We consider pseudo-unitary quantum systems and discuss various properties of pseudo-unitary operators. In particular we prove a characterization theorem for block-diagonalizable pseudo-unitary operators with finite-dimensional diagonal blocks. Furthermore, we show that every pseudo-unitary matrix is the exponential of $i=\sqrt{-1}$ times a pseudo-Hermitian matrix, and determine the structure of the Lie groups consisting of pseudo-unitary matrices. In particular, we present a thorough treatment of $2\times 2$ pseudo-unitary matrices and discuss an example of a quantum system with a $2\times 2$ pseudo-unitary dynamical group. As other applications of our general results we give a proof of the spectral theorem for symplectic transformations of classical mechanics, demonstrate the coincidence of the symplectic group $Sp(2n)$ with the real subgroup of a matrix group that is isomorphic to the pseudo-unitary group U(n,n), and elaborate on an approach to second quantization that makes use of the underlying pseudo-unitary dynamical groups.
Metrics and pseudometrics are defined on the group of unitary operators in a Hilbert space. Several explicit formulas are derived. A special feature of the work is investigation of pseudometrics in unitary groups. The rich classes of pseudometrics have many interesting applications. Three such applications, distinguishibility of unitary operators, quantum coding and quantum search problems are discussed.
Fundamental problems of quantum field theory related to the representation problem of canonical commutation relations are discussed within a gauge field version of a van Hove-type model. The Coulomb field generated by a static charge distribution is described as a formal superposition of time-like pseudo-photons in Fock space with a Krein structure. In this context, a generalization of operator gauge transformations is introduced to generate coherent states of abelian gauge fields interacting with a charged background.
The quantum measurement axiom dictates that physical observables and in particular the Hamiltonian must be diagonalizable and have a real spectrum. For a time-independent Hamiltonian (with a discrete spectrum) these conditions ensure the existence of a positive-definite inner product that renders the Hamiltonian self-adjoint. Unlike for a time-independent Hamiltonian, this does not imply the unitarity of the Schroedinger time-evolution for a general time-dependent Hamiltonian. We give an additional necessary and sufficient condition for the unitarity of time-evolution. In particular, we obtain…
A finite-dimensional pseudo-unitary framework is set up for describing the dynamics of free elementary particles in a purely relativistic quantum mechanical way. States of any individual particles or antiparticles are defined as suitably normalized vectors belonging to the two-complex-dimensional spaces that occur in local orthogonal decompositions of isomorphic copies of Cartan's space. The corresponding dynamical variables thus show up as bounded pseudo-Hermitian operator restrictions that possess real discrete spectra. Any measurement processes have to be performed locally in orthocronous p…
Localized unitary operators are basic probes of locality and causality in quantum systems: localized unitary operators create localized excitations in entangled states. Working with an explicit form, we explore the properties of these operators in quantum mechanics and quantum field theory. We show that, unlike unitary operators, local non-unitary operators generically create non-local excitations. We present a local picture for quantum systems in which localized experimentalists can only act through localized Hamiltonian deformations, and therefore localized unitary operators. We demonstrate …
A Herman-Avila-Bochi type formula is obtained for the average sum of the top d Lyapunov exponents over a one-parameter family of G-cocycles, where G is the group that leaves a certain, non-degenerate hermitian form of signature (c,d) invariant. The generic example of such a group is the pseudo-unitary group U(c,d) or in the case c=d, the hermitian-symplectic group HSp(2d) which naturally appears for cocycles related to Schrödinger operators. In the case d=1, the formula for HSp(2d) cocycles reduces to the Herman-Avila-Bochi formula for SL(2,R) cocycles.
