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arXiv:quant-ph/0509013 · Physical Review A

Limits on entanglement from rotationally invariant scattering of spin systems

Nathan L. Harshman

Published Jun 19, 2006• 8 citations
arXivHTMLPDF

Abstract

This paper investigates the dynamical generation of entanglement in scattering systems, in particular two spin systems that interact via rotationally invariant scattering. The spin degrees of freedom of the in states are assumed to be in unentangled, pure states, as defined by the entropy of entanglement. Because of the restriction of rotationally symmetric interactions, perfectly entangling $S$ matrices, i.e., those that lead to a maximally entangled out state, only exist for a certain class of separable in states. Using Clebsch-Gordan coefficients for the rotation group, the scattering phases that determine the $S$ matrix are determined for the case of spin systems with $\ensuremath{\sigma}=1∕2$, 1, and $3∕2$.

DOI
10.1103/physreva.73.062326

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This paper investigates the dynamical generation of entanglement in scattering systems, in particular two spin systems that interact via rotationally invariant scattering. The spin degrees of freedom of the in states are assumed to be in unentangled, pure states, as defined by the entropy of entanglement. Because of the restriction of rotationally symmetric interactions, perfectly entangling SSS matrices, i.e., those that lead to a maximally entangled out state, only exist for a certain class of separable in states. Using Clebsch-Gordan coefficients for the rotation group, the scattering phases that determine the SSS matrix are determined for the case of spin systems with \ensuremathσ=1∕2\ensuremath{\sigma}=1∕2\ensuremathσ=1∕2, 1, and 3∕23∕23∕2.
This paper explores the connections between particle scattering and quantum information theory in the context of the non-relativistic, elastic scattering of two spin-1/2 particles. An untangled, pure, two-particle in-state is evolved by an S-matrix that respects certain symmetries and the entanglement of the pure out-state is measured. The analysis is phrased in terms of unitary, irreducible representations (UIRs) of the symmetry group in question, either the rotation group for the spin degrees of freedom or the Galilean group for non-relativistic particles. Entanglement may occurs when multiple UIRs appear in the direct sum decomposition of the direct product in-state, but it also depends of the scattering phase shifts.
We study entanglement suppression in sss-wave ΩΩΩΩΩΩ scattering, where each baryon has spin 3/23/23/2. By treating the SSS-matrix as a quantum operator acting on the spin states, we quantify its ability to generate entanglement and identify the conditions on the phase shifts of the spin channels that minimize entanglement generation in the system. In ΩΩΩΩΩΩ scattering, only antisymmetric spin channels are allowed due to Fermi-Dirac statistics. Applying the entanglement-suppression framework to ΩΩΩΩΩΩ scattering, we find two solutions for the phase shifts: one leading to a spin SU(4) symmetry and the other to a nonrelativistic conformal symmetry. We show that the solution associated with the nonrelativistic conformal symmetry originates from the specific structure of the Clebsch-Gordan coefficients in the 3/2⊗3/23/2 \otimes 3/23/2⊗3/2 system.
Elastic scattering can generate entanglement and non-local magic between the spins of the two outgoing particles. The amount of entanglement and magic produced is encoded in the S-matrix; a unitary quantum gate that acts on the joint Hilbert space of the two spin-J particles. Previous work derived the entanglement power in terms of the scattering phase shifts for rotationally invariant S-matrices with J=1/2,1,3/2. In this paper, a closed form solution is derived for all J. We further prove that a broad class of non-local quantum resources depend to leading order only on the squared projective Hilbert-Schmidt distance from the nearest non-entangling gate. This holds for the entanglement power and non-local magic power, and the choice of resource simply changes the overall normalization. This universality establishes a common geometric origin for the leading resource-generating capacity of scattering, independent of the details of the interaction.
The speed limit on entanglement is defined as the maximal rate at which entanglement can be generated or degraded in a physical process. We derive the speed limits on entanglement, using the relative entropy of entanglement and trace-distance entanglement, for unitary and arbitrary quantum dynamics, where we assume that the dynamics of the closest separable state can be approximately described by the closest separable dynamics of the actual dynamics of the system. For the unitary dynamics of isolated bipartite systems which are described by pure states, the rate of entanglement production is bounded by the product of fluctuations of the system's driving Hamiltonian and the surprisal operator, with an additional term reflecting the time-dependent nature of the closest separable state. Removing restrictions on the purity of the input and on the unitarity of the evolution, the two terms in the bound get suitably altered. Furthermore, we find a lower bound on the time required to generate or degrade a certain amount of entanglement by arbitrary quantum dynamics. We demonstrate the tightness of our speed limits on entanglement by considering quantum processes of practical interest.
We consider the bipartite spin entanglement between two identical fermions generated in spin-independent scattering. We show how the spatial degrees of freedom act as ancillas for the creation of entanglement to a degree that depends on the scattering angle $\ensuremath{\theta}$. The number of Slater determinants generated in the process is greater than 1, corresponding to genuine quantum correlations between the identical fermions. The maximal entanglement attainable of 1 ebit is reached at $\ensuremath{\theta}=\ensuremath{\pi}∕2$. We also analyze a simple $\ensuremath{\theta}$-dependent Bell's inequality, which is violated for $\ensuremath{\pi}∕4<\ensuremath{\theta}\ensuremath{\leqslant}\ensuremath{\pi}∕2$. This phenomenon is unrelated to the symmetrization postulate but does not appear for unequal particles.
We study the entanglement property in matrix product spin-ring systems systemically by von Neumann entropy. We find that: (i) the Hilbert space dimension of one spin determines the upper limit of the maximal value of the entanglement entropy of one spin, while for multiparticle entanglement entropy, the upper limit of the maximal value depends on the dimension of the representation matrices. Based on the theory, we can realize the maximum of the entanglement entropy of any spin block by choosing the appropriate control parameter values. (ii) When the entanglement entropy of one spin takes its maximal value, the entanglement entropy of an asymptotically large spin block, i.e. the renormalization group fixed point, is not likely to take its maximal value, and so only the entanglement entropy S n of a spin block that varies with size n can fully characterize the spin-ring entanglement feature. Finally, we give the entanglement dynamics, i.e. the Hamiltonian of the matrix product system.