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Nonlocal Hamiltonians and Information Conservation Law

Jianming Cai, Xiang-Fa Zhou, Zheng-Wei Zhou, Guang‐Can Guo

Published Jul 4, 2005• 0 citations
OpenAlex

Abstract

We use fidelity to characterize the information about a quantum state contained in another quantum state. In a multi-qubit system, each qubit has individual information about a prescribed quantum state. We investigate how the total information changes under the interactions of nonlocal Hamiltonians. We show that for some special physical interactions the total information is conserved. This type of nonlocal Hamiltonians are named as Conservative Hamiltonians, similar to the Conservative Forces. We then apply this Information Conservation Law in the quantum spin system, and propose methods for improving the efficiency of quantum communications through an open end spin chain.

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We use fidelity to characterize the information about a quantum state contained in another quantum state. In a multi-qubit system, each qubit has individual information about a prescribed quantum state. We investigate how the total information changes under the interactions of nonlocal Hamiltonians. We show that for some special physical interactions the total information is conserved. This type of nonlocal Hamiltonians are named as Conservative Hamiltonians, similar to the Conservative Forces. We then apply this Information Conservation Law in the quantum spin system, and propose methods for improving the efficiency of quantum communications through an open end spin chain.
Quantum information is defined by applying the concepts of ordinary (Shannon) information theory to a quantum sample space consisting of a single framework or consistent family. A classical analogy for a spin-half particle and other arguments show that the infinite amount of information needed to specify a precise vector in its Hilbert space is not a measure of the information carried by a quantum entity with a d-dimensional Hilbert space; the latter is, instead, bounded by log2d{\mathrm{log}}_{2}dlog2​d bits (one bit per qubit). The two bits of information transmitted in dense coding are located not in one but in the correlation between two qubits, consistent with this bound. A quantum channel can be thought of as a structure or collection of frameworks, and the physical location of the information in the individual frameworks can be used to identify the location of the channel. Analysis of a quantum circuit used as a model of teleportation shows that the location of the channel depends upon which structure is employed; for ordinary teleportation it is not (contrary to Deutsch and Hayden) present in the two bits resulting from the Bell-basis measurement, but in correlations of these with a distant qubit. In neither teleportation nor dense coding does information travel backwards in time, nor is it transmitted by nonlocal (superluminal) influences. It is (tentatively) proposed that all aspects of quantum information can in principle be understood in terms of the (basically classical) behavior of information in a particular framework, along with the framework dependence of this information.
In this paper, we describe a possible experimental realization of Bose's idea to use spin chains for short-distance quantum communication (Bose 2003 Phys. Rev. Lett. 91 207901). Josephson arrays have been proposed and analysed as transmission channels for systems of superconducting charge qubits. Here, we consider a chain of persistent-current qubits, that is appropriate for state transfer with high fidelity in systems containing flux qubits. We calculate the fidelity of state transfer for this system. In general, the Hamiltonian of this system is not of XXZ-type, and we analyse the magnitude and the effect of the terms that do not conserve the z-component of the total spin.
Quantum integrability is a cornerstone of the exact theory of interacting quantum spin chains. In its standard formulation, however, one starts from R-matrices satisfying the Yang--Baxter equation, rather than from the Hamiltonian itself. It has therefore remained unclear how Yang--Baxter solvability can be characterized directly at the Hamiltonian level, and how it is related to the existence of local conservation laws. Here we prove that, in a broad standard setting, the Reshetikhin condition is not only necessary but also sufficient for Yang--Baxter integrability, thereby reducing the hidden algebraic structure of integrability to a Hamiltonian-level conservation law. Since the Reshetikhin condition is equivalent to conservation of the total energy current, this Hamiltonian-level criterion is also experimentally accessible. This result establishes a quantum counterpart of the Liouville--Arnold theorem for isotropic spin chains, stating that Yang--Baxter solvability is equivalent to an infinite hierarchy of local conserved quantities. Our result also simplifies substantially the search for integrable spin chains by replacing the search for R-matrices with a direct criterion on local Hamiltonians.
Quantum information is nonlocal in the sense that local measurements on a composite quantum system, prepared in one of many mutually orthogonal states, may not reveal in which state the system was prepared. It is shown that in the many copy limit this kind of nonlocality is fundamentally different for pure and mixed quantum states. In particular, orthogonal mixed states may not be distinguishable by local operations and classical communication, no matter how many copies are supplied, whereas any set of N orthogonal pure states can be perfectly discriminated with m copies, where m<N. Thus mixed quantum states can exhibit a new kind of nonlocality absent in pure states. We also argue that a set of orthogonal quantum states may be said to be maximally indistinguishable iff the set is not conclusively locally distinguishable with multiple copies.
Similarly to the system Hamiltonian, a subsystem's reduced density matrix is composed of blocks characterized by symmetry quantum numbers (charge sectors). We present a geometric approach for extracting the contribution of individual charge sectors to the subsystem's entanglement measures within the replica trick method, via threading appropriate conjugate Aharonov-Bohm fluxes through a multisheet Riemann surface. Specializing to the case of 1+1D conformal field theory, we obtain general exact results for the entanglement entropies and spectrum, and apply them to a variety of systems, ranging from free and interacting fermions to spin and parafermion chains, and verify them numerically. We find that the total entanglement entropy, which scales as lnL, is composed of sqrt[lnL] contributions of individual subsystem charge sectors for interacting fermion chains, or even O(L^{0}) contributions when total spin conservation is also accounted for. We also explain how measurements of the contribution to the entanglement from separate charge sectors can be performed experimentally with existing techniques.
Conserved quantities are crucial in quantum physics. Here we discuss a general scenario of Hamiltonians. All the Hamiltonians within this scenario share a common conserved quantity form. For unitary parametrization processes, the characteristic operator of this scenario is analytically provided, as well as the corresponding quantum Fisher information (QFI). As the application of this scenario, we focus on two classes of Hamiltonians: su(2) category and canonical category. Several specific physical systems in these two categories are discussed in detail. Besides, we also calculate an alternative form of QFI in this scenario.