Finite-dimensional Hilbert Space Research Articles

Brief theory of multiqubit measurement

Peculiarities of multiqubit measurement are for the most part similar to peculiarities of measurement for qudit — quantum object with finite-dimensional Hilbert space. Three different interpretations of measurement concept are analyzed. One of those is purely quantum and is in collection, for a given state of the object to be measured, of incompatible observable measurement results in amount enough for reconstruction of the state. Two others make evident the difference between the reduced density matrix and the density matrices of physical objects involved in the measurement. It is shown that the von Neumann projectors, in combination with the concept of qudit phase space, produce an idea of a phase portrait of qudit state as a set of mathematical expectations for projectors on the possible pure states. The phase portrait is not a probability distribution since the projectors on nonorthogonal states are incompatible observables. Along with that, the phase portrait includes probability distributions for all the resolutions of identity of the qudit observable algebra. Additional peculiarities of measurement of the qudit degenerated observables, caused by the possibility of independent measurement of the observables for the particles, make possible to distinguish the local reduction of the qudit particle states from the entanglement of the local measurement results. The phase portrait of a composite system comprised by a qudit pair generates local and conditional phase portraits of particles. The entanglement is represented by the dependence of the shape of conditional phase portrait on the properties of the observable used in the measurement for the other particle. Analysis of the properties of a conditional phase portrait of a multiqubit qubits shows that absence of the entanglement is possible only in the case of substantial restrictions imposed on the method of multiqubit decomposition into qubits. Such a special method for determination of particles exists for each multiqubit state.

Valuation of a financial claim contingent on the outcome of a quantum measurement

We consider a rational agent who at time 0 enters into a financial contract for which the payout is determined by a quantum measurement at some time T > 0. The state of the quantum system is given in the Heisenberg representation by a known density matrix p^ . How much will the agent be willing to pay at time 0 to enter into such a contract? In the case of a finite dimensional Hilbert space H , each such claim is represented by an observable X^T where the eigenvalues of X^T determine the amount paid if the corresponding outcome is obtained in the measurement. We prove, under reasonable axioms, that there exists a pricing state q^ which is equivalent to the physical state p^ such that the pricing function Π0T takes the linear form Π0T(X^T)=P0Ttr(q^X^T) for any claim X^T , where P0T is the one-period discount factor. By ‘equivalent’ we mean that p^ and q^ share the same null space: that is, for any |ξ⟩∈H one has p^|ξ⟩=0 if and only if q^|ξ⟩=0 . We introduce a class of optimization problems and solve for the optimal contract payout structure for a claim based on a given measurement. Then we consider the implications of the Kochen–Specker theorem in this setting and we look at the problem of forming portfolios of such contracts. Finally, we consider multi-period contracts.

On the operators with numerical range in an ellipse

We give new necessary and sufficient conditions for the numerical range W(T) of an operator T∈B(H) to be a subset of the closed elliptical set Kδ⊆C given byKδ=def{x+iy:x2(1+δ)2+y2(1−δ)2≤1}, where 0<δ<1. Here B(H) denotes the collection of bounded linear operators on a Hilbert space H. Central to our efforts is a direct generalization of Berger's well-known criterion for an operator to have numerical radius at most one, his so-called strange dilation theorem. Specifically, we show that, if T acts on a finite-dimensional Hilbert space H and satisfies a certain genericity assumption, then W(T)⊆Kδ if and only if there exists a Hilbert space K⊇H, operators X1 and X2 on H and a unitary operator U acting on K such that(0.1)X1+X2=T,X1X2=δ and(0.2)X1k+X2k=2PHUk|H,k=1,2,…, where PH denotes the orthogonal projection from K to H.We next generalize the lemma of Sarason that describes power dilations in terms of semi-invariant subspaces to operators T that satisfy the relations (0.1) and (0.2). This generalization yields a characterization of the operators T∈B(H) such that W(T) is contained in Kδ in terms of certain structured contractions that act on H⊕H.As a corollary of our results we extend Ando's parametrization of operators having numerical range in a disc to those T such that W(T)⊆Kδ. We prove that, if T acts on a finite-dimensional Hilbert space H, then W(T)⊆Kδ if and only if there exist a pair of contractions A,B∈B(H) such that A is self-adjoint andT=2δA+(1−δ)1+AB1−A. We also obtain a formula for the B. and F. Delyon calcular norm of an analytic function on the inside of an ellipse in terms of the extremal H∞-extension problem for analytic functions defined on a slice of the symmetrized bidisc.

