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arXiv:0808.2226 · Physical Review A

Qubit phase space: SU(n) coherent-state P representations

David Barry, P. D. Drummond

Published Nov 11, 2008• 22 citations
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Abstract

We introduce a phase-space representation for qubits and spin models. The technique uses an $\mathrm{SU}(n)$ coherent-state basis and can equally be used for either static or dynamical simulations. We review previously known definitions and operator identities, and show how these can be used to define an off-diagonal, positive phase-space representation analogous to the positive-$P$ function. As an illustration of the phase-space method, we use the example of the Ising model, which has exact solutions for the finite-temperature canonical ensemble in two dimensions. We show how a canonical ensemble for an Ising model of arbitrary structure can be efficiently simulated using SU(2) or atomic coherent states. The technique utilizes a transformation from a canonical (imaginary-time) weighted simulation to an equivalent unweighted real-time simulation. The results are compared to the exactly soluble two-dimensional case. We note that Ising models in one, two, or three dimensions are potentially achievable experimentally as a lattice gas of ultracold atoms in optical lattices. The technique is not restricted to canonical ensembles or to Ising-like couplings. It is also able to be used for real-time evolution and for systems whose time evolution follows a master equation describing decoherence and coupling to external reservoirs. The case of $\mathrm{SU}(n)$ phase space is used to describe $n$-level systems. In general, the requirement that time evolution be stochastic corresponds to a restriction to Hamiltonians and master equations that are quadratic in the group generators or generalized spin operators.