We study bipartite unitary operators which stay invariant under the local actions of diagonal unitary and orthogonal groups. We investigate structural properties of these operators, arguing that the diagonal symmetry makes them suitable for analytical study. As a first application, we construct large new families of dual unitary gates in arbitrary finite dimensions, which are important toy models for entanglement spreading in quantum circuits. We then analyze the non-local nature of these invariant operators, both in discrete (operator Schmidt rank) and continuous (entangling power) settings. …
Unitary operations are physically implementable. We further the understanding of such operations by studying the possible forms of nonlocal unitary operators, which are bipartite or multipartite unitary operators that are not tensor product operators. They are of broad relevance in quantum information processing. We prove that any nonlocal unitary operator of Schmidt rank three on a $d_A\times d_B$ bipartite system is locally equivalent to a controlled unitary. This operator can be locally implemented assisted by a maximally entangled state of Schmidt rank $\min\{d_A^2,d_B\}$ when $d_A\le d_B$…
Recently, Chau [Quant. Inform. & Comp. 11, 721 (2011)] showed that one can define certain metrics and pseudo-metrics on U(n), the group of all $n\times n$ unitary matrices, based on the arguments of the eigenvalues of the unitary matrices. More importantly, these metrics and pseudo-metrics have quantum information theoretical meanings. So it is instructive to study this kind of metrics and pseudo-metrics on U(n). Here we show that any symmetric norm on ${\mathbb R}^n$ induces a metric on U(n). Furthermore, using the same technique, we prove an inequality concerning the eigenvalues of a product…
The families of quasi-simple or Cayley--Klein algebras associated to antihermitian matrices over R, C and H are described in a unified framework. These three families include simple and non-simple real Lie algebras which can be obtained by contracting the pseudo-orthogonal algebras so(p,q) of the Cartan series $B_l$ and $D_l$, the special pseudo-unitary algebras su(p,q) in the series $A_l$, and the quaternionic pseudo-unitary algebras sq(p,q) in the series $C_l$. This approach allows to study many properties for all these Lie algebras simultaneously. In particular their non-trivial central ext…
We establish a bijection between the self-adjoint extensions of the Laplace operator on a bounded regular domain and the unitary operators on the boundary. Each unitary encodes a specific relation between the boundary value of the function and its normal derivative. This bijection sets up a characterization of all physically admissible dynamics of a nonrelativistic quantum particle confined in a cavity. More- over, this correspondence is discussed also at the level of quadratic forms. Finally, the connection between this parametrization of the extensions and the classical one, in terms of boun…
We consider a reducible unitary representation of Heisenberg-Weyl group in a tensor product of two Hilbert spaces. A non-commutative operator graph generated by this representation is introduced. It is shown that spectral projections of unitaries in the representation are anticliques (quantum error-correcting codes) for this graph. The obtained codes are appeared to be linear envelopes of entangled vectors.
We investigate unitary operators acting on a tensor product space, with the property that the quantum channels they generate, via the Stinespring dilation theorem, are of a particular type, independently of the state of the ancilla system in the Stinespring relation. The types of quantum channels we consider are those of interest in quantum information theory: unitary conjugations, constant channels, unital channels, mixed unitary channels, PPT channels, and entanglement breaking channels. For some of the classes of bipartite unitary operators corresponding to the above types of channels, we p…
Columns of d^2 x N matrices are shown to create different sets of N operators acting on $d$-dimensional Hilbert space. This construction corresponds to a formalism of the star-product of operator symbols. The known bases are shown to be partial cases of generic formulas derived by using d^2 x N matrices as a source for constructing arbitrary bases. The known examples of the SIC-POVM, MUBs, and the phase-space description of qubit states are considered from the viewpoint of the developed unified approach. Star-product schemes are classified with respect to associated d^2 x N matrices. In partic…
A class of non-Dirac-hermitian many-particle quantum systems admitting entirely real spectra and unitary time-evolution is presented. These quantum models are isospectral with Dirac-hermitian systems and are exactly solvable. The general method involves a realization of the basic canonical commutation relations defining the quantum system in terms of operators those are hermitian with respect to a pre-determined positive definite metric in the Hilbert space. Appropriate combinations of these operators result in a large number of pseudo-hermitian quantum systems admitting entirely real spectra …
In this work we uncover the mathematical structure of the Schwinger algebra and introduce an almost unitary Schwinger operators which are derived by considering translation operators on a finite lattice. We calculate mathematical relations between these algebras and show that the almost unitary Schwinger operators are equivalent to the Schwinger algebra. We introduce new representations for MN(C) in terms of these algebras.
The energy bounds condition for intertwining operators of unitary rational vertex operator algebras (VOAs) was studied, first by A.Wassermann for type $A$ affine VOAs, and later by T.Loke for $c<1$ Virasoro VOAs, and by V.Toledano-Laredo for type $D$ affine VOAs. In this paper, we extend their results to affine VOAs of type $B$, $C$, and $G_2$. As a consequence, the modular tensor categories of these unitary vertex operator algebras are unitary.
