Quantum teleportation of spin coherent states: beyond continuous variables teleportation

We introduce a quantum teleportation scheme that can transfer a macroscopic spin coherent state between two locations. In the scheme a large number of copies of a qubit, such as realized in a coherent two-component Bose-Einstein condensate, is teleported onto a distant macroscopic spin coherent state using only elementary operations and measurements. We analyze the error of the protocol with the number of particles N in the spin coherent state under decoherence and find that it scales favorably with N.

We introduce a quantum teleportation scheme that can transfer a macroscopic spin coherent state between two locations. In the scheme a large number of copies of a qubit, such as realized in a coherent two-component Bose-Einstein condensate, is teleported onto a distant macroscopic spin coherent state using only elementary operations and measurements. We analyze the error of the protocol with the number of particles N in the spin coherent state under decoherence and find that it scales favorably with N . Quantum mechanics is typically associated with the microscopic world, where phenomena such as superposition and entanglement occur for few-particle systems, but become difficult to observe when scaled up to macroscopic objects. It has however become clear in recent years that even macroscopic objects can behave quantum mechanically, and possess significant amounts of entanglement [1,2]. Furthermore, it is now well-accepted that the quantum-to-classical transition occurs not inherently because of the large number of particles, but due to fast decoherence of such macroscopic objects [3,4]. Nevertheless, observing and manipulating quantum mechanics beyond the few-qubit level remains a serious challenge in the field of quantum technologies.
Recently there have been several remarkable experimental demonstrations of entanglement generation and teleportation between macroscopic objects [5][6][7][8][9]. These follow from the many successful realizations of teleporation using a variety of systems ranging from those using photons [10][11][12], atoms [13][14][15], and hybrid systems [5,16]. In Ref. [6] spin polarized atomic ensembles were used to form continuous variables (CV) [17] and entanglement was generated between two ensembles. In Ref. [8] such CV variables were teleported between two atomic ensembles, while in Ref. [9] a spin wave qubit state was teleported. While the discrete and CV versions are two particular approaches to quantum teleportation, for other encodings of quantum information, other teleportation schemes need to be devised in general. In this paper we describe a protocol that is suitable for general spin coherent states on any location on the Bloch sphere.
Previously one of the authors proposed an alternative way of performing quantum computation, involving macroscopic spin coherent states [18]. The method encodes qubit states on spin coherent states, which can involve either spinor Bose-Einstein condensates (BECs) or atomic ensembles. Taking the example of the twocomponent BEC, a single qubit is encoded as where creation operators for the two hyperfine states a † , b † obey bosonic commutation relations, and N is the number of bosons in the BEC. The concept of the work of Ref. [18] is that such "BEC qubits" can be used with analogous properties to standard qubits, such that various quantum algorithms can be performed using these states. Although such spin coherent states are generally thought to be "classical" in the sense that the fluctuations of normalized spin variables S j /N decay as ∝ 1 √ N (j = x, y, z) [19], it was shown that quantum effects such as entanglement can be generated between several such BEC qubits [20], and quantum algorithms could be performed provided an equivalent algorithm could be found.
It is clear from the structure of (1) that the quantum information is duplicated N times when encoded in this way. Such a structure is attractive from a quantum technology standpoint since it adds a robustness via duplication of the information. Thus unlike single qubit systems (not involving quantum error-correction) where a single quanta of external noise can destroy the quantum information, even a loss of some fraction of the number of particles in (1) does not result in complete loss of the quantum information, it merely contributes to a diminished signal amplitude. The question is then whether such states (1) can indeed be used to perform quantum information processing tasks, as the Hilbert space is complicated by the fact that we must control all N duplicates at once. In this paper we show that a fundamental protocol in quantum information, quantum teleporation, can be performed using the encoding (1). That is, an unknown state |α, β at one location may be transferred onto another BEC qubit at a different location by the use of shared entanglement.
In our protocol we assume that the following operations are available in order to manipulate the BEC qubits (1): (i) Coherent spin rotations corresponding to applications of the Hamiltonians S x = a † b + b † a, S y = −ia † b + ib † a, and S z = a † a − b † b; (ii) Projective measurements which collapse the coherent state onto the number basis |k = Spin rotations corresponding to (i) have been performed in many contexts, and thus are possible using current experimental techniques [21,22]. Projective measurements (ii) can be performed using Stern-Gerlach methods [23]. Although the two BEC qubit interaction has not been realized experimentally yet, this could either be done via cavity coupling methods [18], or state dependent cold collisions [24]. The realization of strong coupling of a BEC to a cavity [25] and entanglement between a BEC and an atom [26] would suggest that such an interaction is within experimental reach.
