Secret sharing is a procedure for splitting a message into several parts so that no subset of parts is sufficient to read the message, but the entire set is. We show how this procedure can be implemented using GHZ states. In the quantum case the presence of an eavesdropper will introduce errors so that his presence can be detected. We also show how GHZ states can be used to split quantum information into two parts so that both parts are necessary to reconstruct the original qubit.Comment: 6 pages, revtex, revised version, to appear in Phys. Rev.

We analyze a possibility of copying (≡ cloning) of arbitrary states of quantum-mechanical spin-1/2 system. We show that there exists a "universal quantum-copying machine" (i.e. transformation) which approximately copies quantum-mechanical states such that the quality of its output does not depend on the input. We also examine a machine which combines a unitary transformation and a selective measurement to produce good copies of states in the neighborhood of a particular state. We discuss the problem of measurement of the output states. 03.65.Bz

Wootters [Phys. Rev. Lett. 80, 2245 (1998)] has given an explicit formula for the entanglement of formation of two qubits in terms of what he calls the concurrence of the joint density operator. Wootters's concurrence is defined with the help of the superoperator that flips the spin of a qubit. We generalize the spin-flip superoperator to a "universal inverter," which acts on quantum systems of arbitrary dimension, and we introduce the corresponding concurrence for joint pure states of (D1 X D2) bipartite quantum systems. The universal inverter, which is a positive, but not completely positive superoperator, is closely related to the completely positive universal-NOT superoperator, the quantum analogue of a classical NOT gate. We present a physical realization of the universal-NOT superoperator.Comment: Revtex, 25 page

A beam splitter is a simple, readily available device which can act to entangle output optical fields. We show that a necessary condition for the fields at the output of the beam splitter to be entangled is that the pure input states exhibit nonclassical behavior. We generalize this proof for arbitrary ͑pure or impure͒ Gaussian input states. Specifically, nonclassicality of the input Gaussian fields is a necessary condition for entanglement of the field modes with the help of a beam splitter. We conjecture that this is a general property of beam splitters: Nonclassicality of the inputs is a necessary condition for entangling fields in a beam splitter.

It is not a problem to complement a classical bit, i.e. to change the value of a bit, a 0 to a 1 and vice versa. This is accomplished by a NOT gate. Complementing a qubit in an unknown state, however, is another matter. We show that this operation cannot be done perfectly. We define the Universal-NOT (U-NOT) gate which out of N identically prepared pure input qubits generates M output qubits in a state which is as close as possible to the perfect complement. This gate can be realized by classical estimation and subsequent re-preparation of complements of the estimated state. Its fidelity is therefore equal to the fidelity F = (N + 1)/(N + 2) of optimal estimation, and does not depend on the required number of outputs. We also show that when some additional a priori information about the state of input qubit is available, than the fidelity of the quantum NOT gate can be much better than the fidelity of estimation. PACS number: 03.65.Bz, 03.67.-a There was an odd qubit from Donegal, who wanted to become most orthogonal.He went through a gate, but not very straight, and came out instead as a Buckyball.Classical information consists of bits, each of which can be either 0 or 1. Quantum information, on the other hand, consists of qubits which are two-level quantum systems with one level labeled |0 and the other |1 . Qubits can not only be in one of the two levels, but in any superposition of them as well. This fact makes the properties of quantum information quite different from those of its classical counterpart. For example, it is not possible to construct a device which will perfectly copy an arbitrary qubit [1,2] while the copying of classical information presents no difficulties. Another difference between classical and quantum information is as follows: It is not a problem to complement a classical bit, i.e. to change the value of a bit, a 0 to a 1 and vice versa. This is accomplished by a NOT gate. Complementing a qubit, however, is another matter. The complement of a qubit |Ψ is the qubit |Ψ ⊥ which is orthogonal to it. Is it possible to build a device which will take an arbitrary (unknown) qubit and transform it into the qubit orthogonal to it?The best intuition for this problem is obtained by looking at the desired operation as an operation on the Poincaré sphere, which represents the set of pure states of a qubit system. Thus every state, pure or mixed, is represented by a vector in a three-dimensional space, whose components are the expectations of the three Pauli matrices. The full state space is thereby mapped onto the unit ball, whose surface represents the set of pure states. In this picture the ambiguity of choosing an overall phase for |Ψ is already eliminated. The points corresponding to |Ψ and |Ψ ⊥ are antipodes of each other. The desired Universal-NOT (U-NOT) operation is therefore nothing but the inversion of the Poincaré sphere.Note that the inversion preserves angles (related in a simple way to the scalar product | Φ, Ψ | of rays), so by Wigner's Theorem the ideal U-NOT must be implemented eith...

