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Entanglement dynamics in quantum cellular automata
Physics Letters A 326 (2004) 328–332 www.elsevier.com/locate/pla
Entanglement dynamics in quantum cellular automata M. Andrecut ∗ , M.K. Ali Department of Physics, University of Lethbridge, Lethbridge, AB, T1K 3M4, Canada Received 5 January 2004; received in revised form 26 April 2004; accepted 29 April 2004 Available online 11 May 2004 Communicated by P.R. Holland
Abstract A quantum cellular automata architecture is presented. We investigate the complex dynamics of the multi-particle entanglement generated by simple local rules implemented in the described architecture. Also, we show that our model builds up long-range quantum correlations, resulting in generating and distributing entangled states throughout the system. 2004 Elsevier B.V. All rights reserved. PACS: 03.67.Mn; 05.45.Pq Keywords: Quantum cellular automata; Multi-particle entanglement; Regular and stochastic dynamics
1. Introduction Classical cellular automata (CCA) are discrete dynamical systems, which can simulate a wide range of complex physical phenomena including fluid dynamics, non-linear dynamics and phase transitions in many-body systems [1]. The meaning of discrete is, that space, time and properties of the CCA can have only a finite, countable number of states. The basic idea is not to describe a complex system with complex equations, but to let the complexity emerge from the interaction of simple units following simple rules. Formally, a CCA is termed complex if it evolves in a manner that in some sense is computationally irre-
* Corresponding author.
E-mail addresses: [email protected] (M. Andrecut), [email protected] (M.K. Ali). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.04.064
ducible, meaning it cannot be predicted with a compactly written equation [2]. Here we extend the concept of CCA to quantum systems, and we present a quantum cellular automata (QCA) model. The described model combines the quantum parallelism with the parallel architecture corresponding to the CCA. Our motivation for investigating this QCA model is to explore the power of simple local rules, applied uniformly across the quantum system, to produce complex quantum dynamics. We are particularly interested in the complex dynamics of the multi-particle entanglement generated by simple local rules implemented in the described architecture. We show that our QCA model builds up long-range quantum correlations, resulting in generating and distributing entangled states throughout the system. Also, we show that depending on the local update rule, the complex dynamics of the QCA corresponds to regular periodic or pseudo-random regimes.
M. Andrecut, M.K. Ali / Physics Letters A 326 (2004) 328–332
329
2. The QCA architecture
3. The multi-particle entanglement measure
From the mathematical point of view [1], a CCA is defined as a tuple {Qn , S, R, Nq } where: Qn is a regular n-dimensional lattice; each cell of this lattice has a discrete state s ∈ S; R is the local update rule that describe the state of a cell for the next time step, depending on the states of the cells in the neighborhood of the cell Nq . The extension of the CCA model to quantum systems requires a slight modification for implementing the local rules. More specifically, let us consider the case of 1-dimensional QCA with two states S = {0, 1}. The lattice Q corresponds to a 1-dimensional array of qubits |qk ordered from 1 to n. Thus, each cell in the QCA corresponds to a qubit |qk that can be in a superposition of states |0 and |1:
In order to characterize the dynamics of entanglement we use the function on pure states of n qubits introduced in [5] and defined as:
|qk = ak |0 + bk |1,
(1)
where ak2 + bk2 = 1, k ∈ {1, . . . , n} [3]. The local rule is carried out via unitary gate operations on each neighborhood. Here, we consider a quantum system with nearest neighbor interactions, thus the neighborhood is defined over three cells: Nk = {k − 1, k, k + 1}. The unitary rule applied over the neighborhood Nk is given by
Entanglement dynamics in quantum cellular automata M. Andrecut ∗ , M.K. Ali Department of Physics, University of Lethbridge, Lethbridge, AB, T1K 3M4, Canada Received 5 January 2004; received in revised form 26 April 2004; accepted 29 April 2004 Available online 11 May 2004 Communicated by P.R. Holland
Abstract A quantum cellular automata architecture is presented. We investigate the complex dynamics of the multi-particle entanglement generated by simple local rules implemented in the described architecture. Also, we show that our model builds up long-range quantum correlations, resulting in generating and distributing entangled states throughout the system. 2004 Elsevier B.V. All rights reserved. PACS: 03.67.Mn; 05.45.Pq Keywords: Quantum cellular automata; Multi-particle entanglement; Regular and stochastic dynamics
1. Introduction Classical cellular automata (CCA) are discrete dynamical systems, which can simulate a wide range of complex physical phenomena including fluid dynamics, non-linear dynamics and phase transitions in many-body systems [1]. The meaning of discrete is, that space, time and properties of the CCA can have only a finite, countable number of states. The basic idea is not to describe a complex system with complex equations, but to let the complexity emerge from the interaction of simple units following simple rules. Formally, a CCA is termed complex if it evolves in a manner that in some sense is computationally irre-
* Corresponding author.
E-mail addresses: [email protected] (M. Andrecut), [email protected] (M.K. Ali). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.04.064
ducible, meaning it cannot be predicted with a compactly written equation [2]. Here we extend the concept of CCA to quantum systems, and we present a quantum cellular automata (QCA) model. The described model combines the quantum parallelism with the parallel architecture corresponding to the CCA. Our motivation for investigating this QCA model is to explore the power of simple local rules, applied uniformly across the quantum system, to produce complex quantum dynamics. We are particularly interested in the complex dynamics of the multi-particle entanglement generated by simple local rules implemented in the described architecture. We show that our QCA model builds up long-range quantum correlations, resulting in generating and distributing entangled states throughout the system. Also, we show that depending on the local update rule, the complex dynamics of the QCA corresponds to regular periodic or pseudo-random regimes.
M. Andrecut, M.K. Ali / Physics Letters A 326 (2004) 328–332
329
2. The QCA architecture
3. The multi-particle entanglement measure
From the mathematical point of view [1], a CCA is defined as a tuple {Qn , S, R, Nq } where: Qn is a regular n-dimensional lattice; each cell of this lattice has a discrete state s ∈ S; R is the local update rule that describe the state of a cell for the next time step, depending on the states of the cells in the neighborhood of the cell Nq . The extension of the CCA model to quantum systems requires a slight modification for implementing the local rules. More specifically, let us consider the case of 1-dimensional QCA with two states S = {0, 1}. The lattice Q corresponds to a 1-dimensional array of qubits |qk ordered from 1 to n. Thus, each cell in the QCA corresponds to a qubit |qk that can be in a superposition of states |0 and |1:
In order to characterize the dynamics of entanglement we use the function on pure states of n qubits introduced in [5] and defined as:
|qk = ak |0 + bk |1,
(1)
where ak2 + bk2 = 1, k ∈ {1, . . . , n} [3]. The local rule is carried out via unitary gate operations on each neighborhood. Here, we consider a quantum system with nearest neighbor interactions, thus the neighborhood is defined over three cells: Nk = {k − 1, k, k + 1}. The unitary rule applied over the neighborhood Nk is given by