Infinite-dimensional Quantum Research Articles

Semantic Security with Infinite-dimensional Quantum Eavesdropping Channel

Semantic Security with Infinite-dimensional Quantum Eavesdropping Channel

Thermodynamically Ideal Quantum State Inputs to Any Device

We investigate and ascertain the ideal inputs to any finite-time physical process. We demonstrate that the expectation values of entropy flow, heat, and work can all be determined via Hermitian observables of the initial state. These Hermitian operators encapsulate the breadth of behavior and the ideal inputs for common thermodynamic objectives. We show how to construct these Hermitian operators from measurements of thermodynamic output from a finite number of effectively arbitrary inputs. The behavior of a small number of test inputs thus determines the full range of thermodynamic behavior from all inputs. For any process, entropy flow, heat, and work can all be extremized by pure input states—eigenstates of the respective operators. In contrast, the input states that minimize entropy production or maximize the change in free energy are nonpure mixed states obtained from the operators as the solution of a convex-optimization problem. To attain these, we provide an easily implementable gradient-descent method on the manifold of density matrices, where an analytic solution yields a valid direction of descent at each iterative step. Ideal inputs within a limited domain, and their associated thermodynamic operators, are obtained with less effort. This allows analysis of ideal thermodynamic inputs within quantum subspaces of infinite-dimensional quantum systems; it also allows analysis of ideal inputs in the classical limit. Our examples illustrate the diversity of “ideal” inputs: distinct initial states minimize entropy production, extremize the change in free energy, and maximize work extraction. Published by the American Physical Society 2024

On a sharper bound on the stability of non-autonomous Schrödinger equations and applications to quantum control

We study the stability of the Schrödinger equation generated by time-dependent Hamiltonians with constant form domain. That is, we bound the difference between solutions of the Schrödinger equation by the difference of their Hamiltonians. The stability theorem obtained in this article provides a sharper bound than those previously obtained in the literature. This makes it a potentially useful tool for time-dependent problems in Quantum Physics, in particular for Quantum Control. We apply this result to prove two theorems about global approximate controllability of infinite-dimensional quantum systems. These results improve and generalise existing results on infinite-dimensional quantum control.

Path integrals, complex probabilities and the discrete Weyl representation

A discrete formulation of the real-time path integral as the expectation value of a functional of paths with respect to a complex probability on a sample space of discrete valued paths is explored. The formulation in terms of complex probabilities is motivated by a recent reinterpretation of the real-time path integral as the expectation value of a potential functional with respect to a complex probability distribution on cylinder sets of paths. The discrete formulation in this work is based on a discrete version of the Weyl algebra that can be applied to any observable with a finite number of outcomes. The origin of the complex probability in this work is the completeness relation. In the discrete formulation the complex probability exactly factors into products of conditional probabilities and exact unitarity is maintained at each level of approximation. The approximation of infinite dimensional quantum systems by discrete systems is discussed. The method is illustrated by applying it to scattering theory and quantum field theory. The implications of these applications for quantum computing is discussed.

A unifying quantum speed limit for time-independent Hamiltonian evolution

Quantum speed limit (QSL) is the study of fundamental limits on the evolution time of quantum systems. For instance, under the action of a time-independent Hamiltonian, the evolution time between an initial and a final quantum state obeys various mutually complementary lower bounds. They include the Mandelstam–Tamm (MT) bound, the Margolus–Levitin (ML) bound, the Luo–Zhang bound, the Lee–Chau (LC) bound together with the dual ML bound introduced by Ness and coworkers. Here we show that the MT bound can be obtained by optimizing the LC bound over a certain parameter. More importantly, we report a QSL that includes all the above bounds as special cases before optimizing over the physically meaningless reference energy level of a quantum system. This unifying bound depends on a certain parameter p. For any fixed p, we find all pairs of time-independent Hamiltonian and initial pure quantum state that saturate this unifying bound. More importantly, these pairs allow us to compute this bound accurately and efficiently using an oracle that returns certain pth moments related to the absolute value of energy of the quantum state. Moreover, this oracle can be simulated by a computationally efficient and accurate algorithm for finite-dimensional quantum systems as well as for certain infinite-dimensional quantum states with bounded and continuous energy spectra. This makes our computational method feasible in a lot of practical situations. We further compare the performance of this bound for the case of a fixed p as well as the case of optimizing over p with existing QSLs. We find that if the dimension of the underlying Hilbert space is ≲2000 , our unifying bound optimized over p can be computed accurately in a few minutes using Mathematica code with just-in-time compilation in a typical desktop. Besides, this optimized unifying QSL is at least as good as all the existing ones combined and can occasionally be a few percent to a few times better.