We study the relaxation of a quantum system towards the thermal equilibrium using tools developed within the context of quantum information theory. We consider a model in which the system is a qubit, and reaches equilibrium after several successive two-qubit interactions (thermalizing machines) with qubits of a reservoir. We characterize completely the family of thermalizing machines. The model shows a tight link between dissipation, fluctuations, and the maximal entanglement that can be generated by the machines. The interplay of quantum and classical information processes that give rise to practical irreversibility is discussed.

We present a universal algorithm for the optimal quantum state estimation of an arbitrary finite dimensional system. The algorithm specifies a physically realizable positive operator valued measurement (POVM) on a finite number of identically prepared systems. We illustrate the general formalism by applying it to different scenarios of the state estimation of N independent and identically prepared two-level systems (qubits).Suppose we have N quantum objects, each prepared in an unknown pure quantum state described by a density operatorρ = |ψ ψ|. What kind of measurement provides us with the best possible estimation ofρ?Clearly, if we have an unlimited supply of particles in stateρ, i.e. when N approaches infinity, we can estimatê ρ with an arbitrary precision. In practice, however, only finite and usually small ensembles of identically prepared quantum systems are available. This leads to an important problem of the optimal state estimation with fixed physical resources.A particular version of this problem has recently been addressed by Massar and Popescu [1] who have analyzed extraction of information from finite ensembles of spin-1/2 particles. They have shown that there exists a fundamental limit on the fidelity of estimation and they have constructed a POVM consisting of an infinite continuous set of operators which attains this limit. It has been suggested that the optimal finite POVM may exist, however, so far no explicit construction of such finite POVM has been provided. We solve this problem by describing an algorithm which gives the optimal and finite POVM. We also provide a concise derivation of a general bound on the fidelity of a state estimation of arbitrary finitedimensional systems (the Massar-Popescu bound follows as a particular case). Finite POVMs can be, at least in principle, constructed and the optimal state estimation procedures may be applied in many areas of physics ranging from the optimal phase estimation to quantum computation.In fact, in the following we are solving a more general problem of optimal estimation of unitary operations performed on quantum objects (the state estimation follows as a special case). Let us assume that stateρ is generated from a reference stateρ 0 = |ψ 0 ψ 0 | by a unitary operation U (x) which is an element of a particular unitary representation of some group G. Different x denote different points of the group (e.g., different angles of rotation in the case of the SU (2)) and we assume that all values of x are equally probable.Our task is to design the most general POVM, mathematically described as a set {Ô r } R r=1 of positive Hermitian operators such that rÔ r =1 [2,3], which when applied to the combined system of all N copies provides us with the best possible estimation ofρ (and therefore also of U (x)). We quantify the quality of the state estimation in terms of the mean fidelitȳ

We review our recent work on the universal (i.e. input state independent) optimal quantum copying (cloning) of qubits. We present unitary transformations which describe the optimal cloning of a qubit and we present the corresponding quantum logical network. We also present network for an optimal quantum copying "machine" (transformation) which produces N + 1 identical copies from the original qubit. Here again the quality (fidelity) of the copies does not depend on the state of the original and is only a function of the number of copies, N. In addition, we present the machine which universaly and optimally clones states of quantum objects in arbitrary-dimensional Hilbert spaces. In particular, we discuss universal cloning of quantum registers.