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Cordes' characterization of Heisenberg-smooth operators bridges a gap between the theory of pseudo-differential operators and quantum harmonic analysis (QHA). We give a new proof of the result by using the phase space formalism of QHA. Our argument is flexible enough to generalize Cordes' result in several directions: (1) We can admit general quantization schemes, (2) allow for other phase space geometries, (3) obtain Schatten class analogs of the result, and (4) are able to characterize precisely `Heisenberg-analytic' operators. For (3), we use QHA to derive Schatten versions of the Calder\'on-Vaillancourt theorem, which might be of independent interest., Comment: 26 pages, comments are welcome

We develop the theory of local operations and classical communication (LOCC) for bipartite quantum systems represented by commuting von Neumann algebras. Our central result is the extension of Nielsen's Theorem to arbitrary factors. As in the matrix algebra case, the LOCC ordering of bipartite pure states is connected to the majorization of their restrictions. Our theorem implies that, in a bipartite system modeled by commuting factors in Haag duality, a) all states have infinite one-shot entanglement if and only if the local factors are not of type I, b) type III factors are characterized by LOCC transitions of arbitrary precision between any two pure states, and c) the latter holds even without classical communication for type III$_{1}$ factors. In the case of semifinite factors, the usual construction of entanglement monotones carries over using majorization theory. In the appendix, we provide a self-contained treatment of majorization on semifinite von Neumann algebras and $\sigma$-finite measure spaces., Comment: 35+13+5 pages, 1 figure

Embezzlement of entanglement refers to the task of extracting entanglement from an entanglement resource via local operations and without communication while perturbing the resource arbitrarily little. Recently, the existence of embezzling states of bipartite systems of type III von Neumann algebras was shown. However, both the multipartite case and the precise relation between embezzling states and the notion of embezzling families, as originally defined by van Dam and Hayden, was left open. Here, we show that finite-dimensional approximations of multipartite embezzling states form multipartite embezzling families. In contrast, not every embezzling family converges to an embezzling state. We identify an additional consistency condition that ensures that an embezzling family converges to an embezzling state. This criterion distinguishes the embezzling family of van Dam and Hayden from the one by Leung, Toner, and Watrous. The latter generalizes to the multipartite setting. By taking a limit, we obtain a multipartite system of commuting type III$_1$ factors on which every state is an embezzling state. We discuss our results in the context of quantum field theory and quantum many-body physics. As open problems, we ask whether vacua of relativistic quantum fields in more than two spacetime dimensions are multipartite embezzling states and whether multipartite embezzlement allows for an operator-algebraic characterization., Comment: 34 pages, 2 figures, comments welcome

We study convergence rates of the Trotter-Kato splitting $e^{A+L} = \lim_{n \to \infty} (e^{L/n} e^{A/n})^n$ in the strong operator topology. In the first part, we use complex interpolation theory to treat generators $L$ and $A$ of contraction semigroups on Banach spaces, with $L$ relatively $A$-bounded. In the second part, we study unitary dynamics on Hilbert spaces and develop a new technique based on the concept of energy constraints. Our results provide a complete picture of the convergence rates for the Trotter splitting for all common types of Schr\"odinger and Dirac operators, including singular, confining and magnetic vector potentials, as well as molecular many-body Hamiltonians in dimension $d=3$. Using the Brezis-Mironescu inequality, we derive convergence rates for the Schr\"odinger operator with $V(x)=\pm |x|^{-a}$ potential. In each case, our conditions are fully explicit., Comment: 49 pages, 8 figures, comments welcome

Universal embezzlers are bipartite quantum systems from which any entangled state may be extracted to arbitrary precision using local operations while perturbing the state of the system arbitrarily little. Here, we show that universal embezzlers are ubiquitous in many-body physics: The ground state sector of every local, translation-invariant, and critical free-fermionic many-body system on a one-dimensional lattice is a universal embezzler if bi-partitioned into two half-chains. The same property holds in locally-interacting, dual spin chains via the Jordan-Wigner transformation. Universal embezzlement manifests already for finite system sizes, not only in the thermodynamic limit: For any finite error and any targeted entangled state, a finite length of the chain is sufficient to embezzle said state within the given error. On a technical level, our main result establishes that the half-chain observable algebras associated with ground state sectors of the given models are type III$_1$ factors., Comment: 27 pages, 1 figure, comments welcome; minor corrections and improvements, results unaltered (v2)

We consider quantum systems with energy constraints. In general, quantum channels and continuous-time dynamics need not satisfy energy conservation. Physically meaningful channels, however, can only introduce a finite amount of energy to the system, and continuous-time dynamics may only increase the energy gradually over time. We systematically study such "energy-limited" channels and dynamics. For Markovian dynamics, energy-limitedness is equivalent to a single operator inequality in the Heisenberg picture. By tracking the output energy, we observe that the energy-constrained operator and diamond norms of Shirokov and Winter satisfy submultiplicativity estimates with respect to energy-limited channels. This makes for a powerful toolkit for quantitative analyses of dynamical problems in finite and infinite-dimensional systems. As an application, we derive state-dependent bounds for quantum speed limits and related problems that outperform the usual operator/diamond norm estimates, which have to account for fluctuations in high-energy states., Comment: Comments welcome