Phase Spaces, Parity Operators, and the Born–Jordan Distribution

Phase spaces as given by the Wigner distribution function provide a natural description of infinite-dimensional quantum systems. They are an important tool in quantum optics and have been widely applied in the context of time–frequency analysis and pseudo-differential operators. Phase-space distribution functions are usually specified via integral transformations or convolutions which can be averted and subsumed by (displaced) parity operators proposed in this work. Building on earlier work for Wigner distribution functions (Grossmann in Commun Math Phys 48(3):191–194, 1976. https://doi.org/10.1007/BF01617867), parity operators give rise to a general class of distribution functions in the form of quantum-mechanical expectation values. This enables us to precisely characterize the mathematical existence of general phase-space distribution functions. We then relate these distribution functions to the so-called Cohen class (Cohen in J Math Phys 7(5):781–786, 1966. https://doi.org/10.1063/1.1931206) and recover various quantization schemes and distribution functions from the literature. The parity operator approach is also applied to the Born–Jordan distribution which originates from the Born–Jordan quantization (Born and Jordan in Z Phys 34(1):858–888, 1925. https://doi.org/10.1007/BF01328531). The corresponding parity operator is written as a weighted average of both displacements and squeezing operators, and we determine its generalized spectral decomposition. This leads to an efficient computation of the Born–Jordan parity operator in the number-state basis, and example quantum states reveal unique features of the Born–Jordan distribution.

Quantum controllability on graph-like manifolds through magnetic potentials and boundary conditions

We investigate the controllability of an infinite-dimensional quantum system: a quantum particle confined on a Thick Quantum Graph, a generalisation of Quantum Graphs whose edges are allowed to be manifolds of arbitrary dimension with quasi-δ boundary conditions. This is a particular class of self-adjoint boundary conditions compatible with the graph structure. We prove that global approximate controllability can be achieved using two physically distinct protocols: either using the boundary conditions as controls, or using time-dependent magnetic fields. Both cases have time-dependent domains for the Hamiltonians.

Exponential Modalities and Complementarity (extended abstract)

The exponential modalities of linear logic have been used by various authors to model infinite-dimensional quantum systems. This paper explains how these modalities can also give rise to the complementarity principle of quantum mechanics. The paper uses a formulation of quantum systems based on dagger-linear logic, whose categorical semantics lies in mixed unitary categories, and a formulation of measurement therein. The main result exhibits a complementary system as the result of measurements on free exponential modalities. Recalling that, in linear logic, exponential modalities have two distinct but dual components, ! and ?, this shows how these components under measurement become "compacted" into the usual notion of complementary Frobenius algebras from categorical quantum mechanics.

Optimal form of the Kretschmann–Schlingemann–Werner theorem for energy-constrained quantum channels and operations

It is proved that the energy-constrained Bures distance between arbitrary infinite-dimensional quantum channels is equal to the operator E-norm distance from any given Stinespring isometry of one channel to the set of all Stinespring isometries of another channel with the same environment. The same result is shown to be valid for arbitrary quantum operations.

Hamiltonian open quantum system toolkit

We present an open-source software package called “Hamiltonian Open Quantum System Toolkit" (HOQST), a collection of tools for the investigation of open quantum system dynamics in Hamiltonian quantum computing, including both quantum annealing and the gate-model of quantum computing. It features the key master equations (MEs) used in the field, suitable for describing the reduced system dynamics of an arbitrary time-dependent Hamiltonian with either weak or strong coupling to infinite-dimensional quantum baths. We present an overview of the theories behind the various MEs and provide examples to illustrate typical workflows in HOQST. We present an example that shows that HOQST can provide order of magnitude speedups compared to “Quantum Toolbox in Python" (QuTiP), for problems with time-dependent Hamiltonians. The package is ready to be deployed on high performance computing (HPC) clusters and is aimed at providing reliable open-system analysis tools for noisy intermediate-scale quantum (NISQ) devices.