Berndtsson-Lempert-Szőke fields associated to proper holomorphic families of vector bundles

Drawing on work of Berndtsson and of Lempert and Szőke, we define a complex analytic structure for families of (possibly finite-dimensional) Hilbert spaces that might not fit together to form a holomorphic vector bundle but nevertheless have a reasonable definition of curvature that agrees with the curvature of the Chern connection when the family of Hilbert spaces is locally trivial. As one consequence, we obtain a new proof of a theorem of Berndtsson on the curvature of direct images of semi-positively twisted relative canonical bundles (i.e., adjoint bundles), and of its higher-rank generalization by Liu-Yang.

Homotopical foundations of parametrized quantum spin systems

In this paper, we present a homotopical framework for studying invertible gapped phases of matter from the point of view of infinite spin lattice systems, using the framework of algebraic quantum mechanics. We define the notion of quantum state types. These are certain lax-monoidal functors from the category of finite-dimensional Hilbert spaces to the category of topological spaces. The universal example takes a finite-dimensional Hilbert space [Formula: see text] to the pure state space of the quasi-local algebra of the quantum spin system with Hilbert space [Formula: see text] at each site of a specified lattice. The lax-monoidal structure encodes the tensor product of states, which corresponds to stacking for quantum systems. We then explain how to formally extract parametrized phases of matter from quantum state types, and how they naturally give rise to [Formula: see text]-spaces for an operad we call the “multiplicative” linear isometry operad. We define the notion of invertible quantum state types and explain how the passage to phases for these is related to group completion. We also explain how invertible quantum state types give rise to loop-spectra. Our motivation is to provide a framework for constructing Kitaev’s loop-spectrum of bosonic invertible gapped phases of matter. Finally, as a first step toward understanding the homotopy types of the loop-spectra associated to invertible quantum state types, we prove that the pure state space of any UHF algebra is simply connected.

Constrained spin systems and KNdS black holes

Kerr-Newman de Sitter (KNdS) spacetimes have a rich thermodynamic structure that involves multiple horizons, and so differs in key respects from asymptotically flat or AdS black holes. In this paper, we show that certain features of KNdS spacetimes can be reproduced by a constrained system of N non-interacting spins in a magnetic field. Both the KNdS and spin systems have bounded energy and entropy, a maximum of the entropy in the interior of the energy range, and a symmetry that maps lower energy states to higher energy states with the same entropy. Consequently, both systems have a temperature that can be positive or negative, where the gravitational temperature is defined analogously to that of the spins. We find that the number of spins N corresponds to 1/Λ for black holes with very small charge q and rotation parameter a, and scales like a2+q2/Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{\\left({a}^2+{q}^2\\right)/\\Lambda} $$\\end{document} for larger values of a and q. These comparisons show that certain key features of the complex thermodynamics of KNdS spacetimes are reproduced by a simple system of noninteracting spins, suggesting that there may be a relatively simple effective description of quantum de Sitter black holes in terms of a finite dimensional Hilbert space.

Barycentric decomposition for quantum instruments

We present a barycentric decomposition for quantum instruments whose output space is finite-dimensional and input space is separable. As a special case, we obtain a barycentric decomposition for channels between such spaces and for normalized positive-operator-valued measures in separable Hilbert spaces. This extends the known results by Ali and Chiribella et al on decompositions of quantum measurements, and formalizes the fact that every instrument between finite-dimensional Hilbert spaces can be represented using only finite-outcome instruments.

Minimal orthonormal bases for pure quantum state estimation

We present an analytical method to estimate pure quantum states using a minimum of three measurement bases in any finite-dimensional Hilbert space. This is optimal as two bases are insufficient to construct an informationally complete positive operator-valued measurement (IC-POVM) for pure states. We demonstrate our method using a binary tree structure, providing an algorithmic path for implementation. The performance of the method is evaluated through numerical simulations, showcasing its effectiveness for quantum state estimation.

Raznostnye skhemy dekompozitsii na osnove rasshchepleniya resheniya i operatora zadachi

Domain decomposition methods are used for the approximate solution of boundary value problems for partial differential equations on parallel computing systems. The specifics of nonstationary problems is most completely taken into account when using noniterative domain decomposition schemes. Regionally additive schemes are constructed on the basis of various classes of splitting schemes. A new class of domain decomposition schemes with an additive representation of the solution on a new time level is distinguished that is based on splitting the domain into subdomains based on a partition of unity. An example of the Cauchy problem for first-order evolution equations with a positive self-adjoint operator in a finite-dimensional Hilbert space is considered. Unconditionally stable two- and three-level splitting schemes are constructed for the corresponding system of equations.

Quantum detection problem for fusion frames

A quantum injective frame is a frame whose measurements for density operators can be used as a distinguishing feature in a quantum system, and the frame quantum detection problem demands a characterization of all such frames. Very recently, the quantum detection problem for continuous as well as discrete frames in both finite and infinite dimensional Hilbert spaces received significant attention. The quantum detection problem pertaining to the characterization of informationally complete positive operator-valued measures (POVM) can be split into two cases: The quantum injectivity or state separability problem and the rang analysis or quantum state estimation problem. Building upon this notion, this note is aimed at the quantum detection problem for fusion frames. The injectivity of a family of vectors and a family of closed subspaces is characterized in terms of some operator equations in Hilbert–Schmidt and trace classes.

