TelePOVM-A generalized quantum teleportation scheme

IBM Journal of Research and Development, Jan 2004 by Brassard, G, Horodecki, P, Mor, T

In this paper, we show that quantum teleportation is a special case of a generalized Einstein-Podolsky-Rosen (EPR) nonlocality. On the basis of the connection between teleportation and generalized measurements, we define conclusive teleportation. We show that perfect conclusive teleportation can be obtained with any pure entangled state, and it can be arbitrarily approached with a particular mixed state.

1. Introduction

Quantum information processing [1-3] is concerned with the processing of information in which the basic units are two-level quantum systems [4, 5] (such as spin- � particles and the polarization of individual photons) known as quantum bits, or qubits. The state of a qubit is given by |[phi][right angle bracket] = [alpha]|0[right angle bracket] [beta]l1[right angle bracket] = (^sup [alpha]^^sub [beta]^), where [alpha] and [beta] are complex amplitudes subject to the normalization condition |[alpha]^sup 2^| |[beta]|^sup 2^ = 1, and |0[right angle bracket] = (^sup 1^^sub 0^,) and |1[right angle bracket] = (^sup 0^^sub 1^) are the basis state vectors. When the state of two or more qubits cannot be expressed as a tensor product of individual qubits, we say that the system is in an entangled state. Entanglement is at the heart of spectacular phenomena in quantum information theory, such as quantum computation, entanglement-based quantum cryptography, quantum error correction, quantum communication complexity, and more.

One of the most fascinating applications of entanglement is quantum teleportation [9], which is one of the pillars of quantum information theory (see [1, 10]) and has been realized experimentally on several occasions. Quantum teleportation is a process that enables the transmission of an unknown quantum state via a previously shared EPR pair with the help of only two classical bits transmitted on a classical channel. Assume that Alice (the sender) has a qubit in an unknown quantum state, which she wishes to transmit to Bob (the receiver). This would seem to be impossible if a quantum channel were not available to Alice and Bob at the time transmission had to take place. But assume that a quantum channel was available at some point in the past-perhaps when Alice and Bob were in physical contact-and assume that they are capable of storing quantum information faithfully. As a preprocessing step, when the quantum channel is available, Alice can prepare a Bell state, store one particle in her quantum memory, and use the channel to send the other particle to Bob, who stores it in his quantum memory. At this point, we say that Alice and Bob share an EPR pair. Later on, when the quantum channel may no longer be available, Alice receives an unknown quantum state |[phi][right angle bracket] = (^sup [alpha]^^sub [beta]^). To teleport this state to Bob, she performs a joint measurement on the two particles that are in her hands: the unknown quantum state and her share of the EPR pair. This measurement destroys state |[phi][right angle bracket] in her laboratory, but it produces two classical bits. Using a classical channel, she sends her two-bit result to Bob, who performs some unitary operation on his particle, "transforming" it into the (still unknown) original state |[phi][right angle bracket].

The minimal resources required for faithful teleportation are one EPR pair, which is independent of |[phi][right angle bracket], and two classical bits. This may seem rather mysterious because 1) using the Bloch sphere formalism, the state of the particle to be teleported can be described by a point on a unit sphere (assuming that the state to be teleported was pure), hence by two real numbers and certainly not by two bits; and 2) even from those two classical bits, neither Alice nor Bob can learn anything about the unknown parameters of the teleported state because these bits are easily seen to be purely random, and therefore independent of |[phi][right angle bracket].

The alternative approach presented in this paper sheds new light on this mystery. We interpret teleportation in the light of a seminal 1993 paper of Hughston, Jozsa, and Wootters (HJW) [12], which itself was anticipated by an extraordinary paper [13] written by Schrodinger only one year after the original 1935 EPR paper [6]. A slightly more restricted scenario than the one discussed by HJW was previously presented by Gisin [14] in 1989. Specifically, we use the language of generalized measurements to express the ideas of HJW and then we present the teleportation process as a special case of generalized EPR nonlocality.

A positive operator valued measure (POVM) provides the most general physically realizable measurement in quantum mechanics [5]; hence, we also refer to POVMs as "generalized measurements." Formally, a POVM is a collection of positive operators A. on a Hubert space H^sub n^ of dimension n that sum up to the identity A^sub 1^ . . . A^sub r^ = I^sub n^. (An operator is positive if all of its eigenvalues are positive or zero.) Standard measurements, which are usually described by some Hermitian operator in quantum mechanics texts, arise as a special case when A. = |[psi]^sub i^[right angle bracket][left angle bracket][psi]^sub i^|. and A^sub i^ A^sub j^ = [delta]^sub ij^ A^sub i^. We discuss here only rank-one POVMs, in which each of the A^sub i^ = gi|[psi]^sub i^[right angle bracket][left angle bracket][psi]^sub i^| is proportional to a projection operator but the operators are not necessarily orthogonal to each other, so that r can be greater than n. Neumark's theorem states that, at least in principle, any POVM can be implemented by the adjunction of an ancilla in a known state, followed by a standard measurement in the enlarged Hubert space [5].