Finite-dimensional Quantum Systems Research Articles

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.

On complementarity and distribution of imaginarity in finite dimensions

We investigate the complementarity relation among the l1 norm of imaginarity, linear entropy and the Brukner–Zeilinger invariant information in the presence of conjugate symmetry. By using an equality derived from the complementarity relation, we define a quantity named index of imaginarity distribution with respect to the imaginarity, linear entropy and invariant information in finite dimensional quantum systems. We show that the index of imaginarity distribution is nonnegative and upper bounded by the square of the l1 norm of imaginarity, and serves as an invaluable tool in characterizing effectively the imaginarity distribution. Detailed examples are presented to illustrate our theoretical analysis.

Quantum Theory in Finite Dimension Cannot Explain Every General Process with Finite Memory

Arguably, the largest class of stochastic processes generated by means of a finite memory consists of those that are sequences of observations produced by sequential measurements in a suitable generalized probabilistic theory (GPT). These are constructed from a finite-dimensional memory evolving under a set of possible linear maps, and with probabilities of outcomes determined by linear functions of the memory state. Examples of such models are given by classical hidden Markov processes, where the memory state is a probability distribution, and at each step it evolves according to a non-negative matrix, and hidden quantum Markov processes, where the memory is a finite-dimensional quantum system, and at each step it evolves according to a completely positive map. Here we show that the set of processes admitting a finite-dimensional explanation do not need to be explainable in terms of either classical probability or quantum mechanics. To wit, we exhibit families of processes that have a finite-dimensional explanation, defined manifestly by the dynamics of an explicitly given GPT, but that do not admit a quantum, and therefore not even classical, explanation in finite dimension. Furthermore, we present a family of quantum processes on qubits and qutrits that do not admit a classical finite-dimensional realization, which includes examples introduced earlier by Fox, Rubin, Dharmadikari and Nadkarni as functions of infinite-dimensional Markov chains, and lower bound the size of the memory of a classical model realizing a noisy version of the qubit processes.

Duality of averaging of quantum states over arbitrary symmetry groups revealing Schur–Weyl duality

It is a well-established fact in quantum information theory, that uniform averaging over the collective action of a unitary group on a multipartite quantum state projects the state to a form equivalent to a permutation operator of the subsystems. Hence states equivalent to permutation operators are untouched by collective unitary noise. A trivial observation shows that uniform averaging over permutation operators projects the state into a form with block-diagonal structure equivalent to the one of the collective action of the unitary group. We introduce a name for this property: duality of averaging. The mathematical reason behind this duality is the fact that the collective action of the unitary group on the tensor product state space of a multipartite quantum system and the action of the permutation operations are mutual commutants when treated as matrix algebras. Such pairs of matrix algebras are known as dual reductive pairs. In this work we show, that in the case of finite dimensional quantum systems such duality of averaging holds for any pairs of symmetry groups being dual reductive pairs, regardless of whether they are compact or not, as long as the averaging operation is defined via iterated integral over the Cartan decomposition of the group action. Although our result is very general, we focus much attention on the concrete example of a dual reductive pair consisting of collective action of special linear matrices and permutation operations, which physically corresponds to averaging multipartite quantum states over non-unitary SLOCC-type (Stochastic Local Operations and Classical Communication) operations. In this context we show, that noiseless subsystems known from collective unitary averaging persist in the case of SLOCC averaging in a conditional way: upon postselection to specific invariant subspaces.

Renormalized Von Neumann entropy with application to entanglement in genuine infinite dimensional systems

A renormalized version of the von Neumann quantum entropy (which is finite and continuous in general, infinite dimensional case) which obeys several of the natural physical demands (as expected for a “good” measure of entanglement in the case of general quantum states describing bipartite and infinite-dimensional systems) is proposed. The renormalized quantum entropy is defined by the explicit use of the Fredholm determinants theory. To prove the main results on continuity and finiteness of the introduced renormalization, the fundamental Grothendick approach, which is based on the infinite dimensional Grassmann algebra theory, is applied. Several features of majorization theory are preserved under the introduced renormalization as it is proved in this paper. This fact allows to extend most of the known (mainly, in the context of two-partite, finite-dimensional quantum systems) results of the LOCC comparison theory to the case of genuine infinite-dimensional, two-partite quantum systems.

Quantum ergotropy and quantum feedback control.

We study the energy extraction from and charging to a finite-dimensional quantum system by general quantum operations. We prove that the changes in energy induced by unital quantum operations are limited by the ergotropy and charging bounds for unitary quantum operations. This implies that, in order to break the ergotropy bound for unitary quantum operations, one needs to perform a quantum operation with feedback control. We also show that the ergotropy bound for unital quantum operations, applied to initial thermal equilibrium states, is tighter than the inequality representing the standard second law of thermodynamics without feedback control.