Quantitative bounds for unconditional pairs of frames

We formulate a quantitative finite-dimensional conjecture about frame multipliers and prove that it is equivalent to Conjecture 1 in [22]. We then present solutions to the conjecture for certain classes of frame multipliers. In particular, we prove that for all C>0 and N∈N the following is true: Let (xj)j=1N and (fj)j=1N be sequences in a finite dimensional Hilbert space which satisfy ‖xj‖=‖fj‖ for all 1≤j≤N and‖∑j=1Nεj〈x,fj〉xj‖≤C‖x‖,for all x∈ℓ2M and |εj|=1. If the frame operator for (fj)j=1N has eigenvalues λ1≥...≥λM and β>0 is such that λ1≤βM−1∑j=1Mλj then (fj)j=1N has Bessel bound 27β2C. The same holds for (xj)j=1N.

Weak vs. strong breaking of integrability in interacting scalar quantum field theories

The recently proposed classification of integrability-breaking perturbations according to their strength is studied in the context of quantum field theories. Using random matrix methods to diagnose the resulting quantum chaotic behaviour, we investigate the \phi^4ϕ4 and \phi^6ϕ6 interactions of a massive scalar, by considering the crossover between Poissonian and Wigner-Dyson distributions in systems truncated to a finite-dimensional Hilbert space. We find that a naive extension of the scaling of crossover coupling with the volume observed in spin chains does not give satisfactory results for quantum field theory. Instead, we demonstrate that considering the scaling of the crossover coupling with the number of particles yields robust signatures, and is able to distinguish between the strengths of integrability breaking in the \phi^4ϕ4 and \phi^6ϕ6 quantum field theories.

Bounded below composition operators on the space of Bloch functions on the unit ball of a Hilbert space

Let B_E be the open unit ball of a complex finite or infinite dimensional Hilbert space E and consider the space mathcal {B}(B_E) of Bloch functions on B_E. Using Lipschitz continuity of the dilation map on B_E given by x mapsto (1-Vert xVert ^2) mathcal {R}f(x) for x in B_E, where mathcal {R}f denotes the radial derivative of f in mathcal {B}(B_E), we study when a composition operator on mathcal {B}(B_E) is bounded below.

Operator-difference schemes on non-uniform grids for second-order evolutionary equations

Abstract The approximate solution of the Cauchy problem for second-order evolution equations is performed, first of all, using three-level time approximations. Such approximations are easily constructed and relatively uncomplicated to investigate when using uniform time grids. When solving applied problems numerically, we should focus on approximations with variable time steps. When using multilevel schemes on non-uniform grids, we should maintain accuracy by choosing appropriate approximations and ensuring stability of the approximate solution. In this paper, we construct unconditionally stable schemes of the first- and second-order accuracy on a non-uniform time grid for the approximate solution of the Cauchy problem for a second-order evolutionary equation. The novelty of the paper consists in the fact that these stability estimates are obtained without any restrictions on the magnitude of the step change and on the number of step changes. We use a special transformation of the original second-order differential-operator equation to a system of first-order equations. For the system of first-order equations, we apply standard two-level time approximations. We obtained stability estimates for the initial data and the right-hand side in finite-dimensional Hilbert space. Eliminating auxiliary variables leads to three-level schemes for the initial second-order evolution equation. Numerical experiments were performed for the test problem for a one-dimensional in space bi-parabolic equation. The accuracy and stability properties of the constructed schemes are demonstrated on non-uniform grids with randomly varying grid steps.

Equivalent quantum systems

We have studied quantum systems on finite-dimensional Hilbert spaces and found that all these systems are connected through local transformations. Actually, we have shown that these transformations give rise to a gauge group that connects the Hamiltonian operators associated with each quantum system. This bridge allows us to connect different quantum systems, in such a way that studying one of them allows to understand the other through a gauge transformation. Furthermore, we included the case where the Hamiltonian operator can be time-dependent. An application for this construction will be achieved in the theory of control quantum systems.

Passive Quantum Measurement: Arrival Time, Quantum Zeno Effect and Gambler's Fallacy

AbstractClassical measurements are passive, in the sense that they do not affect the physical properties of the measured system. Normally, quantum measurements are not passive in that sense. In the infinite dimensional Hilbert space, however, it is found that quantum projective measurement can be passive in a way which is impossible in finite dimensional Hilbert spaces. Specifically, it is found that expectation value of a hermitian Hamiltonian can have an imaginary part in the infinite dimensional Hilbert space and that such an imaginary part implies a possibility to avoid quantum Zeno effect, which can physically be realized in quantum arrival experiments. The avoidance of quantum Zeno effect can also be understood as avoidance of a quantum version of gambler's fallacy, leading to the notion of passive quantum measurement that updates information about the physical system without affecting its physical properties. The arrival time probability distribution of a particle is found to be given by the flux of the probability current. Possible negative fluxes correspond to regimes at which there is no arrival at all, physically understood as regimes at which the particle departs rather than arrives.

