Non-unitary Similarity Transformation Research Articles

Suppressing the entanglement growth in matrix product state evolution of quantum systems through nonunitary similarity transformations

The possibility of using similarity transformations to alter dynamical entanglement growth in matrix-product-state simulations of quantum systems is explored. By appropriately choosing the similarity transformation, the entanglement growth rate is suppressed, improving the efficiency of numerical simulations of quantum systems. The transformation can be applied to general quantum-many-body systems.

Compact numerical solutions to the two-dimensional repulsive Hubbard model obtained via nonunitary similarity transformations

Similarity transformation of the Hubbard Hamiltonian using a Gutzwiller correlator leads to a non-Hermitian effective Hamiltonian, which can be expressed exactly in momentum-space representation, and contains three-body interactions. We apply this methodology to study the two-dimensional Hubbard model with repulsive interactions near half-filling in the intermediate interaction strength regime ($U/t=4$). We show that at optimal or near optimal strength of the Gutzwiller correlator, the similarity transformed Hamiltonian has extremely compact right eigenvectors, which can be sampled to high accuracy using the Full Configuration Interaction Quantum Monte Carlo (FCIQMC) method, and its initiator approximation. Near-optimal correlators can be obtained using a simple projective equation, thus obviating the need for a numerical optimisation of the correlator. The FCIQMC method, as a projective technique, is well-suited for such non-Hermitian problems, and its stochastic nature can handle the 3-body interactions exactly without undue increase in computational cost. The highly compact nature of the right eigenvectors means that the initiator approximation in FCIQMC is not severe, and that large lattices can be simulated, well beyond the reach of the method applied to the original Hubbard Hamiltonian. Results are provided in lattice sizes upto 50 sites and compared to auxiliary-field QMC. New benchmark results are provided in the off half-filling regime, with no severe sign-problem being encountered. In addition, we show that methodology can be used to calculate excited states of the Hubbard model and lay the groundwork for the calculation of observables other than the energy.

Biorthogonal projected energies of a Gutzwiller similarity transformed Hamiltonian

We present a method incorporating biorthogonal orbital-optimization, symmetry projection, and double-occupancy screening with a non-unitary similarity transformation generated by the Gutzwiller factor , and apply it to the Hubbard model. Energies are calculated with mean-field computational scaling with high-quality results comparable to coupled cluster singles and doubles. This builds on previous work performing similarity transformations with more general, two-body Jastrow-style correlators. The theory is tested on 2D lattices ranging from small systems into the thermodynamic limit and is compared to available reference data.

Inverse eigenvalue problems for extended Hessenberg and extended tridiagonal matrices

In inverse eigenvalue problems one tries to reconstruct a matrix, satisfying some constraints, given some spectral information. Here, two inverse eigenvalue problems are solved.First, given the eigenvalues and the first components of the associated eigenvectors (called the weight vector) an extended Hessenberg matrix with prescribed poles is computed possessing these eigenvalues and satisfying the eigenvector constraints. The extended Hessenberg matrix is retrieved by executing particularly designed unitary similarity transformations on the diagonal matrix containing the eigenvalues. This inverse problem closely links to orthogonal rational functions: the extended Hessenberg matrix contains the recurrence coefficients given the nodes (eigenvalues), poles (poles of the extended Hessenberg matrix), and a weight vector (first eigenvector components) determining the discrete inner product. Moreover, it is also sort of the inverse of the (rational) Arnoldi algorithm: instead of using the (rational) Arnoldi method to compute a Krylov basis to approximate the spectrum, we will reconstruct the orthogonal Krylov basis given the spectral info.In the second inverse eigenvalue problem, we do the same, but refrain from unitarity. As a result we execute possibly non-unitary similarity transformations on the diagonal matrix of eigenvalues to retrieve a (non)-symmetric extended tridiagonal matrix. The algorithm will be less stable, but it will be faster, as the extended tridiagonal matrix admits a low cost factorization of O(n) (n equals the number of eigenvalues), whereas the extended Hessenberg matrix does not. Again there is a close link with orthogonal function theory, the extended tridiagonal matrix captures the recurrence coefficients of bi-orthogonal rational functions. Moreover, it is again sort of inverse of the nonsymmetric Lanczos algorithm: given spectral properties, we reconstruct the two basis Krylov matrices linked to the nonsymmetric Lanczos algorithm.