A non-commutative entropic optimal transport approach to quantum composite systems at positive temperature

This paper establishes new connections between many-body quantum systems, One-body Reduced Density Matrices Functional Theory (1RDMFT) and Optimal Transport (OT), by interpreting the problem of computing the ground-state energy of a finite-dimensional composite quantum system at positive temperature as a non-commutative entropy regularized Optimal Transport problem. We develop a new approach to fully characterize the dual-primal solutions in such non-commutative setting. The mathematical formalism is particularly relevant in quantum chemistry: numerical realizations of the many-electron ground-state energy can be computed via a non-commutative version of Sinkhorn algorithm. Our approach allows to prove convergence and robustness of this algorithm, which, to our best knowledge, were unknown even in the two marginal case. Our methods are based on a priori estimates in the dual problem, which we believe to be of independent interest. Finally, the above results are extended in 1RDMFT setting, where bosonic or fermionic symmetry conditions are enforced on the problem.

Computing the Integrated Information of a Quantum Mechanism.

Originally conceived as a theory of consciousness, integrated information theory (IIT) provides a theoretical framework intended to characterize the compositional causal information that a system, in its current state, specifies about itself. However, it remains to be determined whether IIT as a theory of consciousness is compatible with quantum mechanics as a theory of microphysics. Here, we present an extension of IIT's latest formalism to evaluate the mechanism integrated information (φ) of a system subset to discrete, finite-dimensional quantum systems (e.g., quantum logic gates). To that end, we translate a recently developed, unique measure of intrinsic information into a density matrix formulation and extend the notion of conditional independence to accommodate quantum entanglement. The compositional nature of the IIT analysis might shed some light on the internal structure of composite quantum states and operators that cannot be obtained using standard information-theoretical analysis. Finally, our results should inform theoretical arguments about the link between consciousness, causation, and physics from the classical to the quantum.

Quantum speed limits in arbitrary phase spaces

Quantum speed limits (QSLs) provide an upper bound for the speed of evolution of quantum states in any physical process. Based on the Stratonovich-Weyl correspondence, we derive a universal QSL bound in arbitrary phase spaces that is applicable for both continuous variable systems and finite-dimensional discrete quantum systems. This QSL bound allows the determination of speed limit bounds in specific phase spaces that are tighter than those in Wigner phase space or Hilbert space under the same metric, as illustrated by several typical examples, e.g., a single-mode free field and $N$-level quantum systems in phase spaces. This QSL bound also provides an experimentally realizable way to examine the speed limit in phase spaces relevant to applications in quantum information and quantum optics.

Mixedness timescale in non-Hermitian quantum systems

We discuss the short-time perturbative expansion of the linear entropy for finite-dimensional quantum systems whose dynamics can be effectively described by a non-Hermitian Hamiltonian. We derive a timescale for the degree of mixedness for an input state undergoing non-Hermitian dynamics and specialize these results in the case of a driven-dissipative two-level system. Next, we derive a timescale for the growth of mixedness for bipartite quantum systems that depends on the effective non-Hermitian Hamiltonian. In the Hermitian limit, this result recovers the perturbative expansion for coherence loss in Hermitian systems, while it provides an entanglement timescale for initial pure and uncorrelated states. To illustrate these findings, we consider the many-body transverse-field XY Hamiltonian coupled to an imaginary all-to-all Ising model. We find that the non-Hermitian Hamiltonian enhances the short-time dynamics of the linear entropy for the considered input states. Overall, each timescale depends on minimal ingredients such as the probe state and the non-Hermitian Hamiltonian of the system, and its evaluation requires low computational cost. Our results find applications to non-Hermitian quantum sensing, quantum thermodynamics of non-Hermitian systems, and $\mathcal{PT}$-symmetric quantum field theory.

Lyapunov control of finite-dimensional quantum systems based on bi-objective quantum-behaved particle swarm optimization algorithm

This paper presents a Lyapunov control scheme to drive finite-dimensional closed and Markovian open quantum systems into any target pure state with as high fidelity and as little time as possible. The control law is established by a Lyapunov function with an unknown Hermitian operator, where the eigenvectors of the operator are constructed based on the condition that the LaSalle invariant set contains the desired target state and a set of optimal eigenvalues of the Hermitian operator is searched by the quantum-behaved particle swarm optimization (QPSO) algorithm. In particular, for open systems, when the denominator in the control law is quite small, we propose an improved QPSO algorithm with constraints to enhance the flexibility in implementation. Finally, numerical simulations are carried out on a five-level and a three-qubit closed quantum systems, and a five-level and a ten-level open quantum systems. Simulation results demonstrate that the proposed control scheme can achieve high-fidelity transfer to desired target states for high-dimensional closed and open quantum systems.

