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arXiv:quant-ph/0305194 · International Journal of Quantum Information

Quantum States of Hierarchic Systems

M. V. Altaĭsky

Published Jun 1, 2003• 5 citations
arXivHTMLPDF

Abstract

The density matrix formalism which is widely used in the theory of measurements, quantum computing, quantum description of chemical and biological systems always implies the averaging over all states of the environment. In practice this is impossible because the environment of the system is the complement of this system to the whole Universe and contains infinitely many degrees of freedom. A novel method of construction density matrix which implies the averaging only over the direct environment is proposed. The Hilbert space of state vectors for the hierarchic quantum systems is constructed.

DOI
10.1142/s0219749903000188

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The density matrix formalism which is widely used in the theory of measurements, quantum computing, quantum description of chemical and biological systems always implies the averaging over all states of the environment. In practice this is impossible because the environment of the system is the complement of this system to the whole Universe and contains infinitely many degrees of freedom. A novel method of construction density matrix which implies the averaging only over the direct environment is proposed. The Hilbert space of state vectors for the hierarchic quantum systems is constructed.
Presents the basic postulates of quantum mechanics in terms of the density matrix instead of the usual state vector formalism in the case of an isolated system. Extends it to the case of open systems with the help of the reduced density matrix formalism, and to the case of an imperfect state preparation described by a statistical mixture. Introduces the concept of quantum state purity to characterize the degree of mixture of the state, and shows that one can always "purify" a density matrix by going into a Hilbert space of larger dimension.
A quantum mechanical system that is described by a state vector can be equally described by a density operator, although the inverse is not true. When information about a system is incomplete the only way to describe the system is by the density operator formalism. Incompleteness arises in two general cases: a mixed ensemble of physical systems and a system that interacts with its environment. We shed light on some subtle points that are usually misunderstood. We show explicitly that in the first case the averaged state vector of the system is zero, and in the second case no state vector can be assigned to the system in general.
Matrix product density operators (MPDOs) are tensor network representations of locally purified density matrices where each physical degree of freedom is associated to an environment degree of freedom. MPDOs have interesting properties for mixed state representations: guaranteed positivity by construction, efficient conservation of the trace and computation of local observables. However, they have been challenging to use for noisy quantum circuit simulation, as the application of noise increases the dimension of the environment Hilbert space, leading to an exponential growth of bond dimensions. MPDOs also lack a unique canonical form, due to the freedom in the choice of basis for the environment Hilbert space, which leads to a vast variation of bond dimensions. In this work, we present a systematic way to reduce the bond dimensions of MPDOs by disentangling the purified state. We optimize the basis for the environment Hilbert space by performing density matrix renormalization group (DMRG)-like sweeps of local 2-qubit basis optimization. Interestingly, we find that targeting only the disentanglement of the purified state leads to a reduction of the environment dimension. In other words, a compact MPDO representation requires a low-entanglement purified state. We apply our compression method to the emulation of noisy random quantum circuits. Our technique allows us to keep bounded bond dimensions, and thus bounded memory, contrary to previous works on MPDOs, while keeping reasonable truncation fidelities.
Quantum dynamics (QD) simulation is a powerful tool for interpreting ultrafast spectroscopy experiments and unraveling their microscopic mechanism in out-of-equilibrium excited state behaviors in various chemical, biological, and material systems. Although state-of-the-art numerical QD approaches such as the time-dependent density matrix renormalization group (TD-DMRG) already greatly extended the solvable system size of general linearly coupled exciton-phonon models with up to a few hundred phonon modes, the accurate simulation of larger system sizes or strong system-environment interactions is still computationally highly challenging. Based on quantum information theory (QIT), in this work, we realize that only a small number of effective phonon modes couple to the excitonic system directly regardless of a large or even infinite number of modes in the condensed phase environment. On top of the identified small number of direct effective modes, we propose a hierarchical mapping (HM) approach through performing block Lanczos transformations on the remaining indirect modes, which transforms the Hamiltonian matrix to a nearly block-tridiagonal form and eliminates the long-range interactions. Numerical tests on model spin-boson systems and realistic singlet fission models in a rubrene crystal environment with up to 7000 modes and strong system-environment interactions indicate HM can reduce the system size by 1-2 orders of magnitude and accelerate the calculation by ∼80% without losing accuracy.
In quantum physics, we often need to consider ensembles that are not pure cases, that is, that cannot be described by state vectors. This is so, for instance, when an ensemble is composed of several subensembles that are described by nonequivalent state vectors. It also happens in most cases in which we consider ensembles of definite subsystems of some larger systems. The density matrix—or "statistical operator"—formalism is one that makes it possible to deal with such situations swiftly and efficiently. The rules apply only to ensembles that can be described by state vectors. These rules entail as consequences new expressions for quantum probabilities and mean values, which can be called the—derived— "statistical operator rules." However, it is also possible to formulate the statistical operator rules directly instead. The Heisenberg "uncertainty" relations imply that in general f(x,p) cannot be interpreted as really being a joint probability distribution.