A note on the topological insulator phase in non-Hermitian quantum systems

Examples of non-Hermitian quantum systems admitting a topological insulator phase are presented in one, two and three space dimensions. All of these non-Hermitian Hamiltonians have entirely real bulk eigenvalues and unitarity is maintained with the introduction of appropriate inner products in the corresponding Hilbert spaces. The topological invariant characterizing a particular phase is shown to be identical for a non-Hermitian Hamiltonian and its Hermitian counterpart, to which it is related through a non-unitary similarity transformation. A classification scheme for topological insulator phases in pseudo-Hermitian quantum systems is suggested.

Symmetry and degeneracy of the curved Coulomb potential on the S3 ball

The ‘curved’ Coulomb potential on the S3 ball, whose isometry group is SO(4), takes the form of a cotangent function, and when added to the four-dimensional squared angular momentum operator, one of the so(4) Casimir invariants, a Hamiltonian is obtained which describes a perturbance of the free geodesic motion that results in several peculiar aspects. The spectrum of such a motion has been studied on various occasions and is known to unexpectedly carry so(4) degeneracy patterns despite the non-commutativity of the perturbance with the Casimir operator. We suggest here an explanation for this behavior in designing a set of operators which close the so(4) algebra and whose Casimir invariant coincides with the Hamiltonian of the perturbed motion at the level of the eigenvalue problem. The above operators are related to the canonical geometric SO(4) generators on S3 by a non-unitary similarity transformation of the scaling type. In this fashion, we identify a complementary option to the deformed dynamical so(4) Higgs algebra constructed in terms of the components of the ordinary angular momentum and a related Runge–Lenz vector.

On Lambda and Time Operators: the Inverse Intertwining Problem Revisited

An exact theory of irreversibility was proposed by Misra, Prigogine and Courbage, based on non-unitary similarity transformations Λ that intertwine reversible dynamics and irreversible ones. This would advocate the idea that irreversible behavior would originate at the microscopic level. Reversible evolution with an internal time operator have the intertwining property. Recently the inverse intertwining problem has been answered in the negative, that is, not every unitary evolution allowing such Λ-transformation has an internal time. This work contributes new results in this direction.

Minimum uncertainty wavelets in non-relativistic super-symmetric quantum mechanics

We consider the connection to the harmonic oscillator, super-symmetric quantum mechanics (SUSY-QM) and coherent states of the recently derived constrained Heisenberg “minimum uncertainty” (μ-) wavelets [Phys Rev Lett 85:5263 (2000); Phys Rev A65: 052106-1 (2002); J Phys Chem A107:7318 (2003)]. We analyze several new types of raising and lowering operators,which also can be viewed as arising from a (non-unitary) similarity transformation of the Harmonic Oscillator Hamiltonian and ladder operators. We show that these new ladder operators lead to a new SUSY formalism for harmonic oscillation, so that the μ-wavelets naturally manifest SUSY properties. Using these new ladder operators, we construct coherent and supercoherent states for the oscillator. In the discussion, we consider possible implications of similarity transformations for quantum mechanics. In an appendix we consider the classical limit of the μ-wavelet oscillator.

Nonunitary similarity transformation of conservative to dissipative evolutions: Intertwining without time operator

Reversible evolutions are usually expressed in terms of unitary groups on separable Hilbert spaces, whereas irreversible ones are described by contraction semigroups. In the theory of nonunitary similarity transformations intertwining unitary groups and contraction semigroups, proposed initially in the context of statistical mechanics as part of an exact theory of irreversibility, the unitary groups with such intertwining property have been qualified by the existence of an internal time operator. This work tackles the question of existence of internal time operators for unitary groups with the intertwining property. Equivalent conditions to the existence of internal time operators for such unitary groups are given on the basis of the Sz.-Nagy–Foiaş [Harmonic Analysis of Operators on Hilbert Spaces (North-Holland, Amsterdam, 1970)] dilation theory and the theory of shift invariant subspaces. These conditions permit us to solve the inverse intertwining problem in the negative: there are unitary groups with the intertwining property which do not admit internal time operator. A representative family of such unitary groups is given.