Non-Perturbative Treatment of Open-System Multi-Time Expectation Values in Gaussian Bosonic Environments

We determine the conditions for the equivalence between the multi-time expectation values of a general finite-dimensional open quantum system when interacting with, respectively, an environment undergoing a free unitary evolution or a discrete environment under a free evolution fixed by a proper Gorini-Kossakowski-Lindblad-Sudarshan generator. We prove that the equivalence holds if both environments are bosonic and Gaussian and if the one- and two-time correlation functions of the corresponding interaction operators are the same at all times. This result leads to a non-perturbative evaluation of the multi-time expectation values of operators and maps of open quantum systems interacting with a continuous set of bosonic modes by means of a limited number of damped modes, thus setting the ground for the investigation of open-system multi-time quantities in fully general regimes.

Control of Quantum Noise: On the Role of Dilations

We show that every finite-dimensional quantum system with Markovian (i.e. GKLS-generated) time evolution has an autonomous unitary dilation which can be dynamically decoupled. Since there is also always an autonomous unitary dilation which cannot be dynamically decoupled, this highlights the role of dilations in the control of quantum noise. We construct our dilation via a time-dependent version of Stinespring in combination with Howland’s clock Hamiltonian and certain point-localised states, which may be regarded as a C*-algebraic analogue of improper bra-ket position eigenstates and which are hence of independent mathematical and physical interest.

Moving quantum states without swap via intermediate higher-dimensional qudits

Quantum algorithms can be realized in the form of a quantum circuit. To map a quantum circuit for a specific quantum algorithm to quantum hardware, qubit mapping is an imperative technique based on the qubit topology. Due to the neighborhood constraint of qubit topology, the implementation of the quantum algorithm rightly, is essential for moving information around in a quantum computer. Swapping of qubits using a swap gate moves the quantum state between two qubits and solves the neighborhood constraint of qubit topology. Although, one needs to decompose the swap gate into three controlled-not gates to implement the swap gate efficiently, but unwillingly quantum cost with respect to the gate count and depth increases. In this paper, a formalism of moving quantum states without using swap operation is introduced. Moving quantum states through qubits have been attained with the adoption of temporary intermediate qudit states. This introduction of intermediate qudit states has exhibited a three times reduction in quantum cost with respect to the gate count and approximately two times reduction with respect to circuit depth compared to the state-of-the-art approach of the swap gate insertion. We also exhibit that the adoption of the intermediate qudit makes the approach sublimer than the existing works by obtaining a better fidelity. Furthermore, the proposed approach is generalized to any finite-dimensional quantum system.

A canonical Hamiltonian for open quantum systems

If an open quantum system is initially uncorrelated from its environment, then its dynamics can be written in terms of a Lindblad-form master equation. The master equation is divided into a unitary piece, represented by an effective Hamiltonian, and a dissipative piece, represented by a hermiticity-preserving superoperator; however, the division of open system dynamics into unitary and dissipative pieces is non-unique. For finite-dimensional quantum systems, we resolve this non-uniqueness by specifying a norm on the space of dissipative superoperators and defining the canonical Hamiltonian to be the one whose dissipator is minimal. We show that the canonical Hamiltonian thus defined is equivalent to the Hamiltonian initially defined by Lindblad, and that it is uniquely specified by requiring the dissipator’s jump operators to be traceless, extending a uniqueness result known previously in the special case of Markovian master equations. For a system weakly coupled to its environment, we give a recursive formula for computing the canonical effective Hamiltonian to arbitrary orders in perturbation theory, which we can think of as a perturbative scheme for renormalizing the system’s bare Hamiltonian.

