While the expected time of first visit to a subspace under some quantum evolution is a relevant statistic, it is also important to confront this information with the associated variance, so that one may have a better understanding of how representative such means are. In this work, we study these matters in the context of quantum channels acting on finite-dimensional Hilbert spaces, and we provide practical methods for calculating the above data with respect to goal subspaces of interest. By considering an assumption which is strictly weaker than irreducibility, we are also able to consider unitary quantum walks. Our approach is based on the monitored evolution of positive, trace-preserving operators given by quantum Markov chains as described by Gudder. Then, given any channel and some goal subspace of interest, one may induce an absorbing chain in a natural way so that the monitored evolution of the original map can be examined.

Abstract We present a systematic quantization scheme for bounded symplectic domains of the form D × G ⊂ T ∗ G , where D ⊂ g ∗ is a bounded region defined by algebraic inequalities and G is a compact Lie group with Lie algebra g . The finiteness of the symplectic volume implies that quantization yields a finite-dimensional Hilbert space, with observables represented by Hermitian matrices, for which we provide an explicit realization. Boundary effects necessitate modifications of the standard von Neumann and Dirac conditions, which usually underlie the correspondence principle. Physically, the compact group G plays the role of momentum space, while g ∗ corresponds to the (noncommutative) position space of a particle. The assumption of compact momentum space has profound physical consequences, including the supertunneling phenomenon and the emergence of a maximal fermion density.

In this paper, we propose two projected dynamical systems for solving inverse quasi-variational inequality problems in finite-dimensional Hilbert spaces–one ensuring finite-time stability and the other guaranteeing fixed-time stability. We first establish the connection between these dynamical systems and the solutions of inverse quasi-variational problems. Then, under mild conditions on the operators and parameters, we analyze the finite-time and fixed-time stability of the proposed systems. Both approaches offer accelerated convergence; however, while the settling time of a finite-time stable dynamical system depends on initial conditions, the fixed-time stable system achieves convergence within a predefined time, independent of initial conditions. Furthermore, we consider an explicit forward Euler discretization of the dynamical system, ensuring a consistent discretization of the fixed-time stable dynamics. This leads to a novel forward-backward algorithm, for which we present a detailed convergence analysis. To demonstrate their effectiveness, we provide numerical experiments, including an application to the traffic assignment problem.

Abstract Traced monoidal categories model processes that can feed their outputs back to their own inputs, abstracting iteration. The category of finite-dimensional Hilbert spaces with the direct sum tensor is not traced. But surprisingly, in 2014, Bartha showed that the monoidal subcategory of isometries is traced. The same holds for coisometries, unitary maps, and contractions. This suggests the possibility of feeding outputs of quantum processes back to their own inputs, analogous to iteration. In this paper, we show that Bartha’s result is not specifically tied to Hilbert spaces, but works in any dagger additive category with Moore–Penrose pseudoinverses (a natural dagger-categorical generalization of inverses).

While quantum statistical mechanics triumphs in explaining many equilibrium phenomena, there is an increasing focus on going beyond conventional scenarios of thermalization. Traditionally examples of nonthermalizing systems are either integrable or disordered. Recently, examples of translationally invariant physical systems have been discovered whose excited energies avoid thermalization either due to local constraints (whether exact or emergent) or due to higher-form symmetries. In this article, we extend these investigations for the case of 3D U ( 1 ) quantum dimer models, which are lattice gauge theories with finite-dimensional local Hilbert spaces (also generically called quantum link models) with staggered charged static matter. Using a combination of analytical and numerical methods, we uncover a class of athermal states that arise in large winding sectors, when the system is subjected to external electric fields. The polarization of the dynamical fluxes in the direction of applied field traps excitations in 2D planes, while an interplay with the Gauss law constraint in the perpendicular direction causes exotic athermal behavior due to the emergence of new conserved quantities. This causes a geometric fragmentation of the system. We provide analytical arguments showing that the scaling of the number of fragments is exponential in the linear system size, leading to weak fragmentation. Further, we identify sectors which host fractonic excitations with severe mobility restrictions. The unitary evolution of fragments dominated by fractons is qualitatively different from the one dominated by nonfractonic excitations.

Given a quantum state in the finite-dimensional Hilbert space Cn, the range of possible values of a quantum observable is usually identified with the discrete spectrum of eigenvalues of a corresponding Hermitian matrix. Here any such observable is identified with (i) an 'ortho-measurable' function defined on the Boolean 'ortho-algebra' generated by the eigenspaces that form an orthogonal decomposition of Cn, and (ii) a 'numerically identified' orthogonal decomposition of Cn. The latter means that each subspace of the orthogonal decomposition can be uniquely identified by its own attached real number, just as each eigenspace of a Hermitian matrix can be uniquely identified by the corresponding eigenvalue. Furthermore, any density matrix on Cn is identified with a Bayesian prior 'ortho-probability' measure defined on the linear subspaces that make up the Boolean ortho-algebra induced by its eigenspaces. Then any pure quantum state is identified with a degenerate density matrix, and any mixed state with a probability measure on a set of orthogonal pure states. Finally, given any quantum observable, the relevant Bayesian posterior probabilities of measured outcomes can be found by the usual trace formula that extends Born's rule. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.

