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Find it at other libraries via WorldCat Limited preview. Contributor Nikolopoulos, Georgios M. Jex, I. Igor , editor. Nikolopoulos and Stelios Tzortzakis. Summary Faithful communication is a necessary precondition for large-scale quantum information processing and networking, irrespective of the physical platform. Thus, the problems of quantum-state transfer and quantum-network engineering have attracted enormous interest over the last years, and constitute one of the most active areas of research in quantum information processing. The present volume introduces the reader to fundamental concepts and various aspects of this exciting research area, including links to other related areas and problems.
The implementation of state-transfer schemes and the engineering of quantum networks are discussed in the framework of various quantum optical and condensed matter systems, emphasizing the interdisciplinary character of the research area. Each chapter is a review of theoretical or experimental achievements on a particular topic, written by leading scientists in the field.
The volume aims at both newcomers as well as experienced researchers. Subject Quantum communication. Quantum computers. Quantum Information Technology, Spintronics. Communications Engineering, Networks. Quantum Physics. Special Purpose and Application-Based Systems. Quantum Computing. Quantum communication. Bibliographic information.
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Create citation alert. Buy this article in print. Journal RSS feed. Sign up for new issue notifications. The XX model with uniform couplings represents the most natural choice for quantum state transfer through spin chains. Given that it has long been established that single-qubit states cannot be transferred with perfect fidelity in this model, the notion of pretty good state transfer has been recently introduced as a relaxation of the constraints on fidelity.
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In this paper, we study the transfer of multi-qubit entangled and unentangled states through unmodulated spin chains, and we prove that it is possible to have pretty good state transfer of any multi-particle state. This significantly generalizes the previous results on single-qubit state transfer and opens the way to using uniformly coupled spin chains as short-distance quantum channels for the transfer of arbitrary states of any dimension. Our results could be tested with current technology.
Content from this work may be used under the terms of the Creative Commons Attribution 3. Any further distribution of this work must maintain attribution to the author s and the title of the work, journal citation and DOI. The transfer of quantum states from one site to another is a key task in quantum information processing. A straightforward approach could be to apply a sequence of SWAP gates, but that would be too demanding in terms of control and very prone to errors [ 2 ].
This has stimulated the proposal of passive quantum networks as transmission devices that do not require control except during the preparation and readout. In particular, the one-dimensional half-integer spin chain has been extensively studied as a quantum wire for the transfer of qubit states [ 3 — 5 ].
Much attention has been given to the ideal scenario where we have perfect state transfer PST , i.
The experimental realization of spin chains with PST would be possible if the system were engineered according to non-natural coupling schemes [ 7 — 9 ], but this would unfortunately be a highly difficult task in practice. Furthermore, it has been argued that the condition of perfect transfer is far too demanding in comparison with the level of fidelity required for the implementation of most quantum information processing tasks [ 10 ].
Therefore, it is important to investigate if a relaxation of the constraints on the fidelity can lead to better results. It would be useful to establish that quantum state transfer can be accomplished in chains with minimal variation of the coupling strengths. This is so because the unmodulated chain is much easier to fabricate as compared to chains where local engineering is required [ 1 ], and, furthermore, there exist experimental quantum information transfer platforms whose natural dynamics are governed by Hamiltonians with uniform couplings [ 11 , 12 ].
Motivated by this, several alternatives have been proposed, such as designing weakly varying coupling configurations which also display the PST property [ 13 ], obtaining high quality ballistic state transfer or dynamic generation of entanglement in chains where the bulk is uniform and the couplings between the bulk and the boundary qubits are tuned to optimal values [ 14 — 17 ], using an iterative measurement procedure to perform state transfer in unmodulated chains with low probability of failure [ 18 , 19 ], and improving the quality of the transmission by initializing the channel qubits to a specific state [ 20 ].
In our work, we focus on the notion of pretty good state transfer PGST , which has recently been introduced as a significant alternative to PST [ 21 , 22 ]. Here, the requirement is that the fidelity of transfer gets arbitrarily close to unity. By applying this concept to the unmodulated XX-type chain, which represents the model of spin chain whose construction requires less control over the individual parts [ 1 ], Godsil et al proved that there is a PGST of single-particle states in such a chain if and only if the length of the chain is , , where p is prime, or [ 23 ].
This result means that the quality of the state transfer protocol depends on conditions which are purely number-theoretic and shows that there is a surprising connection between the dynamics of quantum spin chains and primality. Since the discovery by Shor of a polynomial time algorithm for prime factorization on a quantum computer [ 24 ], the application of quantum algorithms to problems in prime number theory has been a topic of special interest. This continues to be an active research direction: for instance, a recent contribution is that of Latorre and Sierra, who propose the creation of a single quantum state made of the superposition of prime numbers to study primality problems [ 25 ].
In this context, the result of [ 23 ] is of high significance, since it establishes a nontrivial link between quantum dynamics and primality that is outside the scope of the traditional algorithmic applications. It would therefore be desirable to generalize this result to other protocols with practical relevance for quantum state transfer. The work in [ 23 ] is restricted to single-particle qubit states.