On the NCCS model of the quantum Hall fluid

Area non-preserving transformations in the non-commutative plane are introduced with the aim to map the ν=1 integer quantum Hall effect (IQHE) state on the \(\nu=\frac{1}{2p+1}\) fractional quantum Hall effect (FQHE) states. Using the hydrodynamical description of the quantum Hall fluid, it is shown that these transformations are generated by vector fields satisfying the Gauss law in the interacting non-commutative Chern–Simons gauge theory, and the corresponding field-theory Lagrangian is reconstructed. It is demonstrated that the geometric transformations induce quantum-mechanical non-unitary similarity transformations, establishing the interplay between integral and fractional QHEs.

Kinetic equation, non-perturbative approach and decoherence free subspace for quantum open system

A Schrödinger (Liouville) type of equation for an quantum open system is presented. The equation has a correlated part and many Master equations can be derived from it as special cases. Most significantly, it can be applied to construct a decoherence-free subspace for quantum computing. The original Schrödinger (Liouville) equation for the total system is related to it by a non-unitary similarity transformation, which enables us to propose a non-perturbative method for solving the eigenvalue problem for the total Hamiltonian. In addition, it also enables one to uncover a simple procedure to treat the eigenvalue problem of an open system under strong interaction. The correlated part of the equation is not necessarily self-adjoint, so that there exists a complex spectrum for the corresponding Hamiltonian (Liouvillian) which enables the time evolution of states to be asymmetric. This then exposes just the correlation required to produce evolution, which coincides with the second law of thermodynamics.

Schrödinger Equation for an Open System

We present a Schrödinger (Liouville) type of equation for a quantum opensystem. It has a correlated part, and various masterequations may be its special cases. It also has significant applicationsfor constructing decoherence-free subspace for quantum computation. It isrelated to the original Schrödinger (Liouville) equation for thetotal system through a non-unitary similarity transformation. It isunnecessary for its correlated part to be self-adjoint, so there is acomplex spectrum for the corresponding Hamiltonian (Liouvillian), whichenables the time evolution of states to be asymmetric. This shows justthe correlation to produce evolution of world.

Confined Subspaces: A Schematic Model

An essential feature of confining systems is the existence of channels into which probability flux can enter only for short spacetime intervals, but where it can never, irrespective of the total energy involved, propagate to asymptotic distances. It is demonstrated in a schematic quantum-mechanical model that this situation, which is unknown in ordinary scattering theory, does arise in a representation obtained through a non-unitary similarity transformation, and therefore involving a nontrivial metric. In the “confined” channels of this representation, propagation is described by Green’s functions having a complex conjugate pair of poles in momentum space instead of the usual real-axis pole. On the asymptotically accessible subspace, the S matrix is unitary, but finite-time unitarity for the total system becomes visible only when using the nontrivial metric. It is suggested that field-theory propagators with complex-conjugate pairs of singularities as obtained in several QCD studies, which seem to contradict finite-time unitarity, can in fact be imbedded into a unitary framework along the line illustrated by this model. A further conjecture is that interacting QCD fields differing from the perturbative ones by a non-unitary similarity transformation form an essential ingredient for a theory of confinement.

Nonlocality of the Misra-Prigogine-Courbage semigroup

We show that the Markov semigroups constructed by Misra, Prigogine, and Courbage through nonunitary similarity transformations of Kolmogorov systems are not implementable by local point transformations, i.e., they are not the Frobenius-Perron semigroups associated with noninvertible point transformations, in contrast with the semigroups obtained by coarse-graining projections. Our result is a straightforward generalization of the proof of the nonlocality of the similarity transformation given by Goldstein, Misra, and Courbage and also of the previous illustration by Misra and Prigogine for the baker transformation and completes the characterization of the Misra-Prigogine-Courbage semigroups.