We describe a subclass of the class of normal operators on Banach spaces over non-Archimedean fields (A. N. Kochubei, J. Math. Phys. 51 (2010), article 023526) consisting of operators whose properties resemble those of unitary operators. In particular, an analog of Stone's theorem about one-parameter groups of unitary operators is proved.
The most general possible central extensions of two whole families of Lie algebras, which can be obtained by contracting the special pseudo-unitary algebras su(p,q) of the Cartan series A_l and the pseudo-unitary algebras u(p,q), are completely determined and classified for arbitrary p,q. In addition to the su(p,q) and u({p,q}) algebras, whose second cohomology group is well known to be trivial, each family includes many non-semisimple algebras; their central extensions, which are explicitly given, can be classified into three types as far as their properties under contraction are involved. A …
While quantum circuits built from two-particle dual-unitary (maximally entangled) operators serve as minimal models of typically nonintegrable many-body systems, the construction and characterization of dual-unitary operators themselves are only partially understood. A nonlinear map on the space of unitary operators was proposed in PRL.~125, 070501 (2020) that results in operators being arbitrarily close to dual unitaries. Here we study the map analytically for the two-qubit case describing the basins of attraction, fixed points, and rates of approach to dual unitaries. A subset of dual-unitar…
We compute an explicit formula for the one-parameter unitary group of the single-particle Rashba spin-orbit coupled operator in dimension three. As an application, we derive the formula for the Green function for the two-particle operator, and then prove that the spin-dependent point-interaction is of class $\mathcal{H}_{-4}$. The latter is thus the example of a supersingular perturbation for which no self-adjoint operator can be constructed.
The Bott index of two unitary operators on an infinite dimensional Hilbert space is defined. homotopic invariance with respect to multiplicative unitary perturbations of the type identity plus trace class and the "logarithmic" law for the index are proven. The index and its properties are then extended to the case of a pair of invertible operators. An application to the physics of two dimensional quantum systems proves that the index is equal to the Chern number therefore showing that the transverse Hall conductance is an integer.
We present a new technique to obtain polynomial decay estimates for the matrix coefficients of unitary operators. Our approach, based on commutator methods, applies to nets of unitary operators, unitary representations of topological groups, and unitary operators given by the evolution group of a self-adjoint operator or by powers of a unitary operator. Our results are illustrated with a wide range of examples in quantum mechanics and dynamical systems, as for instance Schrödinger operators, Dirac operators, quantum waveguides, horocycle flows, adjacency matrices, Jacobi matrices, quantum walk…
We present a suitable framework for the definition of quantum time delay in terms of sojourn times for unitary operators in a two-Hilbert spaces setting. We prove that this time delay defined in terms of sojourn times (time-dependent definition) exists and coincides with the expectation value of a unitary analogue of the Eisenbud-Wigner time delay operator (time-independent definition). Our proofs rely on a new summation formula relating localisation operators to time operators and on various tools from functional analysis such as Mackey's imprimititvity theorem, Trotter-Kato Formula and commu…
This is the first part in a two-part series of papers constructing a unitary structure for the modular tensor category (MTC) associated to a unitary rational vertex operator algebra (VOA).
