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arXiv:1207.3737 · New Journal of Physics

Simulating symmetric time evolution with local operations

Borzu Toloui, Gilad Gour

Published Dec 14, 2012• 13 citations
arXivHTMLPDF

Abstract

In closed systems, dynamical symmetries lead to conservation laws. However, conservation laws are not applicable to open systems that undergo irreversible transformations. More general selection rules are needed to determine whether, given two states, the transition from one state to the other is possible. The usual approach to the problem of finding such rules relies heavily on group theory and involves a detailed study of the structure of the respective symmetry group. We approach the problem in a completely new way by using entanglement to investigate the asymmetry properties of quantum states. To this end, we embed the space state of the system in a tensor product Hilbert space, whereby symmetric transformations between two states are replaced with local operations on their bipartite images.The embedding enables us to use the well-studied theory of entanglement to investigate the consequences of dynamic symmetries. Moreover, under reversible transformations, the entanglement of the bipartite image states becomes a conserved quantity. These entanglement-based conserved quantities are new and different from the conserved quantities based on expectation values of the Hamiltonian symmetry generators. Our method is not group-specific and applies to general symmetries associated with any semi-simple Lie group.

DOI
10.1088/1367-2630/14/12/123026

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In closed systems, dynamical symmetries lead to conservation laws. However, conservation laws are not applicable to open systems that undergo irreversible transformations. More general selection rules are needed to determine whether, given two states, the transition from one state to the other is possible. The usual approach to the problem of finding such rules relies heavily on group theory and involves a detailed study of the structure of the respective symmetry group. We approach the problem in a completely new way by using entanglement to investigate the asymmetry properties of quantum states. To this end, we embed the space state of the system in a tensor product Hilbert space, whereby symmetric transformations between two states are replaced with local operations on their bipartite images.The embedding enables us to use the well-studied theory of entanglement to investigate the consequences of dynamic symmetries. Moreover, under reversible transformations, the entanglement of the bipartite image states becomes a conserved quantity. These entanglement-based conserved quantities are new and different from the conserved quantities based on expectation values of the Hamiltonian symmetry generators. Our method is not group-specific and applies to general symmetries associated with any semi-simple Lie group.
For open systems, symmetric dynamics do not always lead to conservation laws. We show that, for a dynamic symmetry associated with a compact Lie group, one can derive new selection rules from entanglement theory. These selection rules apply to both closed and open systems as well as reversible and irreversible time evolutions. Our approach is based on an embedding of the system’s Hilbert space into a tensor product of two Hilbert spaces allowing for the symmetric dynamics to be simulated with local operations. The entanglement of the embedded states determines which transformations are forbidden because of the symmetry. In fact, every bipartite entanglement monotone can be used to quantify the asymmetry of the initial states. Moreover, where the dynamics is reversible, each of these monotones becomes a new conserved quantity.
Quantifying how much a quantum state breaks a symmetry is essential for characterizing phases, nonequilibrium dynamics, and open-system behavior. For mixed states, however, conventional diagnostics of weak symmetry breaking can miss a stronger form of symmetry breaking associated with the possibility of exchanging conserved charges with an environment. We develop a resource theory of strong symmetry breaking by identifying the appropriate free states and free operations, and, for channels on a fixed system, by showing that the latter are precisely the operations realizable without exchanging the conserved charge with the environment. We systematically construct measures of strong symmetry breaking, including strong entanglement asymmetry and covariance matrices of symmetry generators, for a broad class of symmetry groups. We further completely characterize i.i.d. convertibility between arbitrary states under any compact Lie group strong symmetry. This identifies the combinations of resource-theoretically valid measures that quantify strong symmetry breaking for arbitrary states, including all mixed states. In particular, for U(1)U(1)U(1) symmetry, if the states have positive variances of the conserved quantity and equal strong-symmetry periods, the conversion rate is completely determined by the variance ratio whenever the input is pure or the output is weak symmetric. Thus, for these state conversions, the variance of the conserved quantity plays an operational role analogous to that of entanglement entropy in entanglement theory or quantum Fisher information in the resource theory of weak asymmetry. We further show how weak symmetry breaking is irreversibly converted into strong symmetry breaking in open-system dynamics. We illustrate the framework with examples from quantum field theory, strong-to-weak spontaneous symmetry breaking, and a strong-symmetry analogue of Mpemba-type dynamics.