A note on the infinite-dimensional quantum Strassen’s theorem

In Ref. 3, the quantum Strassen’s theorem has been extended to the infinite-dimensional case. This theorem consists in the solution of the coupling problem for two states on the algebra of bounded operators on two Hilbert spaces [Formula: see text], [Formula: see text] with the additional constraint that the coupling state has support in a pre-assigned sub-space of [Formula: see text]. In this paper, we give an alternative proof of the main theorem in Ref. 3 that allows such extension.

Classical and Quantum Controllability of a Rotating Asymmetric Molecule

We study both the classical and quantum rotational dynamics of an asymmetric top molecule, controlled through three orthogonal electric fields that interact with its dipole moment. The main difficulties in studying the controllability of these infinite-dimensional quantum systems are the presence of severe spectral degeneracies in the drift Hamiltonian and the nonsolvability of the stationary free Schrödinger equation, which lead us to apply a perturbative Lie algebraic approach. In this paper we show that, while the classical equations given by the Hamiltonian system on $$\mathrm{SO}(3)\times {\mathbb {R}}^3$$ are controllable for all values of the rotational constants and all dipole configurations, the Schödinger equation for the quantum evolution on $$L^2(\mathrm{SO}(3){,{\mathbb {C}}})$$ is approximately controllable for almost all values of the rotational constants if and only if the dipole is not parallel to any of the principal axes of inertia of the asymmetric rigid body.

Становление методологии сложностно-семиотического мышления в диалоге с квантовой механикой

It is stated that the process of human evolution on planet Earth has entered a permanent crisis mode. The Earth’s civilization is facing a growing set of challenges, threats of uncertainty, even in the very near future. These chal­lenges are associated with the onset of the epoch of the Anthropocene, when human intervention in the bio-geo-sphere leads to its irreversible transforma­tions, which with a high degree of probability may not be compatible with his own existence. On the other hand, on the horizon of the crisis evolution of mankind, a qualitatively new but still ghostly virtual entity appears, called artificial intelligence, which carries a potential threat of its enslavement, turn­ing it into a new pet at best. In this context, a request is rapidly being formed for a new way (or mode) of integratively oriented polyocular thinking, a new conceptual optics that allows us to comprehend the current situation here and now, in order to build a variety of possible scenarios for future human co­evolution, technology and the environment endowed with semiotic agency. Complexity thinking is evolutionary thinking together with complexity, think­ing quantum-relativistic, mediating, procedural-temporal, thinking immersed in a non-equilibrium flow of becoming distinctions, along with the retention of the distinguishable and indistinguishable, as a kind of figure and background, text and context, self-organizing autopoietic system and structurally related en­vironment. Thus, this mode of thinking becomes a recursive semiotically loaded contingent process in which the dialectic of necessity and randomness generates its qualitatively new states. Summing up, we can say that complexity acts as a kind of potentially active background, not fully manifested process, an infinite-dimensional quantum mechanical function, contact-meeting with which brings to life the appearance of a figure, a sign and its observer, referred to by us as a cross-border observer of network complexity.

Operational Characterization of Infinite-Dimensional Quantum Resources.

Recently, various nonclassical properties of quantum states and channels have been characterized through an advantage they provide in quantum information tasks over their classical counterparts. Such advantage can be typically proven to be quantitative, in that larger amounts of quantum resources lead to better performance in the corresponding tasks. So far, these characterizations have been established only in the finite-dimensional setting, hence, leaving out central resources in continuous variable systems such as entanglement and nonclassicality of states as well as entanglement breaking and broadcasting channels. In this Letter, we present a fully general framework for resource quantification in infinite-dimensional systems. The framework is applicable to a wide range of resources with the only premises being that classical randomness cannot create a resource and that the resourceless objects form a closed set in an appropriate sense. As the latter may be hard to establish for the abstract topologies of continuous variable systems, we provide a relaxation of the condition with no reference to topology. This envelopes the aforementioned resources and various others, hence, giving them an interpretation as performance enhancement in so-called input-output games.