Abstract We present a self-testing protocol in a constrained prepare-measure scenario, based on a communication game known as parity-oblivious multiplexing (POM) task. In this scenario, a parity-oblivious constraint is imposed on the preparations, allowing us to define a classical bound derived from a preparation noncontextual ontological model. We demonstrate that the optimal quantum value exceeds preparation noncontextual bound and therefore enables self-testing of the preparations and the measurement devices. Contrasting the prevailing consensus that the self-testing in prepare-measure scenario inevitably requires an upper bound on the dimension of the quantum system, we derive the optimal quantum success probability of the POM task devoid of assuming the dimension of the quantum system. Furthermore, by proving the existence of a unitary, we show that the optimal preparations and measurements in an unknown but finite dimensional Hilbert space, responsible for the observed input–output correlations, can be mapped, via a unitary, onto a known finite-dimensional quantum system. Our results thus pave the way for scalable, single system based certification protocols in the prepare-measure scenario.

As a contribution to the field of quantum mereology, we study how a change of tensor product structure in a finite-dimensional Hilbert space affects its entanglement properties. In particular, we ask whether, given a time-evolving state, there exists a tensor product structure in which no entanglement is generated. We give a concrete, constructive example of disentangling tensor product structure in the case of a C-NOT gate evolution between two qbits, before showing that this cannot be achieved for most time-evolving quantum states.

We continue to study Pythagorean unitary representations of Richard Thompson’s groups F,T,V and their extension to the Cuntz(–Dixmier) algebra \mathcal{O} . Any linear isometry from a Hilbert space to its direct sum square produces such. We focus on those arising from a finite-dimensional Hilbert space. We show that they decompose as a direct sum of a so-called diffuse part and an atomic part. We previously proved that the diffuse part is Ind-mixing: it does not contain induced representations of finite-dimensional ones. In this article, we fully describe the atomic part it is a finite direct sum of irreducible monomial representations arising from a precise family of parabolic subgroups.

We study the Hamiltonian formulation of SU(2) Yang-Mills theory with staggered fermions in a (2+1)-dimensional small lattice system. We construct a gauge-invariant and finite-dimensional Hilbert space for the theory by applying the loop-string-hadron formulation and specifically map the model to a spin system. We classically emulate digital quantum simulation and observe the real-time evolution of the single-site entanglement entropy, the fermion entanglement entropy, and the fermion pair production.

ZX-calculus is Complete for Finite-Dimensional Hilbert Spaces

Abstract We introduce and study some special classes of ladder operators in finite-dimensional Hilbert spaces. In particular we consider a truncated version of quons, their pseudo-version, and a third family of operators acting on a closed chain. In this latter situation, we discuss the existence of what could be considered discrete coherent states, as suitable eigenvectors of the annihilation operator of the chain. We see that, under reasonable assumptions, a resolution of the identity can be recovered, involving these states, together with a biorthogonal family of vectors, which turn out to be eigenstates of the raising operator of the chain.

Corresponding to a finite dimensional Hilbert space $H$ with $\dim H=n$, we define a geometric algebra $\mathcal{G}(H)$ with $\dim\left[\mathcal{G}(H)\right]=2^n$. The algebra $\mathcal{G}(H)$ is a Hilbert space that contains $H$ as a subspace. We interpret the unit vectors of $H$ as states of individual fermions of the same type and $\mathcal{G}(H)$ as a fermion quantum field whose unit vectors represent states of collections of interacting fermions. We discuss creation operators on $\mathcal{G}(H)$ and provide their matrix representations. Evolution operators provided by self-adjoint Hamiltonians on $H$ and $\mathcal{G}(H)$ are considered. Boson-Fermion quantum fields are constructed. Extensions of operators from $H$ to $\mathcal{G}(H)$ are studied. Finally, we present a generalization of our work to infinite dimensional separable Hilbert spaces. Quanta 2025; 14: 48–65.

In this paper, a quantum computational framework for algebraic topology based on simplicial set theory is presented. This extends previous work, which was limited to simplicial complexes and aimed mostly at topological data analysis. The proposed set-up applies to any parafinite simplicial set and proceeds by associating with it a finite dimensional simplicial Hilbert space, whose simplicial operator structure is studied in some depth. It is shown in particular how the problem of determining the simplicial set’s homology can be solved within the simplicial Hilbert framework. Further, the conditions under which simplicial set theoretic algorithms can be implemented in a quantum computational setting with finite resources are examined. Finally a quantum algorithmic scheme capable of computing the simplicial homology spaces and Betti numbers of a simplicial set combining a number of basic quantum algorithms is outlined.