We consider unitary analogs of $1-$dimensional Anderson models on $l^2(\Z)$ defined by the product $U_ω=D_ωS$ where $S$ is a deterministic unitary and $D_ω$ is a diagonal matrix of i.i.d. random phases. The operator $S$ is an absolutely continuous band matrix which depends on a parameter controlling the size of its off-diagonal elements. We prove that the spectrum of $U_ω$ is pure point almost surely for all values of the parameter of $S$. We provide similar results for unitary operators defined on $l^2(\N)$ together with an application to orthogonal polynomials on the unit circle. We get almo…
Following the various statements of [DW16] to their logical conclusion, this note explicitly argues the following statement, implicit in [DW16]: for positive semi-definite operators $C_{1},\ldots,\,C_{L} $, a unitary $V_{C_{i}}$ commuting with $C_{i}$, and $p\geq1$, the quantity $$ \max_{V_{C_{1}},\ldots,V_{C_{L}}}\left\Vert C_{1}^{1/p}V_{C_{1}}\cdots C_{L}^{1/p}V_{C_{L}}\right\Vert _{p}^{p} $$ is monotone non-increasing with respect to $p$. The idea from [DW16] is that by allowing unitary swivels connecting a long chain of positive semi-definite operators together, we can establish such a sta…
We present an improved version of commutator methods for unitary operators under a weak regularity condition. Once applied to a unitary operator, the method typically leads to the absence of singularly continuous spectrum and to the local finiteness of point spectrum. Large families of locally smooth operators are also exhibited. Half of the paper is dedicated to applications, and a special emphasize is put on the study of cocycles over irrational rotations. It is apparently the first time that commutator methods are applied in the context of rotation algebras, for the study of their generator…
We present a general account on the stationary scattering theory for unitary operators in a two-Hilbert spaces setting. For unitary operators $U_0,U$ in Hilbert spaces ${\cal H}_0,{\cal H}$ and for an identification operator $J:{\cal H}_0\to{\cal H}$, we give the definitions and collect properties of the stationary wave operators, the strong wave operators, the scattering operator and the scattering matrix for the triple $(U,U_0,J)$. In particular, we exhibit conditions under which the stationary wave operators and the strong wave operators exist and coincide, and we derive representation form…
In this paper, the Moyal star-product quantization is used to construct the unitary irreducible representations of the Euclidean motion group on 3-dimensions. These unitary representations will come from the representation of its Lie algebra whose operators are defined by the left Moyal star-product multiplication. In fact, these representations of the Lie algebra is the infinitisimal representation. Hence, the exponentiation of these operators gives rise to unitary operators that defines the desired unitary representations.
This paper continues the previous work (Quantum Inf. Process (2019)) by two authors of the present paper about a spectral mapping property of chiral symmetric unitary operators. In physics, they treat non-unitary time-evolution operators to consider quantum walks in open systems. In this paper, we generalize the above result to include a chiral symmetric non-unitary operator whose coin operator only has two eigenvalues. As a result, the spectra of such non-unitary operators are included in the (possibly non-unit) circle and the real axis in the complex plane. We also give some examples of our …
We consider unitary analogs of $d-$dimensional Anderson models on $l^2(\Z^d)$ defined by the product $U_ω=D_ωS$ where $S$ is a deterministic unitary and $D_ω$ is a diagonal matrix of i.i.d. random phases. The operator $S$ is an absolutely continuous band matrix which depends on parameters controlling the size of its off-diagonal elements. We adapt the method of Aizenman-Molchanov to get exponential estimates on fractional moments of the matrix elements of $U_ω(U_ω-z)^{-1}$, provided the distribution of phases is absolutely continuous and the parameters correspond to small off-diagonal elements…
We consider the iteration of a unitary operator on a separable Hilbert space and study the spreading rates of the associated discrete-time dynamical system relative to a given orthonormal basis. We prove lower bounds for the transport exponents, which measure the time-averaged spreading on a power-law scale, in terms of dimensional properties of the spectral measure associated with the unitary operator and the initial state. These results are the unitary analog of results established in recent years for the dynamics of the Schrödinger equation, which is a continuum-time dynamical system associ…
In this paper we use the canonical complex structure $\mathbb{J}$ on $\mathbb{R}^{2n}$ to introduce a twist of the symplectic Dirac operator. As a matter of fact, these operators can be interpreted as the bosonic analogues of the Dirac operators on a Hermitian manifold. Moreover, we prove that the algebra of these symplectic Dirac operators is isomorphic to the Lie algebra $\mathfrak{su}(1,2)$ which leads to the Howe dual pair $(\mathsf{U}(n),\mathfrak{su}(1,2))$.
Let $H$ be a complex Hilbert space and let ${\mathcal C}$ be a conjugacy class of finite rank self-adjoint operators on $H$ with respect to the action of unitary operators. We suppose that ${\mathcal C}$ is formed by operators of rank $k$ and for every $A\in {\mathcal C}$ the dimensions of distinct maximal eigenspaces are distinct. Under the assumption that $\dim H\ge 4k$ we establish that every bijective transformation $f$ of ${\mathcal C}$ preserving the commutativity in both directions is induced by a unitary or anti-unitary operator, i.e. there is a unitary or anti-unitary operator $U$ suc…