Similarity transformation for Fermi operators

The normally ordered Hilbert space image of the complex linear similarity transformation in Grassmann number phase space is derived in fermionic coherent state representation. Though keeping the anticommutator of Fermi operators invariant, the quantum mechanical images of classical transformations are generally not unitary. Remarkably, although kets and bras produced by the non-unitary similarity transformations are not Hermitian conjugates, they still form a complete basis. The derivation of the normally ordered operator which engenders the similarity transformation is facilitated by virtue of the technique of integration within order products.

Similarity transformations in one- and two-mode Fock space

The normally ordered Hilbert space image of the general (complex) linear similarity transformation in phase space is obtained in a coherent state representation. Although preserving the commutator of a and a+, these quantum mechanical images of classical transformations are generally not unitary. Remarkably, although the kets and bras produced by the non-unitary similarity transformations are not Hermitian conjugates, squeezed state analogues produced using the similarity transformation still satisfy an overcompleteness relation. The results are extended to two-mode Fock space and simple examples of the utility of the similarity transformation are presented. The evaluation of the normally ordered operators is greatly facilitated by the use of integration within ordered products.

New Variational Principle for Decaying States

The Hamiltonian of a system containing decaying states is subjected to a nonunitary similarity transformation which introduces the wave functions of these decaying states as well as those of the "antidecaying" dual states explicitly into the representation. The dependence of the Green's-function pole on these wave functions is used to formulate a variational principle for their determination, yielding a non-Hermitian, nonlinear wave equation including self-energy contributions. Explicit solutions are given in a simple example.

From deterministic dynamics to probabilistic descriptions

The present work is devoted to the following question: What is the relation between the deterministic laws of dynamics and probabilistic description of physical processes? It is generally accepted that probabilistic processes can arise from deterministic dynamics only through a process of “coarse graining” or “contraction of description” which inevitably involves a loss of information. In this work we present an alternative point of view toward the relation between deterministic dynamics and probabilistic descriptions. Speaking in general terms, we demonstrate the possibility of obtaining (stochastic) Markov processes from deterministic dynamics simply through a “change of representation” which involves no loss of information provided the dynamical system under consideration has a suitably high degree of instability of motion. The fundamental implications of this finding for statistical mechanics and other areas of physics are discussed. From a mathematical point of view, the theory we present is a theory of invertible, positivity preserving and necessarily nonunitary similarity transformations that convert the unitary groups associated with deterministic dynamics to contraction semigroups associated with stochastic Markov processes. We explicitly construct such similarity transformations for the so-called Bernoulli systems. This construction illustrates also the construction of the so-called Lyapounov variables and the operator of “internal time” which play an important role in our approach to the problem of irreversibility. The theory we present can also be viewed as a theory of entropy increasing evolutions and their relation to deterministic dynamics. The possibility of obtaining entropy increasing evolutions from deterministic dynamics through a nonunitary similarity transformation is concretely illustrated in the present work.

From deterministic dynamics to probabilistic descriptions

THE PRESENT WORK IS DEVOTED TO THE FOLLOWING QUESTION: What is the relationship between the deterministic laws of dynamics and probabilistic description of physical processes? It is generally accepted that probabilistic processes can arise from deterministic dynamics only through a process of "coarse graining" or "contraction of description" that inevitably involves a loss of information. In this work we present an alternative point of view toward the relationship between deterministic dynamics and probabilistic descriptions. Speaking in general terms, we demonstrate the possibility of obtaining (stochastic) Markov processes from deterministic dynamics simply through a "change of representation" that involves no loss of information provided the dynamical system under consideration has a suitably high degree of instability of motion. The fundamental implications of this finding for statistical mechanics and other areas of physics are discussed. From a mathematical point of view, the theory we present is a theory of invertible, positivity-preserving, and necessarily nonunitary similarity transformations that convert the unitary groups associated with deterministic dynamics to contraction semigroups associated with stochastic Markov processes. We explicitly construct such similarity transformations for the so-called Bernoulli systems. This construction illustrates also the construction of the so-called Lyapounov variables and the operator of "internal time," which play an important role in our approach to the problem of irreversibility. The theory we present can also be viewed as a theory of entropy-increasing evolutions and their relationship to deterministic dynamics.