DOI
10.1103/physreva.78.052108

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We introduce a phase-space representation for qubits and spin models. The technique uses an SU(n)\mathrm{SU}(n)SU(n) coherent-state basis and can equally be used for either static or dynamical simulations. We review previously known definitions and operator identities, and show how these can be used to define an off-diagonal, positive phase-space representation analogous to the positive-PPP function. As an illustration of the phase-space method, we use the example of the Ising model, which has exact solutions for the finite-temperature canonical ensemble in two dimensions. We show how a canonical ensemble for an Ising model of arbitrary structure can be efficiently simulated using SU(2) or atomic coherent states. The technique utilizes a transformation from a canonical (imaginary-time) weighted simulation to an equivalent unweighted real-time simulation. The results are compared to the exactly soluble two-dimensional case. We note that Ising models in one, two, or three dimensions are potentially achievable experimentally as a lattice gas of ultracold atoms in optical lattices. The technique is not restricted to canonical ensembles or to Ising-like couplings. It is also able to be used for real-time evolution and for systems whose time evolution follows a master equation describing decoherence and coupling to external reservoirs. The case of SU(n)\mathrm{SU}(n)SU(n) phase space is used to describe nnn-level systems. In general, the requirement that time evolution be stochastic corresponds to a restriction to Hamiltonians and master equations that are quadratic in the group generators or generalized spin operators.
Grand-canonical inverse-temperature calculations of a single mode Bose-Hubbard model are presented, using the Gaussian phase space representation. Simulation of 100 particles is achieved in the ground state, having started with a low-particle-number thermal state. A preliminary foray into a three-mode lattice is made, but the sampling error appears to be too large for the simple approach taken here to be successful in larger systems. The quantum (real-time) dynamics of a one-dimensional Bose gas with two-particle losses are investigated. The Positive-P equations for this system are unstable, and this causes Positive-P simulations to `die' after a certain amount of time. Gauges are used to (sometimes partially) stabilise the equations. The effects on simulation times of various gauges, branching methods, and non-square diffusion matrix factorisations on simulation times are investigated. Despite the absence of repulsive inter-particle interactions, it is observed that g(2)g^{(2)}g(2) rises above 1 at a finite particle separation. A phase space method for spin systems is introduced, based on SU(2) coherent states. This is essentially a spin analogue of the Positive-P method. The system of stochastic differential equations arising out of this method require weighted averages to be taken, and the weights can vary exponentially, leading to inefficient sampling. For the case of the Ising model, a transform is made to a set of equations which relaxes (in a dummy time variable) to the partition function at a given temperature, and allows unweighted ensemble averages to be taken. This allows accurate simulations to be achieved at a range of temperatures, with nearest-neighbour correlation functions agreeing with theory. This represents a proof of principle for the use of stochastic phase space methods in spin systems, and furthermore the method should be suited to open spin systems, at least for a small number of qubits.
We reformulate the full quantum dynamics of spin systems using a phase-space representation based on SU(2) coherent states which generates an exact mapping of the dynamics of any spin system onto a set of stochastic differential equations. This representation is superior in practice to an earlier phase-space approach based on Schwinger bosons, with the numerical effort scaling only linearly with system size. By also implementing extrapolation techniques from quasiclassical equations to the full quantum limit, we are able to extend useful simulation times severalfold. This approach is applicable in any dimension including cases where frustration is present in the spin system. The method is demonstrated by simulating quenches in the transverse-field Ising model in one and two dimensions.
We introduce a positive phase-space representation for fermions, using the most general possible multimode Gaussian operator basis. The representation generalizes previous bosonic quantum phase-space methods to Fermi systems. We derive equivalences between quantum and stochastic moments, as well as operator correspondences that map quantum operator evolution onto stochastic processes in phase space. The representation thus enables first-principles quantum dynamical or equilibrium calculations in many-body Fermi systems. Potential applications are to strongly interacting and correlated Fermi gases, including coherent behavior in open systems and nanostructures described by master equations. Examples of an ideal gas and the Hubbard model are given, as well as a generic open system, in order to illustrate these ideas.
This thesis presents our work on two separate subfields and has for this reason been divided into two parts: (I) the development and implementation of a numerical technique for non-equilibrium dynamics and a (II) detailed investigation of the dirty-boson problem. In part (I), we optimize and develop the quantum phase space method known as the positive-P representation (PPR) specifically for the simulations of the real time quench dynamics of quantum spin systems. The main benefit of this approach is that the dynamics of the density operator is mapped onto ∼N\sim N∼N stochastic variables (where NNN is the size of the system) that obey Langevin-type stochastic differential equations, thereby greatly reducing the complexity of the problem. The first publication presents our initial use of the PPR on spin systems by using a Schwinger mapping for spin operators on to Bosonic operators since the underlying basis comprises of Bosonic coherent states. We simulate the quench dynamics of the generalized transverse-field spin-1/2 XXZ model in 1d showing that simulations of up to 100 spins are possible, albeit for relatively short simulation times. In our second publication, we reformulate the PPR using SU(2) spin coherent states and further optimize simulation lifetimes by implementing an extrapolation scheme, to achieve several-fold improvements over the results of our first publication. We focus solely on the transverse Ising model and simulate its quench dynamics in 1d and 2d using up to 10410^4104 spins, while significantly extending simulation lifetimes. The second part of this thesis is a numerical study of the universality class of the Superfluid-Bose Glass transition of the dirty-boson system. Recently the longstanding exact result z=dz=dz=d, where zzz is the dynamic critical exponent and ddd is the dimensionality of the system, has been challenged by a series of numerical studies, suggesting the alternative scenario that zzz should be instead unconstrained. To address this controversy, we use large scale quantum Monte-Carlo simulations on two independent quantum models, and average over 5×104−1055\times10^4-10^55×104−105 disorder realizations to numerically determine the universality class of the dirty-boson transition, paying particular attention to the dynamic critical exponent, zzz and without any biasing assumptions.
We introduce a classical limit of the dynamics of quantum spin systems based on coherent states of SU(N)\mathrm{SU}(N)SU(N), where NNN is the dimension of the local Hilbert space. This approach, which generalizes the well-known Landau-Lifshitz dynamics from SU(2) to SU(N)\mathrm{SU}(N)SU(N), provides a better approximation to the exact quantum dynamics for a large class of realistic spin Hamiltonians, including $S\ensuremath{\ge}1$ systems with large single-ion anisotropy and weakly coupled multispin units, such as dimers or trimers. We illustrate this idea by comparing the spin structure factors of a single-ion S=1S=1S=1 model that are obtained with the SU(2) and SU(3) classical spin dynamics against the exact solution.
This thesis presents a comprehensive study of phase-space measurements, decoherence, and the emergence of classicality in quantum systems. Motivated by limitations of position-based localisation in standard decoherence models, we develop a phase-space decoherence formalism using coherent-state positive operator-valued measures (POVMs). These generalised measurements, constructed from overcomplete sets of coherent states, provide a natural and physically motivated framework for modelling coarse-grained quantum observations. We develop an iterative measurement scheme for phase-space decoherence in systems with Heisenberg-Weyl, SU(2)\mathrm{SU}(2)SU(2), SU(1,1)\mathrm{SU}(1,1)SU(1,1), and SU(N)\mathrm{SU}(N)SU(N) symmetries. In each case, repeated weak measurements gradually suppress quantum interference, driving the system toward a maximally mixed state. A universal structure is shown to emerge: a single iteration maps the Wigner function to the Husimi QQQ-function, removing negativity, while full decoherence maps the singular Glauber-Sudarshan PPP-function to the QQQ-function, eliminating all singularities. This establishes a group-independent mechanism by which phase-space measurements induce classicality. We compare this model to Lindblad-type dynamics, which describe open quantum systems via completely positive, trace-preserving semigroups. For Heisenberg-Weyl and tomographic SU(N)\mathrm{SU}(N)SU(N) systems, the measurement channel coincides exactly with isotropic Lindblad evolution. In SU(2)\mathrm{SU}(2)SU(2) and SU(1,1)\mathrm{SU}(1,1)SU(1,1) cases, the correspondence is only partial, with exact agreement in asymptotic regimes. A universal decoherence timescale, t∗t_*t∗​, emerges beyond which quasi-probability distributions become strictly positive. We then construct a stochastic unravelling based on Brownian motion on phase space, offering a geometric alternative to standard stochastic Schr\"odinger equations. This reveals that decoherence occurs without GKSL-type dynamics in noncompact systems. Such unravellings preserve the intrinsic phase-space symmetry and provide a natural setting for studying semiclassical limits and quantum-to-classical transitions. Finally, we extend the framework to relativistic quantum systems by defining a covariant coherent-state POVM on the future tube and constructing a Bargmann transform to the forward light cone. Preliminary results on relativistic phase-space decoherence are presented, along with possible directions for future research.