Our quantum teleportation protocol for BEC qubit states is as follows. We continue the tradition and call the heroes of our protocol as Alice (sender) and Bob (receiver). Alice is in possession of BEC qubits 1 and 2, and Bob has BEC qubit 3. BEC qubit 1 is in an unknown state (1) on the surface of the Bloch sphere which we parameterize α = cos For simplicity we assume that θ = π/2, such that the initial state is along the equator of the Bloch sphere. Teleporting a BEC qubit for arbitrary state on the Bloch sphere can be performed by adapting the method for the two coordinates (θ, π). The initial state of whole system is The protocol then follows the following sequence: 1) Apply the entangling gate S z 2 S z 3 on BEC qubits 2 and 3 for a time T = 1/ √ 2N ; 2) Apply the entangling gate −S z 1 S z 2 on BEC qubits 1 and 2 for a time τ = 1/ √ 2N ; 3) Apply a Hadamard gate on BEC qubit 1; 4) Measure BEC qubits 1 and 2 in the |k basis; 5) Classically transmit measurement result of BEC qubit 1 to Bob; 6) Rotate BEC qubit 3 by an angle arccos(1 − 2k 1 /N ). At this point Bob now has possession of Alice's BEC qubit completing the teleportation.
We now explain how the protocol works. After the first entangling gate the state is where we have used the binomial representation the state of two BEC qubits. The state between BEC qubits 2 and 3 is discussed in Ref. [18] in detail. This is analog of Bell state for coherent spin states in this representation. On application of the entangling gate for a time τ between BEC qubits 1 and 2 we have The Hadamard operation consists of consists of making the replacement a → (a Expanding BEC qubit 1 in the |k basis gives The resulting state after the measurement on Bob's BEC qubit is Here, k 2 is the measurement outcome of measuring BEC qubit 2. The probability distribution for the measurement in step 4 is To obtain some insight into this expression let us use the  The first factor in the probability distribution states that k 2 = N/2 and random variations to this occur with standard deviation √ N /2. The second factor involves a correlation between Alice's angle φ to be teleported and k 2 . This correlation gives rise to the ability to teleport φ, because after the measurement Bob holds the state (7), which must be correlated to the original state. The probability distribution p(k 1 , k 2 ) has a maximum when Dependences of p(k 1 , k 2 ) at fixed k 1 versus φ and k 2 are presented on Figure 1. We have sharp correlations between φ and k 2 which obey the linear function (10). This arises due to the standard deviations of the second term in (9) being ∼ O(1), thanks to the choice τ = 1/ √ 2N . In order to match the state of Bob's BEC qubit (7) to the initial state, by substituting (10) into (7) we see that the entangling time T should be set to T = 1/ √ 2N , as given in our prescription. The state of Bob's BEC qubit at the end of step 4 is then with probability approaching unity for large N This is equal to the initial state, up to some known angle. The random variable k 1 must be classically transmitted to Bob in order to correct the angle by the angle   Figure 1c. This shows a broad distribution which is mostly peaked at k 1 = 0 or N . In terms of standard qubit teleportation, k 1 plays the role of the outcome of the Bell measurement, which must be classically communicated to Bob in order to make the "corrections" to the final state [27]. Performing these corrections in Step 6 completes the teleportation and Bob's is now in possession of Alice's initial state.
An important issue when discussing quantum teleportation is evaluating the "quantumness" of the protocol [17,28,29]. That is, in comparison to an purely classical strategy of transmitting information, what improvement does the quantum protocol achieve in the fidelity of transfer of the original state? Such bounds are useful in experimentally verifying that a quantum teleportation has indeed taken place, and places a boundary between classical and quantum communication. In our case, let us compare our protocol with the following all-classical strategy: 1) Alice measures the initial unknown state in the spin coherent basis |α m , β m α m , β m |; 2) Alice classically communicates the measurement result (θ m , φ m ) to Bob; 3) Bob prepares the state | e −iφm /2 . As a measure of the quality of the state transfer, we use the trace distance ε We derive that such a classical strategy averaged over all initial states and measurement outcomes achieves for large N The inequality follows from the fact that the best estimation fidelity in step 1 is obtained by a measurement in the spin coherent state basis [30]. In Figure 2a we compare the classical and quantum strategies. We see that for the zero dephasing case, the protocol presented here outperforms the classical boundary, showing a quantum enhanced state transfer using entanglement. The superior performance of the quantum protocol can be understood qualitatively in the following way. In the classical strategy, the location of the initial state on the Bloch sphere can be estimated with standard deviation S j /N ∼ 1/ √ N . Meanwhile, the sharp correlations in the second term of (9) give variations of order S j /N ∼ 1/N , which allows for smaller fluctuations of the teleported state, giving rise to a lower error ε.
