Phase-Conjugate-State Pairs in Entangled States

We consider the probability that a bipartite quantum state contains phase-conjugate-state (PCS) pairs and/or identical-state pairs as signatures of quantum entanglement. While the fraction of the PCS pairs directly indicates the property of a maximally entangled state, the fraction of the identical-state pairs negatively determines antisymmetric entangled states such as singlet states. We also consider the physical limits of these probabilities. This imposes fundamental restrictions on the pair appearance of the states with respect to the local access of the physical system. For continuous-variable system, we investigate similar relations by employing the pairs of phase-conjugate coherent states. We also address the role of the PCS pairs for quantum teleportation in both discrete-variable and continuous-variable systems. 1. Introduction The spooky action induced by the entangled particles at a distance is an interesting starting point in studies of quantum entanglement [1–5]. For the singlet state, the spins are always antiparallel to each other independently of the measured spin angle. Another familiar example is the Einstein-Podolski-Rosen (EPR) state. For the EPR states, not only the positions of the two particles but also the momentums of the two particles have perfect correlation. When we consider the standard form of the maximally entangled state in two -level (qudit) systems , an interesting property is the appearance of a phase-conjugate state when a subsystem is projected onto a local state , namely, for any state , we have the relation An example with two spin systems is schematically shown in Figure 1. A natural question is whether this coherent appearance of the phase-conjugate-state (PCS) pairs can be a signature of entanglement. Figure 1: The pair of phase-conjugate states in the Bloch sphere when the fixed basis is the basis with the parameterization . In this representation the phase-conjugation changes the sign of the -coordinate and the phase-conjugate state vector of the system is given by . The directions of the Bloch vectors are in the reflection relation with respect to the plane. The maximally entangled state is a coherent superposition of these pairs and produces the phase-conjugate pairs associated with the local projection in any direction as described in ( 1). It has been known that the use of PCS pairs is more efficient than the use of identical-state pairs, , in the transmission of unknown quantum states. Gisin and Popescu showed that the set of antiparallel spins stores the quantum information better than the set of