Teleporting a quantum state from a subset of the whole Hilbert space

Abstract

We investigate the lower bound of the amount of entanglement for faithfully teleporting a quantum state belonging to a subset of the whole Hilbert space. Moreover, when the quantum state belongs to a set composed of two states, a probabilistic teleportation scheme is presented using a non-maximally entangled state as the quantum channel.

Introduction

Quantum entanglement is one of the most striking features of quantum mechanics and has been widely used as an essential resource in the quantum information processing. Some physical incidents such as quantum teleportation [1], quantum key distribution [2], quantum computation [3], [4], [5] and quantum secure direct communication [6], [7], [8], [9], [10] use it essentially. Since quantum entanglement is the essential resource, we always expect to complete a task using as small amount of entanglement as possible or to complete more tasks when the amount of entanglement is certain. The amount of entanglement of the state $|\Psi 〉$ of the joint system of the two subsystems A and B is defined as the Von Neumann entropy of either of the two subsystems:

$$E(|\Psi 〉)=-\mathrm{Tr}({\rho }_A{\log }_2{\rho }_A)=-\mathrm{Tr}({\rho }_B{\log }_2{\rho }_B),$$

where ${\rho }_A$ is the partial trace of $|\Psi 〉〈\Psi |$ over subsystem B, and ${\rho }_B$ has a similar meaning.

Since an arbitrary pure state of two-body has the form of Schmidt decomposition

$$|\Psi 〉=\sum _{i=1}^n\sqrt{p_i}|i〉|i〉,$$

where n is the Schmidt number and $p_i$ is the Schmidt coefficient (we have included in the sum only the non-zero $p_i$'s). Thus it follows easily that $E=-{\sum }_i{p}_i{\log }_2{p}_i$. Particularly, $E={\log }_{2}d$ when $|\Psi 〉$ is the maximally entangled state, where d is the dimension of the Hilbert space of the subsystem.

Quantum teleportation is one of the most important applications of quantum entanglement. In quantum teleportation process, an unknown quantum state can be transmitted from a sender (called Alice) to a receiver (Bob) without transmission of carrier of quantum state. Since Bennett et al. [1] presented a quantum teleportation scheme, there have been great developments in theoretical and experimental studies. Now quantum teleportation has been generalized to many cases such as continuous variable quantum teleportation [11], probabilistic teleportation [12], [13], [14], [15], [16], [17], controlled quantum teleportation [18], [19], [20] and so on [21], [22], [23], [24], [25], [26], [27], [28]. Moreover, quantum teleportation has been demonstrated with the polarization photon [29] and a single coherent mode of fields [30] in the experiments. The teleportation of a coherent state corresponding to continuous variable system was also realized in the laboratory [31].

In the Bennett's protocol [1], the quantum state to be teleported belongs to the whole Hilbert state vector space. A quantum state of d-state particle (or qudit) can be faithfully teleported using a pair of d-state particles in a maximally entangled state, in which, the amount of entanglement $E={\log }_{2}d$ is used. Gour [32] pointed out that ${\log }_{2}d$ is the minimal amount of entanglement for faithfully teleporting an arbitrary d-dimensional quantum state. When knowing that the quantum state to be teleported is from the subspace of the whole Hilbert space, we can complete the quantum teleportation using less amount of entanglement. For instance, Gorbachev and Trubilko [33] considered the quantum teleportation of two-particle entangled state by a three-particle GHZ state. The amount of entanglement required is $E={\log }_{2}2=1$ instead of ${\log }_{2}4=2$, the amount of entanglement for teleporting a general two-qubit state. Yan and Yang discussed the economical teleportation of multiparticle quantum state [34].

In this Letter, we will study how many amount of entanglement must be used at least when the quantum state belongs to a subset of the whole Hilbert space. The lower bound of the amount of entanglement for completing faithful teleportation in this case is calculated. Moreover, when we know the quantum state is coming from a two-state set, a probabilistic teleportation scheme is presented using a non-maximally entangled state as the quantum channel. The transmission efficiency of this scheme is calculated also.