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arXiv:1707.09396 · Physical Review A

Sequentially generated entanglement, macroscopicity, and squeezing in a spin chain

Tahereh Abad, Klaus Mølmer, Vahid Karimipour

Published Oct 26, 2017• 3 citations
arXivHTMLPDF

Abstract

We study quantum states generated by a sequence of nearest neighbor bipartite entangling operations along a one-dimensional chain of spin qubits. After a single sweep of such a set of operations, the system is effectively described by a matrix product state (MPS) with the same virtual dimension as the spin qubits. We employ the explicit form of the MPS to calculate expectation values and two-site correlation functions of local observables, and we use the results to study fluctuations of collective observables. Through the so-called macroscopicity and the squeezing properties of the collective spin variables they witness the quantum correlations and multiparticle entanglement within the chain. Macroscopicity only occurs over the entire chain if the nearest neighbor interaction is maximally entangling, while a finite, sequential interaction between nearest neighbor particles leads to squeezing of the collective spin.

DOI
10.1103/physreva.96.042337

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We study quantum states generated by a sequence of nearest neighbor bipartite entangling operations along a one-dimensional chain of spin qubits. After a single sweep of such a set of operations, the system is effectively described by a matrix product state (MPS) with the same virtual dimension as the spin qubits. We employ the explicit form of the MPS to calculate expectation values and two-site correlation functions of local observables, and we use the results to study fluctuations of collective observables. Through the so-called macroscopicity and the squeezing properties of the collective spin variables they witness the quantum correlations and multiparticle entanglement within the chain. Macroscopicity only occurs over the entire chain if the nearest neighbor interaction is maximally entangling, while a finite, sequential interaction between nearest neighbor particles leads to squeezing of the collective spin.
We consider two distant spin-12\frac{1}{2}21​ particles (or qubits) and a number of interacting objects, all with the same value $S\ensuremath{\gg}1$ of their respective spin, distributed on a one-dimensional lattice (or large-SSS spin chain). The quantum states of the chain are constructed by linearly combining tensor products of single-spin coherent states, whose evolution is determined accordingly, i.e., via classical-like equations of motions. We show that the quantum superposition of the above product states resulting from a local interaction between the first qubit and one spin of the chain evolves so that the second qubit, after having itself interacted with another spin of the chain, can be entangled with the first qubit. Obtaining such an outcome does not imply imposing constraints on the length of the chain or the distance between the qubits, which demonstrates the possibility of generating quantum correlations at a distance by means of a macroscopic system, as far as local interactions with just a few of its components are feasible.
Abstract For our proposed composite parity-conserved matrix product state (MPS), if only a spin block length is larger than 1, any two such spin blocks have correlation including classical correlation and quantum correlation. Both the total correlation and the classical correlation become larger than that in any subcomponent; while the quantum correlations of the two nearest-neighbor spin blocks and the two next-nearest-neighbor spin blocks become smaller and for other conditions the quantum correlation becomes larger, i.e., the increase or the production of the long-range quantum correlation is at the cost of reducing the short-range quantum correlation, which deserves to be investigated in the future; and the ration of the quantum correlation to the total correlation monotonically decreases to a steady value as the spacing spin length increasing.
Abstract A systematic and compact treatment of arbitrary su (2) invariant spin- s quantum chains with nearest-neighbour interactions is presented. The ground-state is derived in terms of matrix product states (MPS). The fundamental MPS calculations consist of taking products of basic tensors of rank 3 and contractions thereof. The algebraic su (2) calculations are carried out completely by making use of Wigner calculus. As an example of application, the spin-1 bilinear-biquadratic quantum chain is investigated. Various physical quantities are calculated with high numerical accuracy of up to 8 digits. We obtain explicit results for the ground-state energy, entanglement entropy, singlet operator correlations and the string order parameter. We find an interesting crossover phenomenon in the correlation lengths.
We consider a one-parameter family of matrix-product states of spin-1 particles on a periodic chain and study in detail the entanglement properties of such a state. In particular, we calculate exactly the entanglement of one site with the rest of the chain, and the entanglement of two distant sites with each other, and show that the derivative of both these properties diverge when the parameter ggg of the states passes through a critical point. Such a point can be called a point of quantum phase transition, since at this point the character of the matrix-product state, which is the ground state of a Hamiltonian, changes discontinuously. We also study the finite size effects and show how the entanglement depends on the size of the chain. This later part is relevant to the field of quantum computation where the problem of initial state preparation in finite arrays of qubits or qutrits is important. It is also shown that the entanglement of two sites have scaling behavior near the critical point.
Matrix product states can be defined as the family of quantum states that can be sequentially generated in a one-dimensional system [Sch\"on et al., Phys. Rev. Lett. 95, 110503 (2005)]. We introduce a family of states that extends this definition to two dimensions. Like in matrix product states, expectation values of few body observables can be efficiently evaluated and, for the case of translationally invariant systems, the correlation functions decay exponentially with the distance. We show that such states are a subclass of projected entangled pair states and investigate their suitability for approximating the ground states of local Hamiltonians.
For the matrix product system of a one-dimensional spin-½ chain, we present a new model of quantum phase transitions and find that in the thermodynamic limit, both sides of the critical point are respectively described by phases |Ψ a 〉 = |1...1〉 representing all particles spin up and |Ψ b 〉 = |0...0〉 representing all particles spin down, while the phase transition point is an isolated intermediate-coupling point where the two phases coexist equally, which is described by the so-called N -qubit maximally entangled GHZ state . At the critical point, the physical quantities including the entanglement are not discontinuous and the matrix product system has long-range correlation and N -qubit maximal entanglement. We believe that our work is helpful for having a comprehensive understanding of quantum phase transitions in matrix product states of one-dimensional spin chains and of potential directive significance to the preparation and control of one-dimensional spin lattice models with stable coherence and N -qubit maximal entanglement.