We unravel a deep connection between limits of real numbers and limits in category theory. Using a new variant of the classical characterisation of the real numbers, we characterise the category of finite-dimensional Hilbert spaces and linear contractions in terms of simple category-theoretic structures and properties that do not refer to norms, continuity, or real numbers. This builds on Heunen, Kornell, and Van der Schaaf’s easier characterisation of the category of all Hilbert spaces and linear contractions.

Quantum skyrmions as topologically structured entangled states have the potential to be a pathway toward robustness against external perturbations, but so far no theoretical framework exists to validate this. Here, we introduce the notion of a new entanglement observable based on such topologies and develop a theoretical framework for studying its evolution in general quantum channels. Using photons entangled in orbital angular momentum and polarization as an example, we show that the noise affecting both photons can be recast as a position-dependent perturbation affecting only the photon in the polarization state, restricting the noise to a finite-dimensional Hilbert space. From this we predict complete resilience for both non-depolarizing and depolarizing noise, the former by rigorous arguments based on homotopic maps and the latter by numerical simulation. Finally, we identify sources of local noise that can destabilize the topology and suggest why this may be ignored in practical situations. Our work sets a solid foundational framework for understanding how and why topology enhances the resilience of such entanglement observables, with immediate relevance to the distribution of information through entanglement in noisy environments, such as quantum computers and quantum networks.

The exact response theory based on the Dissipation Function applies to general dynamical systems and has yielded excellent results in various applications. In this article, we propose a method to apply it to quantum mechanics. In many quantum systems, it has not yet been possible to overcome the perturbative approach, and the most developed theory is the linear one. Extensions of the exact response theory developed in the field of nonequilibrium molecular dynamics could prove useful in quantum mechanics, as perturbations of small systems or far-from-equilibrium states cannot always be taken as small perturbations. Here, we introduce a quantum analogue of the classical Dissipation Function. We then derive a quantum expression for the exact calculation of time-dependent expectation values of observables, in a form analogous to that of the classical theory. We restrict our analysis to finite-dimensional Hilbert spaces, for the sake of simplicity, and we apply our method to specific examples, like qubit systems, for which exact results can be obtained by standard techniques. This way, we prove the consistency of our approach with the existing methods, where they apply. Although not required for open systems, we propose a self-adjoint version of our Dissipation Operator, obtaining a second equivalent expression of response, where the contribution of an anti-self-adjoint operator appears. We conclude by using new formalism to solve the Lindblad equations, obtaining exact results for a specific case of qubit decoherence, and suggesting possible future developments of this work.

We investigate the possibility of reproducing the continuum physics of 2D SU(N) gauge theory coupled to a single flavor of massless Dirac fermion using qubit regularization. The continuum theory is described by N free fermions in the ultraviolet (UV) and a coset Wess-Zumino-Witten (WZW) model in the infrared (IR). In this work, we first explore how well these features can be reproduced using the Kogut-Susskind (KS) Hamiltonian with a finite-dimensional link Hilbert space and a generalized Hubbard coupling. We do this by analyzing the renormalization group (RG) flow diagram of the continuum theory and identifying important phases of the theory. Using strong coupling expansions, we show that our lattice model exhibits a gapped dimer phase and a spin-chain phase. Furthermore, for N=2, using tensor network methods, we show that there is a second-order phase transition between these two phases, which we identify as the critical surface of the continuum theory that connects the IR and UV fixed points. In the IR, we identify the critical theory at the transition as the expected SU(2)1 WZW model. Lastly, we argue that modifications of our model may allow the study of the UV physics of free fermions.

Chirality is a property of a gapped phase of matter in two spatial dimensions that can be manifested through nonzero thermal or electrical Hall conductance. In this Letter, we prove two no-go theorems that forbid such chirality for a quantum state in a finite dimensional local Hilbert space with strict area law entanglement entropies. We also show that the finite dimensional local Hilbert space condition can be relaxed to the condition that the state has finite local entanglement entropies. As a crucial ingredient in the proofs, we introduce a new quantum information-theoretic primitive called instantaneous modular flow, which has many other potential applications.

The effectiveness of a quantum battery relies on a robust charging process, yet these are often sensitive to the initial state of the battery. We introduce the concept of a universally charging (UC) protocol, defined as one that either increases or maintains the average battery energy for all initial states, without ever decreasing it. We show that UC protocols are impossible for closed quantum batteries, thus necessitating interactions with auxiliary quantum systems. To that end, we prove a no-go theorem that prohibits UC protocols for closed quantum batteries with finite-dimensional Hilbert spaces. Leveraging a no-go theorem for topological quantum walks, we argue that even for infinite-dimensional Hilbert spaces, while unitary UC operators exist, they cannot be generated by physically reasonable Hamiltonians. However, regardless of the dimension, non-unitary UC protocols can be achieved in open quantum batteries. To illustrate this, we present a general model with a control qubit, whose state interpolates between universal-charging and universal-discharging protocols.