We study the quantum information theoretic task of embezzlement of entanglement in the setting of von Neumann algebras. Given a shared entangled resource state, this task asks to produce arbitrary entangled states using local operations without communication while perturbing the resource arbitrarily little. We quantify the performance of a given resource state by the worst-case error. States for which the latter vanishes are 'embezzling states' as they allow to embezzle arbitrary entangled states with arbitrarily small error. The best and worst performance among all states defines two algebraic invariants for von Neumann algebras. The first invariant takes only two values. Either it vanishes and embezzling states exist, which can only happen in type III, or no state allows for nontrivial embezzlement. In the case of factors not of finite type I, the second invariant equals the diameter of the state space. This provides a quantitative operational interpretation of Connes' classification of type III factors within quantum information theory. Type III$_1$ factors are 'universal embezzlers' where every state is embezzling. Our findings have implications for relativistic quantum field theory, where type III algebras naturally appear. For instance, they explain the maximal violation of Bell inequalities in the vacuum. Our results follow from a one-to-one correspondence between embezzling states and invariant probability measures on the flow of weights. We also establish that universally embezzling ITPFI factors are of type III$_1$ by elementary arguments., Comment: See arXiv:2401.07292 for an overview article; 68 pages + 1 table + 1 figure; comments welcome; v3: resolved open problems

Embezzlement of entanglement refers to the counterintuitive possibility of extracting entangled quantum states from a reference state of an auxiliary system (the "embezzler") via local quantum operations while hardly perturbing the latter. We uncover a deep connection between the operational task of embezzling entanglement and the mathematical classification of von Neumann algebras. Our result implies that relativistic quantum fields are universal embezzlers: Any entangled state of any dimension can be embezzled from them with arbitrary precision. This provides an operational characterization of the infinite amount of entanglement present in the vacuum state of relativistic quantum field theories., Comment: Overview paper accompanying the technical manuscript arXiv:2401.07299; v2: improved presentation and added an End Matter including a short introduction to von Neumann algebraic Quantum Information Theory

In quantum information theory, the Schmidt rank is a fundamental measure for the entanglement dimension of a pure bipartite state. Its natural definition uses the Schmidt decomposition of vectors on bipartite Hilbert spaces, which does not exist (or at least is not canonically given) if the observable algebras of the local systems are allowed to be general C*-algebras. In this work, we generalize the Schmidt rank to the commuting operator framework where the joint system is not necessarily described by the minimal tensor product but by a general bipartite algebra. We give algebraic and operational definitions for the Schmidt rank and show their equivalence. We analyze bipartite states and compute the Schmidt rank in several examples: The vacuum in quantum field theory, Araki-Woods-Powers states, as well as ground states and translation invariant states on spin chains which are viewed as bipartite systems for the left and right half chains. We conclude with a list of open problems for the commuting operator framework., Comment: 44 pages, 3 figures

Many features of physical systems, both qualitative and quantitative, become sharply defined or tractable only in some limiting situation. Examples are phase transitions in the thermodynamic limit, the emergence of classical mechanics from quantum theory at large action, and continuum quantum field theory arising from renormalization group fixed points. It would seem that few methods can be useful in such diverse applications. However, we here present a flexible modeling tool for the limit of theories: soft inductive limits constituting a generalization of inductive limits of Banach spaces. In this context, general criteria for the convergence of dynamics will be formulated, and these criteria will be shown to apply in the situations mentioned and more., Comment: Comments welcome

A set of classical or quantum states is equivalent to another one if there exists a pair of classical or quantum channels mapping either set to the other one. For dichotomies (pairs of states), this is closely connected to (classical or quantum) R\'enyi divergences (RD) and the data-processing inequality: If a RD remains unchanged when a channel is applied to the dichotomy, then there is a recovery channel mapping the image back to the initial dichotomy. Here, we prove for classical dichotomies that equality of the RDs alone is already sufficient for the existence of a channel in any of the two directions and discuss some applications. In the quantum case, all families of quantum RDs are seen to be insufficient because they cannot detect anti-unitary transformations. Thus, including anti-unitaries, we pose the problem of finding a sufficient family. It is shown that the Petz and maximal quantum RD are still insufficient in this more general sense and we provide evidence for sufficiency of the minimal quantum RD. As a side result of our techniques, we obtain an infinite list of inequalities fulfilled by the classical, the Petz quantum, and the maximal quantum RDs. These inequalities are not true for the minimal quantum RDs. Our results further imply that any sufficient set of conditions for state transitions in the resource theory of athermality must be able to detect time-reversal., Comment: Comments welcome, 40 pages, major changes, modified conjecture

