Remote state preparation in higher dimension and the parallelizable manifold S n − 1

This paper proves that the remote state preparation (RSP) scheme in real Hilbert space can only be implemented when the dimension of the space is 2,4 or 8. This fact is shown to be related to the parallelazablity of the n-1 dimensional sphere Sn−1. When the dimension is 4 and 8 the generalized scheme is explicitly presented. It is also shown that for a given state with components having the same norm, RSP can be generalized to arbitrary dimension case. Remote state preparation (RSP) [1][2][3]is called “teleportation of a known state”. Unlike quantum teleportation [4][5][6][7][8], in RSP, Alice knows the state which she will transmit to Bob. Her task is to help Bob to construct a state which is unknown to him by means of a prior shared entanglement and a classical communication channel. Recently, Pati has shown that a state of a qubit chosen from equatorial or polar great circles on the Bloch sphere (i.e. a state with the components of the same amplitude or with real components) can be remotely prepared with one cbit from Alice to Bob if they share one ebit of entanglement[1]. Here, qubit stands for quantum bit whose state is a superposition of two orthonormal basis |0〉 and |1〉 ; cbit is classical bit carrying classical information; ebit is the so called entanglement bit usually carrying a Bell state. It is noted that in Pati’s special case, to 1 remotely prepare a state of one qubit, the entanglement cost is the same as that in teleportation but the classical information cost is only half of that in teleportation. Most recently, Lo and Bennett et al have studied the classical information cost for general state preparation in the scheme of RSP [2][3], using the concepts of entanglement dilution [9][10], high-entanglement limit and low-entanglement RSP[3]. They have also investigated the trade-off between entanglement cost and classical communication cost in RSP [2][3]. However, in Lo or Bennett et al’s protocols, either the entanglement cost or the classical information cost is more than that in Pati’s special case. This fact can be well understood by considering the geometry of Pati’s case: Pati’s states lie on the equatorial or polar great circles on a Bloch sphere. For this reason, we call the case treated by Pati the “minimum” case. As Pati presents his result only in the qubit case, it is natural to ask whether his result can be generalized to higher dimension case. It is well known that as far as teleportation, which transmits an unknown state, is concerned, the generalization from the qubit case to higher dimension case is straightforward. In fact, the first n-dimensional teleportation protocol is just given by Bennett et al in their first paper that introduced the celebrated concept of quantum teleportation [4]. Later the n-dimensional case of teleportation and its mathematical background were studied in more detail by many other authors [11][12][13][14]. Even in the case concerning continous variable [15], it can well be tackled [16]. The purpose of this paper is to seek a generalization of Pati’s result to higher dimension case. It will be shown that one can directly generalize the equatorial case. On the other hand,the generalization of the polar great circle case is highly nontrivial. We first consider the generalization of the polar great circle case (i.e. the case that the state has real components ). Precisely, we formulate our problem as follows. Suppose that Alice and Bob can share entangled state between two identical quantum systems the dimension of the state space of which is n.Choose an orthonormal basis {φi|i = 0, 1, · · · , n− 1} of the state space. By measuring the system with respect to a certain basis, Alice wishes to prepare a quantum state of the form |Ψ〉 = n−1 ∑ i=0 ai |φi〉 at Bob, where the coefficients are real numbers. Between Alice and Bob there is a classical channel capable of transmitting information carried by a 2 “classical bit” that can take n different values, say, 0, 1, · · · .n − 1. By prior agreement, each value carried by the “classical bit” can be corresponded to a unitary operation on the quantum system at Bob. That is to say, when Bob receives a value i he will exert a certain unitary operation Ui on his system. Now our question is: for the above minimum RSP procedure to be realizable what condition should the dimension n satisfy? By convention,in the procedure of RSP the maximally entangled state shared by Alice and Bob, will be the EPR state |Φ〉AB = 1 √ n ( n−1 ∑