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Quantum Information and Computation

Extension of remotely creatable region via local unitary transformation on receiver side

G. A. Bochkin, Alexander Ivanovich Zenchuk

Published Nov 1, 2016• 4 citations
Publisher

Abstract

We show that the length of the effective remote state creation via the homogeneous spin1/2 chain can be increased more than three times using the local unitary transformation of the so-called extended receiver (i.e., receiver joined with the nearest node(s)). This transformation is most effective in the models with all-node interactions. We consider an example of communication lines with the two-qubit sender, one-qubit receiver and two-qubit extended receiver.

DOI
10.26421/qic16.15-16-6

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We show that the length of the effective remote state creation via the homogeneous spin1/2 chain can be increased more than three times using the local unitary transformation of the so-called extended receiver (i.e., receiver joined with the nearest node(s)). This transformation is most effective in the models with all-node interactions. We consider an example of communication lines with the two-qubit sender, one-qubit receiver and two-qubit extended receiver.
We study the optimization problem for remote one- and two-qubit state creation via a homogeneous spin-1/2 communication line using the local unitary transformations of the multi-qubit sender and extended receiver. We show that the maximal length of a communication line used for the needed state creation (the critical length) increases with an increase in the dimensionality of the sender and extended receiver. The model with the sender and extended receiver consisting of up to 10 nodes is used for the one-qubit state creation and we consider two particular states: the almost pure state and the maximally mixed one. Regarding the two-qubit state creation, we numerically study the dependence of the critical length on a particular triad of independent eigenvalues to be created, the model with four-qubit sender without an extended receiver is used for this purpose.
We consider a method of remote mixed state creation of a one-qubit subsystem (receiver) in a spin-1/2 chain governed by the nearest-neighbor XYXYXY Hamiltonian. Owing to the evolution of the chain along with the variable local unitary transformation of the one- or two-qubit sender, a large variety of receiver states can be created during a specific time interval starting with a fixed initial state of the whole quantum system. These states form the creatable region of the receiver's state-space. It is remarkable that, with a two-qubit sender, a large creatable region may be covered at a properly fixed time instant t0{t}_{0}t0​ using just the variable local unitary transformation of the sender. In this case we have completely local control of remote state creation. In general, for a given initial state, there are some receiver states that may not be created using the above tool. These states form the unavailable region. In turn, this unavailable region might be the creatable region of another sender. Thus, in future, we have a way to share the whole receiver's state-space among the creatable regions of several senders. The effectiveness of remote state creation is characterized by the density function of the creatable region.
We study the optimization problem for remote one- and two-qubit state creation via a homogeneous communication line of spin-1/2 particles using local unitary transformations of the multiqubit sender and extended receiver. We show that the maximum length of a communication line used for the needed state creation (the critical length) increases as the dimensionality of the sender and extended receiver increases. We use the model with the sender and extended receiver comprising up to ten qubits for the one-qubit state creation and consider the creation of two particular states, the almost pure state and the maximally mixed state. Regarding the two-qubit state creation, we numerically study the dependence of the critical length on a particular triad of independent eigenvalues to be created using the model with a four-qubit sender without an extended receiver.
We study the problem of remote one-qubit mixed state creation using a pure initial state of two-qubit sender and spin-1/2 chain as a connecting line. We express the parameters of creatable states in terms of transition amplitudes. We show that the creation of complete receiver's state-space can be achieved only in the chain engineered for the one-qubit perfect state transfer (PST) (for instance, in the fully engineered Ekert chain), the chain can be arbitrarily long in this case. As for the homogeneous chain, the creatable receiver's state region decreases quickly with the chain length. Both homogeneous chains and chains engineered for PST can be used for the purpose of selective state creation, when only the restricted part of the whole receiver's state space is of interest. Among the parameters of the receiver's state, the eigenvalue is the most hard creatable one and therefore deserves the special study. Regarding the homogeneous spin chain, an arbitrary eigenvalue can be created only if the chain is of no more then 34 nodes. Alternating chain allows us to increase this length up to 68 nodes.
We study the problem of remote one-qubit mixed state creation using a pure initial state of two-qubit sender and spin-1/2 chain as a connecting line. We express the parameters of creatable states in terms of transition amplitudes. We show that the creation of a complete receiver's state space can be achieved only in the chain engineered for the one-qubit perfect state transfer (PST) (for instance, in the fully engineered Ekert chain); the chain can be arbitrarily long in this case. As for the homogeneous chain, the creatable receiver's state region decreases quickly with the chain length. Both homogeneous chains and chains engineered for PST can be used for the purpose of selective state creation, when only the restricted part of the whole receiver's state space is of interest. Among the parameters of the receiver's state, the eigenvalue is the most hard to create and therefore deserves special study. Regarding the homogeneous spin chain, an arbitrary eigenvalue can be created only if the chain is of no more than 34 nodes. The alternating chain allows us to increase this length to up to 68 nodes.