Effective operators and their variational principles for discrete electrical network problems

Using a Hilbert space framework inspired by the methods of orthogonal projections and Hodge decompositions, we study a general class of problems (called Z-problems) that arise in effective media theory, especially within the theory of composites, for defining the effective operator. A new and unified approach is developed, based on block operator methods, for obtaining solutions of the Z-problem, formulas for the effective operator in terms of the Schur complement, and associated variational principles (e.g., the Dirichlet and Thomson minimization principles) that lead to upper and lower bounds on the effective operator. In the case of finite-dimensional Hilbert spaces, this allows for a relaxation of the standard hypotheses on positivity and invertibility for the classes of operators usually considered in such problems by replacing inverses with the Moore–Penrose pseudoinverse. As we develop the theory, we show how it applies to the classical example from the theory of composites on the effective conductivity in the periodic conductivity problem in the continuum (2d and 3d) under the standard hypotheses. After that, we consider the following three important and diverse examples (increasing in complexity) of discrete electrical network problems in which our theory applies under the relaxed hypotheses. First, an operator-theoretic reformulation of the discrete Dirichlet-to-Neumann (DtN) map for an electrical network on a finite linear graph is given and used to relate the DtN map to the effective operator of an associated Z-problem. Second, we show how the classical effective conductivity of an electrical network on a finite linear graph is essentially the effective operator of an associated Z-problem. Finally, we consider electrical networks on periodic linear graphs and develop a discrete analog to the classical example of the periodic conductivity equation and effective conductivity in the continuum.

Path-connectedness and topological closure of some sets related to the non-compact Stiefel manifold

If H is a Hilbert space, the non-compact Stiefel manifold St(n, H) consists of independent n-tuples in H. In this article, we contribute to the topological study of non-compact Stiefel manifolds, mainly by proving two results on the path-connectedness and topological closure of some sets related to the non-compact Stiefel manifold. In the first part, after introducing and proving an essential lemma, we prove that ∩j∈J (U(j) + St(n, H)) is path-connected by polygonal paths under a condition on the codimension of the span of the components of the translating J-family. Then, in the second part, we show that the topological closure of St(n, H)∩S contains all polynomial paths contained in S and passing through a point in St(n, H). As a consequence, we prove that St(n, H) is relatively dense in a certain class of subsets which we illustrate with many examples from frame theory coming from the study of the solutions of some linear and quadratic equations which are finite-dimensional continuous frames. Since St(n, L2(X, μ; F)) is isometric to, FF(X, μ), n, this article is also a contribution to the theory of finite-dimensional continuous Hilbert space frames.

Mixed discrete-variable Gaussian states

The quantum systems with finite-dimensional Hilbert space have several applications and are intensively explored theoretically and experimentally. The mathematical description of these systems follows the analogy with the usual infinite-dimensional case. There exist finite versions for most of the elements used in the continuous case, but (to our knowledge) there does not exist a finite version corresponding to the mixed Gaussian states. Our aim is to fill this gap. The definition we propose for the mixed discrete Gaussian states is based on the explicit formulas we have obtained in the case of pure discrete variable Gaussian states.

Selective and Efficient Quantum Process Tomography for Non-Trace-Preserving Maps: Implementation with a Superconducting Quantum Processor

Alternatively to the full reconstruction of an unknown quantum process, the so-called selective and efficient quantum process tomography (SEQPT) allows estimating, individually and up to the required accuracy, a given element of the matrix that describes such an operation with a polynomial amount of resources. The implementation of this protocol has been carried out with success to characterize the evolution of a quantum system that is well described by a trace-preserving quantum map. Here, we deal with a more general type of quantum process that does not preserve the trace of the input quantum state, which naturally arises in the presence of imperfect devices and system-environment interactions, in the context of quantum information science or quantum dynamics control. In that case, we show with the aid of a priori information on the losses structure of the quantum channel that the SEQPT reconstruction can be adapted to reconstruct the non-trace-preserving map. We explicitly describe how to implement the reconstruction in an arbitrary Hilbert space of finite dimension $d$. The method is experimentally verified on a superconducting quantum processor provided by IBM Quantum services, by estimating several non-trace-preserving quantum processes in dimensions up to $d=6$. Our results show that it is possible to efficiently reconstruct non-trace-preserving processes, with high precision, and with significantly higher fidelity than when the process is assumed to be trace preserving.