Characters of infinite-dimensional quantum classical groups: BCD cases

We study the character theory of inductive limits of [Formula: see text]-deformed classical compact groups. In particular, we clarify the relationship between the representation theory of Drinfeld–Jimbo quantized universal enveloping algebras and our previous work on the quantized characters. We also apply the character theory to construct Markov semigroups on unitary duals of [Formula: see text], [Formula: see text], and their inductive limits.

Inductive limits of compact quantum groups and their unitary representations

We introduce inductive limits of compact quantum groups and their unitary representation theories. Those give a more explicit representation-theoretic meaning to our previous study of quantization of characters and central probability measures in the asymptotic representation theory. We also study tensor product representations of the infinite-dimensional quantum unitary group. As a by-product, we give an explicit representation-theoretic interpretation to certain transformations that play an important role in analyzing q-central probability measures on the paths in the q-Gelfand–Tsetlin graph.

Uniform continuity bounds for characteristics of multipartite quantum systems

We consider universal methods for obtaining (uniform) continuity bounds for characteristics of multipartite quantum systems. We pay special attention to infinite-dimensional multipartite quantum systems under the energy constraints. By these methods, we obtain continuity bounds for several important characteristics of a multipartite quantum state: the quantum (conditional) mutual information, the squashed entanglement, the c-squashed entanglement, and the conditional entanglement of mutual information. The continuity bounds for the multipartite quantum mutual information are asymptotically tight for large dimension/energy. The obtained results are used to prove the asymptotic continuity of the n-partite squashed entanglement, the c-squashed entanglement, and the conditional entanglement of mutual information under the energy constraints.

Wigner Measures and Coherent Quantum Control

We introduce Wigner measures for infinite-dimensional open quantum systems; important examples of such systems are encountered in quantum control theory. In addition, we propose an axiomatic definition of coherent quantum feedback.

Efficient Verification of Continuous-Variable Quantum States and Devices without Assuming Identical and Independent Operations.

Continuous-variable quantum information, encoded into infinite-dimensional quantum systems, is a promising platform for the realization of many quantum information protocols, including quantum computation, quantum metrology, quantum cryptography, and quantum communication. To successfully demonstrate these protocols, an essential step is the certification of multimode continuous-variable quantum states and quantum devices. This problem is well studied under the assumption that multiple uses of the same device result in identical and independently distributed (i.i.d.) operations. However, in realistic scenarios, identical and independent state preparation and calls to the quantum devices cannot be generally guaranteed. Important instances include adversarial scenarios and instances of time-dependent and correlated noise. In this Letter, we propose the first set of reliable protocols for verifying multimode continuous-variable entangled states and devices in these non-i.i.d scenarios. Although not fully universal, these protocols are applicable to Gaussian quantum states, non-Gaussian hypergraph states, as well as amplification, attenuation, and purification of noisy coherent states.

Energy-Constrained Discrimination of Unitaries, Quantum Speed Limits, and a Gaussian Solovay-Kitaev Theorem.

We investigate the energy-constrained (EC) diamond norm distance between unitary channels acting on possibly infinite-dimensional quantum systems, and establish a number of results. First, we prove that optimal EC discrimination between two unitary channels does not require the use of any entanglement. Extending a result by Acín, we also show that a finite number of parallel queries suffices to achieve zero error discrimination even in this EC setting. Second, we employ EC diamond norms to study a novel type of quantum speed limits, which apply to pairs of quantum dynamical semigroups. We expect these results to be relevant for benchmarking internal dynamics of quantum devices. Third, we establish a version of the Solovay-Kitaev theorem that applies to the group of Gaussian unitaries over a finite number of modes, with the approximation error being measured with respect to the EC diamond norm relative to the photon number Hamiltonian.