Dynamical symmetries in closed systems lead to conservation laws and to selection rules that are based on those conservation laws. However, conservation laws are not applica- ble to open systems that undergo irreversible transformations. More general selection rules are needed to determine whether, given two states, the transition from one state to the other is possible. Characterizing the asymmetry properties of quantum states using quantum information theoretical tools and methods is particularly fruitful for this purpose. This thesis is concerned with the problem of finding new general selection rules that hold for both open and closed systems. The usual approach to this problem relies heavily on group theory and involves a detailed study of the structure of the symmetry group. In this thesis, we approach the problem in a completely new way. Our approach is to use entanglement to investigate the asymmetry properties of quantum states. In order to do that, we embed the system’s Hilbert space in a larger tensor product Hilbert space, whereby all symmetric states are mapped to separable states, asymmetric states are mapped to entangled states, and the symmetric transformations between two states are replaced by local operations on their bipartite images. Our method is not restricted to only a specific group but applies, in general, to all symmetries that are associated with semi-simple compact Lie groups and their associated Lie algebras. The mapping of the original states to bipartite states that act on the larger Hilbert space enables us to use the well studied theory of entanglement to investigate the consequences of dynamic symmetries. For example, the monotonicity condition on measures of entanglement provide us with new selection rules. Under reversible transfor- mations, the entanglement of the bipartite image states becomes a conserved quantity. These entanglement-based conserved quantities are new and different from the conserved quantities based on expectation values of the Hamiltonian symmetry generators.
The analysis of symmetry in quantum systems is of utmost theoretical importance, useful in a variety of applications and experimental settings, and difficult to accomplish in general. Symmetries imply conservation laws, which partition Hilbert space into invariant subspaces of the time-evolution operator, each of which is demarcated according to its conserved quantity. We show that, starting from a chosen basis, any invariant, symmetry-protected subspaces which are diagonal in that basis are discoverable using transitive closure on graphs representing state-to-state transitions under kkk-local unitary operations. Importantly, the discovery of these subspaces relies neither upon the explicit identification of a symmetry operator or its eigenvalues nor upon the construction of matrices of the full Hilbert space dimension. We introduce two classical algorithms, which efficiently compute and elucidate features of these subspaces. The first algorithm explores the entire symmetry-protected subspace of an initial state in time complexity linear to the size of the subspace by closing local basis state-to-basis state transitions. The second algorithm determines, with bounded error, if a given measurement outcome of a dynamically generated state is within the symmetry-protected subspace of the state in which the dynamical system is initialized. We demonstrate the applicability of these algorithms by performing postselection on data generated from emulated noisy quantum simulations of three different dynamical systems: the Heisenberg-XXX model and the T6{T}_{6}T6​ and F4{F}_{4}F4​ quantum cellular automata. Due to their efficient computability and indifference to identifying the underlying symmetry, these algorithms lend themselves to the postselection of quantum computer data, optimized classical simulation of quantum systems, and discovery of previously hidden symmetries in quantum mechanical systems.
Symmetry principles have played a crucial role in the development of modern physics and underpin our most fundamental theories of nature. The present thesis is concerned with the analysis of symmetries in the context of quantum information theory. Whenever a quantum mechanical system interacts with its environment it is subjected to decoherence, a process that is typically irreversible and most generally treated abstractly using the concept of quantum operations. If there is an underlying symmetry principle, additional structures emerge. The central question we address is, What are the consequences of global or local gauge symmetry on the structure of many-body quantum processes?This leads us to a diagrammatic framework of decomposing quantum operations into terms that respond to the symmetry principle in particular ways and respect the causal structures involved. We present two core applications. First, we address the interplay between irreversibility and repeatable use of coherent resources under symmetry constraints. Second, we give an information-theoretic perspective on gauging globally symmetric dynamics to a local symmetry applicable even in the presence of irreversibility and thus it goes beyond the usual Lagrangian formulation. Finally, we analyse the departure from conservation laws under symmetric dynamics subject to decoherence.
The asymmetry properties of a state relative to some symmetry group specify how and to what extent the given symmetry is broken by the state. Characterizing these is found to be surprisingly useful for addressing a very common problem: to determine what follows from a system's dynamics (possibly open) having that symmetry. We demonstrate and exploit the fact that the asymmetry properties of a state can be understood in terms of information-theoretic concepts. We show that for a pure state $\ensuremath{\psi}$ and a symmetry group GGG, they are completely specified by the characteristic function of the state, defined as ${\ensuremath{\chi}}_{\ensuremath{\psi}}(g)\ensuremath{\equiv}\ensuremath{\langle}\ensuremath{\psi}|U(g)|\ensuremath{\psi}\ensuremath{\rangle}$, where $g\ensuremath{\in}G$ and UUU is the unitary representation of interest. Based on this observation, we study several important problems about the interconversion of pure states under symmetric dynamics, such as determining the conditions for reversible transformations, deterministic irreversible transformations, and asymptotic transformations.