However, given the practical challenges inherent in the implementation of quantum systems, the usefulness of the experimental realization of the uniformly coupled XX chain as a quantum channel clearly depends on whether or not many-particle qubit states and, in particular, entangled states, can also be transferred with arbitrarily high fidelity through a single chain. This topic has already been addressed for PST protocols [ 13 , 26 ], and it has been established that, in the XX model, PST of arbitrary single-qubit states is a sufficient condition for PST of multi-qubit states 3. While completing this manuscript, we also became aware of an independent work in chains with Rabi-like dynamics [ 28 ] that further endorses the relevance of the problem of multi-particle state transfer through a single chain.
However, to the best of our knowledge, the study of PGST has never encompassed multi-particle state transfer. In this paper, we extend the definition of PGST to arbitrary multi-qubit states, and we show that regardless of the number of qubits of the input state, the fidelity of the transfer is arbitrarily high if and only if the length of the chain is , , or. From a strictly physical point of view, this confirms that the uniformly coupled quantum spin chain is a versatile channel for the construction of quantum communication systems.
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Additionally, our work widens the number of protocols for which PGST is characterized as a function of primality conditions. This highlights that the link between quantum dynamics and primality goes beyond the established applications in the field of quantum algorithms. By employing the Jordan—Wigner transformation [ 29 ], this system can be mapped to a local fermionic Hamiltonian,.
The total z -spin operator, given by , commutes with the Hamiltonian of the system, ; therefore, the Hilbert space of the register can be diagonalized and decomposed into invariant subspaces consisting of the distinct eigenstates of S tot z. Each of these subspaces can be characterized by the number of spins on the excited state, i. Since the Hamiltonian 2 describes a system of non-interacting spinless fermions, the initial quantum states that are in each of the individual subspaces of H will remain there under time evolution. The state transfer scheme we will consider is a natural generalization of the usual single-qubit scheme: the state sender S, located on one end of the chain, wishes to transmit an m -qubit composite state which will generally consist of an entangled state to the receiver R, located on the other end see figure 1.
A generic state of the system is therefore of the form.
After initialization of the n spins to the eigenstate. Figure 1. Since he is located on the antipodal site, we assume that the receiver reads the state of his subsystem in the opposite order. If we let be the mirrored state of the whole system, the output state is thus given by. As will be seen later, this reversal enables us to exploit the symmetry properties of the system.
Pretty good state transfer of entangled states through quantum spin chains
We will start by restricting our problem to the single-excitation subspace, since this represents the simplest scenario for transfer of entangled states. This subspace is described by the class of states , where only the j th qubit is in the excited state. We are thus assuming that the state of the whole system immediately after the sender places its state on the chain is of the form. The two definitions of PGST we will adopt in this paper are straightforward generalizations of the notion introduced by Godsil in [ 21 ].
We start by stating the first, which applies to the single-excitation scenario: we say that there is PGST of the state if for any , there is a time such that. The effects of the absolute values taken above can be easily corrected through the application of appropriate phase gates [ 27 ]; therefore, condition 8 implies that our figure of merit can be made arbitrarily close to 1. Under the single-excitation framework, the state vectors are equivalent to the n vectors of the usual basis of the space n , and the unitary operator is equivalent to an continuous-time quantum walk on the uniformly coupled path graph.
This enables us to undertake a graph-theoretic approach in the study of PGST, as in [ 23 , 30 ]. The following theorem is the main result of [ 23 ]: Theorem 1. Suppose the Hamiltonian of the chain is given by 1. Then, given an , there is a time such that.
We will now invoke a particular case of a theorem which was proved by Cameron et al see theorem 3 in [ 30 ]. Theorem 2. Then, condition 9 is true if and only if for any given , there is a time such that. Corollary 3. If the multi-qubit input state is restricted to the single-excitation manifold, then there is PGST of if and only if , , or , where p is a prime and.
If , , or , then by theorem 2, for each there is a time such that. This proves that condition 8 is satisfied. We remark that, even though we have taken the magnetic field to be zero in the XX Hamiltonian 1 , the results of corollary 3 are also valid if a uniform magnetic field is added to the Hamiltonian. This is so because, in such a case, the single-excitation transfer amplitudes would only be shifted by an overall phase factor [ 31 ].
This result per se is already quite interesting, but the fact that we have restricted the analysis to the single-excitation subspace is a major limitation to its practical applicability. Hence, we will now turn our attention to the higher-excitation framework. The r -excitation subspaces are spanned by the set of states with and pairwise distinct , where the qubits are in the state and the remaining qubits are in the state ; the dimension of these subspaces is thus given by the binomial coefficients. For example, denotes the state , which belongs to the three-excitation manifold.
In this unrestricted case, the sender wishes to transmit a state , which corresponds to a superposition of states with up to m excitations. Therefore, the state of the whole system after placement of the input state is. In other words, there is PGST of if there is a solution for the nonlinear system of inequalities composed by the m inequalities 13 and by the inequalities 14 ; this assures that all of the conditions are simultaneously verified at time t.
This definition reduces to the former one in the single-excitation case, and it likewise implies that can be made arbitrarily close to one.