We prove new criteria for essential self-adjointness of pseudodifferential operators which do not involve ellipticity type assumptions. For example, we show that self-adjointness holds in case that the symbol is $C^{2d+3}$ with derivatives of order two and higher being uniformly bounded. These results also apply to hermitian operator-valued symbols on infinite-dimensional Hilbert spaces which are important to applications in physics. Our method relies on a phase space differential calculus for quadratic forms on $L^2(\mathbb{R}^d)$, Calder\'on-Vaillancourt type theorems and a recent self-adjointness result for Toeplitz operators on the Segal-Bargmann space., Comment: 7 pages. Comments welcome. v2: improved presentation

A fundamental problem in quantum engineering is determining the lowest time required to ensure that all possible unitaries can be generated with the tools available, which is one of a number of possible quantum speed limits. We examine this problem from the perspective of quantum control, where the system of interest is described by a drift Hamiltonian and set of control Hamiltonians. Our approach uses a combination of Lie algebra theory, Lie groups and differential geometry, and formulates the problem in terms of geodesics on a differentiable manifold. We provide explicit lower bounds on the quantum speed limit for the case of an arbitrary drift, requiring only that the control Hamiltonians generate a topologically closed subgroup of the full unitary group, and formulate criteria as to when our expression for the speed limit is exact and not merely a lower bound. These analytic results are then tested and confirmed using a numerical optimization scheme. Finally we extend the analysis to find a lower bound on the quantum speed limit in the common case where the system is described by a drift Hamiltonian and a single control Hamiltonian., Comment: 13 pages

Catalysts are quantum systems that open up dynamical pathways between quantum states which are otherwise inaccessible under a given set of operational restrictions while, at the same time, they do not change their quantum state. We here consider the restrictions imposed by symmetries and conservation laws, where any quantum channel has to be covariant with respect to the unitary representation of a symmetry group, and present two results. First, for an exact catalyst to be useful, it has to build up correlations to either the system of interest or the degrees of freedom dilating the given process to covariant unitary dynamics. This explains why catalysts in pure states are useless. Second, if a quantum system ("reference frame") is used to simulate to high precision unitary dynamics (which possibly violates the conservation law) on another system via a global, covariant quantum channel, then this channel can be chosen so that the reference frame is approximately catalytic. In other words, a reference frame that simulates unitary dynamics to high precision degrades only very little., Comment: 12+1 pages. Accepted for Publication at Quantum - the open journal for quantum science

We provide state-dependent error bounds for strongly continuous unitary representations of connected Lie groups. That is, we bound the difference of two unitaries applied to a state in terms of the energy with respect to a reference Hamiltonian associated to the representation and a left-invariant metric distance on the group. Our method works for any connected Lie group and the metric is independent of the chosen representation. The approach also applies to projective representations and allows us to provide bounds on the energy constrained diamond norm distance of any suitably continuous channel representation of the group., Comment: 31 pages, 2 figures

Despite its popularity, several empirical and theoretical studies suggest that the quantum approximate optimization algorithm (QAOA) has persistent issues in providing a substantial practical advantage. So far, those findings mostly account for a regime of few qubits and shallow circuits. We find clear evidence for a `no free lunch'-behavior of QAOA on a general optimization task with no further structure; individual cases have, however, to be analyzed more carefully. We propose and justify a performance indicator for the deep-circuit QAOA that can be accessed by solely evaluating statistical properties of the classical objective function. We further discuss the various favorable properties a generic QAOA instance has in the asymptotic regime of infinitely many gates, and elaborate on the immanent drawbacks of finite circuits. We provide several numerical examples of a deep-circuit QAOA method based on local search strategies and find that - in alignment with our performance indicator - some special function classes, like QUBO, admit a favorable optimization landscape., Comment: 19 pages, 13 figures

The Trotter product formula is a key instrument in numerical simulations of quantum systems. However, computers cannot deal with continuous degrees of freedom, such as the position of particles in molecules, or the amplitude of electromagnetic fields. It is therefore necessary to discretize these variables to make them amenable to digital simulations. Here, we give sufficient conditions to conclude the validity of this approximate discretized physics. Essentially, it depends on the state-dependent Trotter error, for which we establish explicit bounds that are also of independent interest., Comment: 16 pages, 5 figures

We prove a new criterion that guarantees self-adjointness of Toeplitz operator with unbounded operator-valued symbols. Our criterion applies, in particular, to symbols with Lipschitz continuous derivatives, which is the natural class of Hamiltonian functions for classical mechanics. For this we extend the Berger-Coburn estimate to the case of vector-valued Segal-Bargmann spaces. Finally, we apply our result to prove self-adjointness for a class of (operator-valued) quadratic forms on the space of Schwartz functions in the Schr\"odinger representation., Comment: 17 pages