In any experimental situation decoherence is inevitably present, and it is an important question how well the teleportation can be carried out under such conditions. Typically, quantum effects such as entanglement, which teleportation is based on, disappear at a timescale extremely quickly for macroscopic objects. Remarkably for the entangled states that we consider in this protocol, this is not the case. We consider decoherence in the form of a generic dephasing process, which can be described by the master equation where H(t) is the Hamiltonian sequence described above.
For concreteness we assume that the system starts in the initial state (2), and the dephasing takes place for the duration of the entangling gates (Steps 1 and 2). We do not consider dephasing to take place during the Hadamard step since this can be combined with the measurement step as a projection on the x basis. Figure 1d shows the effect of the dephasing term on the probability distribution p(k 1 , k 2 ). We show only k 1 = N since this is the most probable outcome for −π/2 < φ < π/2, and other cases give similar results up to a angular correction factor. We see that the sharp correlation between φ and k 2 is reduced and the distribution is broadened due to the dephasing, due to random variations in the correlations between BEC qubit 1 and 2. Figure 2b shows the output state at Bob's BEC qubit which shows a degradation of the fidelity from the ideal zero decoherence case. The effect of the decoherence can be seen to diminish the amplitude of the output spins due to a combination of degradation of the correlations between Alice and Bob's states, as well as a direct decoherence on Bob's qubit. Figure 2a shows the scaling of the error with N , as measured by the trace distance between the ideal state | e −iφm /2 √ 2 , e iφm /2 √ 2 and the teleported state. Interestingly, we see that the case including dephasing tends to improve with N . There are several contributing factors to this. The first reason is that the fidelity of the protocol itself improves with N , as can be observed from the ideal γ = 0 case. A second reason is due to the fact that as N grows, the gate times that are necessary actually decreases as τ, T = 1/ √ 2N . This means that there is less time for the dephasing to destroy the states for large N , due to the shorter times that the decoherence acts. A third reason is that in our protocol we never use states that are extremely sensitive to decoherence. Decoherence is intrinsically a state-dependent process, and the time that quantum coherence can survive depends upon the encoding. For example, a qubit that is encoded on a Schrodinger cat state α|1, 0 + β|0, 1 dephases to a mixed state under (13) in time ∼ 1/(N 2 γ), whereas the spin coherent state dephases in time ∼ 1/γ [31]. Thus although such cat states could in principle be used to encode quantum information for teleportation, from a decoherence point of view this is generally not a wise choice. For example, evolving (3) for a time T = π/4 gives which is an entangled Schrodinger cat state. Such a state decoheres in time ∼ 1/(N 2 γ), and would not have the favorable scaling behavior as seen in Figure 2b. Thus the use of the entangled states corresponding to times T = τ = 1/ √ 2N can be seen as the key to allowing teleportation to be performed without the very fast degradation of coherence that cat states would suffer.
We have introduced a protocol that can teleport a macroscopic BEC qubit in a spin coherent state to another macroscopic BEC qubit. As with standard qubit teleportation, a classical random variable needs to be sent in order to "correct" the transmitted state. Although we have used the language of BEC qubits, the algebra used is that of spin coherent states, so in principle should apply to other systems, such as atomic ensembles as the only operations that are used are the total spin operators and projective measurements. The interesting aspects of the protocol are that (i) true macroscopic states involving N copies of a quantum state are entangled and teleported to a distant location; and (ii) decoherence scales favorably with N due to the use of non-maximally entangled states, such that it should be applicable to realistically sized systems. This work makes clear that with carefully prepared states it is possible to perform useful quantum mechanical protocols beyond the microscopic level. From a quantum technology viewpoint this is important as by distributing resources such as entanglement over many particles this adds a natural resilience to unavoidable experimental imperfections.
T. B. thanks Eugene Polzik and Mikhail Lukin for discussions. This work is supported by the Transdisciplinary Research Integration Center and Japan Russia Youth Exchange Center.