Abstract—The entanglement-assisted classical capacity of a noisy quantum channel ( ) is the amount of information per channel use that can be sent over the channel in the limit of many uses of the channel, assuming that the sender and receiver have access to the resource of shared quantum entanglement, which may be used up by the communication protocol. We show that the capacity is given by an expression parallel to that for the capacity of a purely classical channel: i.e., the maximum, over channel inputs, of the entropy of the channel input plus the entropy of the channel output minus their joint entropy, the latter being defined as the entropy of an entangled purification of after half of it has passed through the channel. We calculate entanglement-assisted capacities for two interesting quantum channels, the qubit amplitude damping channel and the bosonic channel with amplification/attenuation and Gaussian noise. We discuss how many independent parameters are required to completely characterize the asymptotic behavior of a general quantum channel, alone or in the presence of ancillary resources such as prior entanglement. In the classical analog of entanglement-assisted communication—communication over a discrete memoryless channel (DMC) between parties who share prior random information—we show that one parameter is sufficient, i.e., that in the presence of prior shared random information, all DMCs of equal capacity can simulate one another with unit asymptotic efficiency. Index Terms—Channel capacity, entanglement, quantum information, Shannon theory. I.

The manner in which states of some quantum systems become effectively classical is of great significance for the foundations of quantum physics, as well as for problems of practical interest such as quantum engineering. In the past two decades it has become increasingly clear that many (perhaps all) of the symptoms of classicality can be induced in quantum systems by their environments. Thus decoherence is caused by the interaction in which the environment in effect monitors certain observables of the system, destroying coherence between the pointer states corresponding to their eigenvalues. This leads to environment-induced superselection or einselection, a quantum process associated with selective loss of information. Einselected pointer states are stable. They can retain correlations with the rest of the universe in spite of the environment. Einselection enforces classicality by imposing an effective ban on the vast majority of the Hilbert space, eliminating especially the flagrantly nonlocal ‘‘Schrödinger-cat states.’ ’ The classical structure of phase space emerges from the quantum Hilbert space in the appropriate macroscopic limit. Combination of einselection with dynamics leads to the idealizations of a point and of a classical trajectory. In measurements, einselection replaces quantum entanglement between the apparatus and the measured system with the classical correlation. Only the preferred pointer observable of the apparatus can store information

We explore in the framework of Quantum Computation the notion of computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm that exploits the quantum adiabatic which is equivalent to the Turing halting problem and known to be mathematically noncomputable. Generalised quantum algorithms are also considered for some other mathematical noncomputables in the same and of different noncomputability classes. The key element of all these algorithms is the measurability of both the values of physical observables and of the quantum-mechanical probability distributions for these values. It is argued that computability, and thus the limits of Mathematics, ought to be determined not

We present a proof of unconditional security of a practical quantum key distribution protocol. It is an extension of a previous result obtained by Mayers [1, 2], which proves unconditional security provided that a perfect single photon source is used. In present days, perfect single photon sources are not available and, therefore, practical implementations use either dim laser pulses or post-selected states from parametric

Abstract. The use of quantum bits (qubits) in cryptography holds the promise of secure cryptographic quantum key distribution schemes. Unfortunately, the implemented schemes are often operated in a regime which excludes unconditional security. We provide a thorough investigation of security issues for practical quantum key distribution, taking into account channel losses, a realistic detection process, and modifications of the “qubits ” sent from the sender to the receiver. We first show that even quantum key distribution with perfect qubits might not be achievable over long distances when fixed channel losses and fixed dark count errors are taken into account. Then we show that existing experimental schemes (based on weak pulses) currently do not offer unconditional security for the reported distances and signal strength. Finally we show that parametric downconversion offers enhanced performance compared to its weak coherent pulse counterpart. 1

and their relation to thermodynamics

The engineering and control of devices at the quantum mechanical level—such as those consisting of small numbers of atoms and photons—is a delicate business. The fundamental uncertainty that is inherently present at this scale manifests itself in the unavoidable presence of noise, making this a novel field of application for stochastic estimation and control theory. In this expository paper we demonstrate estimation and feedback control of quantum mechanical systems in what is essentially a noncommutative version of the binomial model that is popular in mathematical finance. The model is extremely rich and allows a full development of the theory while remaining completely within the setting of finite-dimensional Hilbert spaces (thus avoiding the technical complications of the continuous theory). We introduce discretized models of an atom in interaction with the electromagnetic field, obtain filtering equations for photon counting and homodyne detection, and solve a stochastic control problem using dynamic programming and Lyapunov function methods.

This paper is organised as follows. Sections 1 and 2 give the necessary background. Sections 3 and 4 develop criteria for the existence of scattering operators. They are applied to various situations in Sections 5, 6 and 7. x1. Probability spaces and stochastic processes.

We study the complexity of the Local Hamiltonian Problem (denoted as LH-MIN) in the special case when a Hamiltonian obeys conditions of the Perron-Frobenius theorem: all off-diagonal matrix elements in the standard basis are real and non-positive. We will call such Hamiltonians, which are common in the natural world, stoquastic. An equivalent characterization of stoquastic Hamiltonians is that they have an entry-wise non-negative Gibbs density matrix for any temperature. We prove that LH-MIN for stoquastic Hamiltonians belongs to the complexity class AM — a probabilistic version of NP with two rounds of communication between the prover and the verifier. We also show that 2-local stoquastic LH-MIN is hard for the class MA. With the additional promise of having a polynomial spectral gap, we show that stoquastic LH-MIN belongs to the class PostBPP=BPPpath — a generalization of BPP in which a post-selective readout is allowed. This last result also shows that any problem solved by adiabatic quantum computation using stoquastic Hamiltonians is in PostBPP ∩ AM. 1

Decoherence of a quantum system (which then starts to display classical features) results from the interaction of the system with the environment, and is well described in the framework of the theory of continuous quantum measurements (CQM). Reviewed are the various approaches to the CQM theory, and the approach based on the effective complex Hamiltonians is discussed in greater detail. The effective complex Hamiltonian is derived from the restricted path integral, which emphasizes the role of information in the dynamics of the system being measured. The complex Hamiltonian is used for analyzing the CQM of energy in a two-level system. Such measurement is demonstrated to be capable of monitoring the quantum transition, and the back effect of monitoring on the probability of transition is analyzed. The realization of this type of measurement by a long series