obtain simple polynomial expressions for the (`Gill–Massar') traces of the products of the inverse of the quantum Helstrom information matrix and these Fisher information matrices. The six traces all have minima of 2N − 1 in the pure state limit—while for separable measurements, the traces can equal N, but not exceed it. Then, the result of an analysis for m = 3, N = 2 leads us to conjecture that for optimal measurements for all m and N, the Gill–Massar trace achieves a minimum of (2N − 1)(m − 1) in the pure state limit.

The entropy H_T (ρ) of a state with respect to a channel T and the Holevo capacity of the channel require the solution of difficult variational problems. For a class of 1-qubit channels, which contains all the extremal ones, the problem can be significantly simplified by attaching a unique Hermitian antilinear operator ϑ to every channel of the considered class. The channel's concurrence C_T can be expressed by ϑ and turns out to be a flat roof. This allows to write down an explicit expression for H_T. Its maximum would give the Holevo (one-shot) capacity.

Certain quantum gates, such as the controlled-NOT gate, are symmetric in terms of the operation of the control system upon the target system and vice versa. However, no operational criteria yet exist for establishing whether or not a given quantum gate is symmetrical in this sense. We consider a restricted, yet broad, class of two-party controlled gate operations for which the gate transforms a reference state of the target into one of an orthogonal set of states. We show that for this class of gates it is possible to establish a simple necessary and sufficient condition for the gate operation to be symmetric.

The notion of a qubit is ubiquitous in quantum information processing. In spite of the simple abstract definition of qubits as two-state quantum systems, identifying qubits in physical systems is often unexpectedly difficult. There is an astonishing variety of ways in which qubits can emerge from devices. What essential features are required for an implementation to properly instantiate a qubit? We give three typical examples and propose an operational characterization of qubits based on quantum observables and subsystems.

We establish a one-to-one correspondence between (1) quantum teleportation schemes, (2) dense coding schemes, (3) orthonormal bases of maximally entangled vectors, (4) orthonormal bases of unitary operators with respect to the Hilbert–Schmidt scalar product and (5) depolarizing operations, whose Kraus operators can be chosen to be unitary. The teleportation and dense coding schemes are assumed to be `tight' in the sense that all Hilbert spaces involved have the same finite dimension d, and the classical channel involved distinguishes d ² signals. A general construction procedure for orthonormal bases of unitaries, involving Latin squares and complex Hadamard matrices is also presented.

The fidelity of two pure states (also known as transition probability) is a symmetric function of two operators, and well founded operationally as an event probability in a certain preparation-test pair. Motivated by the idea that the fidelity is the continuous quantum extension of the combinatorial equality function, we enquire whether there exists a symmetric operational way of obtaining the fidelity. It is shown that this is impossible. Finally, we discuss the optimal universal approximation by a quantum operation.

An experimental test of relativistic wave-packet collapse is presented. The tested model assumes that the collapse takes place in the reference frame determined by the massive measuring detectors. Entangled photons are measured at 10 km distance within a time interval of less than 5 ps. The two apparatuses are in relative motion so that both detectors, each in its own inertial reference frame, are first to perform the measurement. The data always reproduces the quantum correlations and thus rule out a class of collapse models. The results also set a lower bound on the `speed of quantum information' to 2/3 ×10⁷ and 3/2 ×10⁴ times the speed of light in the Geneva and the background radiation reference frames, respectively. The very difficult and deep question of where the collapse takes place—if it takes place at all—is considered in a concrete experimental context.

We analyse several product measures in the space of mixed quantum states. In particular, we study measures induced by the operation of partial tracing. The natural, rotationally invariant measure on the set of all pure states of a N×K composite system, induces a unique measure in the space of N×N mixed states (or in the space of K×K mixed states, if the reduction takes place with respect to the first subsystem). For K = N the induced measure is equal to the Hilbert-Schmidt measure, which is shown to coincide with the measure induced by singular values of non-Hermitian random Gaussian matrices pertaining to the Ginibre ensemble. We compute several averages with respect to this measure and show that the mean entanglement of N×N pure states behaves as lnN-1/2.