Embedding classical dynamics in a quantum computer

We develop a framework for simulating measure-preserving, ergodic dynamical systems on a quantum computer. Our approach provides an operator-theoretic representation of classical dynamics by combining ergodic theory with quantum information science. The resulting quantum embedding of classical dynamics (QECD) enables efficient simulation of spaces of classical observables with exponentially large dimension using a quadratic number of quantum gates. The QECD framework is based on a quantum feature map that we introduce for representing classical states by density operators on a reproducing kernel Hilbert space, $\mathcal{H}$. Furthermore, an embedding of classical observables into self-adjoint operators on $\mathcal{H}$ is established, such that quantum mechanical expectation values are consistent with pointwise function evaluation. In this scheme, quantum states and observables evolve unitarily under the lifted action of Koopman evolution operators of the classical system. Moreover, by virtue of the reproducing property of $\mathcal{H}$, the quantum system is pointwise-consistent with the underlying classical dynamics. To achieve a quantum computational advantage, we project the state of the quantum system onto a finite-rank density operator on a ${2}^{n}$-dimensional tensor product Hilbert space associated with $n$ qubits. By employing discrete Fourier-Walsh transforms of spectral functions, the evolution operator of the finite-dimensional quantum system is factorized into tensor product form, enabling implementation through an $n$-channel quantum circuit of size $O(n)$ and no interchannel communication. Furthermore, the circuit features a state preparation stage, also of size $O(n)$, and a quantum Fourier transform stage of size $O({n}^{2})$, which makes predictions of observables possible by measurement in the standard computational basis. We prove theoretical convergence results for these predictions in the large-qubit limit, $n\ensuremath{\rightarrow}\ensuremath{\infty}$. In light of these properties, QECD provides a consistent simulator of the evolution of classical observables, realized through projective quantum measurement, which is able to simulate spaces of classical observables of dimension ${2}^{n}$ using circuits of size $O({n}^{2})$. We demonstrate the consistency of the scheme in prototypical dynamical systems involving periodic and quasiperiodic oscillators on tori. These examples include simulated quantum circuit experiments in Qiskit Aer, as well as actual experiments on the IBM Quantum System One.

Dynamics of Open Quantum Systems II, Markovian Approximation

A finite-dimensional quantum system is coupled to a bath of oscillators in thermal equilibrium at temperatureT>0. We show that for fixed, small values of the coupling constantλ, the true reduced dynamics of the system is approximated by the completely positive, trace preserving Markovian semigroup generated by the Davies-Lindblad generator. The difference between the true and the Markovian dynamics isO(|λ|1/4)for all times, meaning that the solution of the Gorini-Kossakowski-Sudarshan-Lindblad master equation is approximating the true dynamics to accuracyO(|λ|1/4)for all times. Our method is based on a recently obtained expansion of the full system-bath propagator. It applies to reservoirs with correlation functions decaying in time as1/t4or faster, which is a significant improvement relative to the previously required exponential decay.

Parameterizing qudit states

Quantum systems with a finite number of states at all times have been a primary element of many physical models in nuclear and elementary particle physics, as well as in condensed matter physics. Today, however, due to a practical demand in the area of developing quantum technologies, a whole set of novel tasks for improving our understanding of the structure of finite-dimensional quantum systems has appeared. In the present article we will concentrate on one aspect of such studies related to the problem of explicit parameterization of state space of an NN-level quantum system. More precisely, we will discuss the problem of a practical description of the unitary SU(N){SU(N)}-invariant counterpart of the NN-level state space BN{\mathcal{B}_N}, i.e., the unitary orbit space BN/SU(N){B_N/SU(N)}. It will be demonstrated that the combination of well-known methods of the polynomial invariant theory and convex geometry provides useful parameterization for the elements of BN/SU(N){B_N/SU(N)}. To illustrate the general situation, a detailed description ofBN/SU(N){B_N/SU(N)} for low-level systems: qubit (N=2{N= 2}), qutrit (N=3{N=3}), quatrit (N=4{N= 4}) - will be given.

Canonical structure of A and B maps

In their seminal 1961 paper, Sudarshan, Mathews and Rau investigated properties of the dynamical A and B maps acting on n-dimensional quantum systems. The nature of dynamical maps in open quantum system evolutions has attracted great deal of attention in the later years. However, the novel paper on the A and B dynamical maps has not received its due attention. In this tutorial article, we review the properties of A and B forms associated with the dynamics of finite dimensional quantum systems. In particular, we investigate a canonical structure associated with the A form and establish its equivalence with the associated B form. We show that the canonical structure of the A form captures the completely positive (not completely positive) nature of the dynamics in a succinct manner. This feature is illustrated through physical examples of qubit channels.Quanta 2021; 10: 34–41.

Combinations of quantum observables and instruments

This article points out that observables and instruments can be combined in many ways that have natural and physical interpretations. We shall mainly concentrate on the mathematical properties of these combinations. After an introduction in section , section reviews the basic definitions and observables are considered in section . We study parts of observables, post-processing, generalized convex combinations, sequential products and tensor products. These combinations are extended to instruments in section . We consider properties of observables measured by combinations of instruments. We introduce four special types of instruments, namely Kraus, Lüders, trivial and semitrivial instruments. We study when these types are closed under various combinations. In this work, we only consider finite-dimensional quantum systems. A few of the results presented here have appeared in the author’s previous articles Gudder (2020 arXiv:2005.08117 [quant-ph], arXiv:2005.13642 [quant-ph], arXiv:2009.07371 [quant-ph]).