Noncommutative Measure Research Articles

An inductive bias from quantum mechanics: learning order effects with non-commuting measurements

There are two major approaches to building good machine learning algorithms: feeding lots of data into large models or picking a model class with an “inductive bias” that suits the structure of the data. When taking the second approach as a starting point to design quantum algorithms for machine learning, it is important to understand how mathematical structures in quantum mechanics can lead to useful inductive biases in quantum models. In this work, we bring a collection of theoretical evidence from the quantum cognition literature to the field of quantum machine learning to investigate how non-commutativity of quantum observables can help to learn data with “order effects,” such as the changes in human answering patterns when swapping the order of questions in a survey. We design a multi-task learning setting in which a generative quantum model consisting of sequential learnable measurements can be adapted to a given task — or question order — by changing the order of observables, and we provide artificial datasets inspired by human psychology to carry out our investigation. Our first experimental simulations show that in some cases the quantum model learns more non-commutativity as the amount of order effect present in the data is increased and that the quantum model can learn to generate better samples for unseen question orders when trained on others — both signs that the model architecture suits the task.

Quantum energetics of a noncommuting measurement

When a measurement observable does not commute with a quantum system's Hamiltonian, the energy of the measured system is typically not conserved during the measurement. Instead, energy can be transferred between the measured system and the meter. In this work, we experimentally investigate the energetics of noncommuting measurements in a circuit quantum electrodynamics system containing a transmon qubit embedded in a 3D microwave cavity. We show through spectral analysis of the cavity photons that a frequency shift is imparted on the probe, in balance with the associated energy changes of the qubit. Our experiment provides new insights into foundations of quantum measurement, as well as a better understanding of the key mechanisms at play in quantum energetics. Published by the American Physical Society 2024

Local quantum uncertainty and non-commutativity measure discord in two-mode photon-added entangled coherent states

Local quantum uncertainty and non-commutativity measure discord in two-mode photon-added entangled coherent states

Decoding the projective transverse field Ising model

The competition between noncommuting projective measurements in discrete quantum circuits can give rise to entanglement transitions. It separates a regime where initially stored quantum information survives the time evolution from a regime where the measurements destroy the quantum information. Here we study one such system---the projective transverse field Ising model---with a focus on its capabilities as a quantum error correction code. The idea is to interpret one type of measurement as an error and the other type as a syndrome measurement. We demonstrate that there is a finite threshold below which quantum information encoded in an initially entangled state can be retrieved reliably. In particular, we implement the maximum likelihood decoder to demonstrate that the error correction threshold is distinct from the entanglement transition. This implies that there is a finite regime where quantum information is protected by the projective dynamics, but cannot be retrieved by using syndrome measurements.

Entanglement transitions with free fermions

We use Majorana operators to study entanglement dynamics under random free fermion unitary evolution and projective measurements in one dimension. For certain choices of unitary evolution, namely, those which swap neighboring Majorana operators, and measurements of neighboring Majorana bilinears, one can map the evolution to the statistical model of completely packed loops with crossings (CPLC) and study the corresponding phase diagram. We generalize this model using the language of fermionic Gaussian states to a general free fermion unitary evolution acting on neighboring Majorana operators and numerically compute its phase diagram. We find that both the Goldstone and area-law phases persist in this new phase diagram, but with a shifted phase boundary. One important qualitative aspect of the new phase boundary is that even for the case of commuting measurements, the Goldstone phase persists up to a finite nonzero measurement rate. This is in contrast with the CPLC, in which noncommuting measurements are necessary for realizing the Goldstone phase. We also numerically compute the correlation length critical exponent at the transition, which we find to be near to that of the CPLC, and give a tentative symmetry-based explanation for some differences in the phase transition line between the CPLC and generalized models.

Noncommutative rational Clark measures

Abstract We characterize the noncommutative Aleksandrov–Clark measures and the minimal realization formulas of contractive and, in particular, isometric noncommutative rational multipliers of the Fock space. Here, the full Fock space over $\mathbb {C} ^d$ is defined as the Hilbert space of square-summable power series in several noncommuting (NC) formal variables, and we interpret this space as the noncommutative and multivariable analogue of the Hardy space of square-summable Taylor series in the complex unit disk. We further obtain analogues of several classical results in Aleksandrov–Clark measure theory for noncommutative and contractive rational multipliers. Noncommutative measures are defined as positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz algebra, the unital $C^*$ -algebra generated by the left creation operators on the full Fock space. Our results demonstrate that there is a fundamental relationship between NC Hardy space theory, representation theory of the Cuntz–Toeplitz and Cuntz algebras, and the emerging field of noncommutative rational functions.

Non-commutative measure theory: Henkin and analytic functionals on $$\mathrm {C}^*$$-algebras

Henkin functionals on non-commutative $$\mathrm {C}^*$$ -algebras have recently emerged as a pivotal link between operator theory and complex function theory in several variables. Our aim in this paper is characterize these functionals through a notion of absolute continuity, inspired by a seminal theorem of Cole and Range. To do this, we recast the problem as a question in non-commutative measure theory. We develop a Glicksberg–König–Seever decomposition of the dual space of a $$\mathrm {C}^*$$ -algebra into an absolutely continuous part and a singular part, relative to a fixed convex subset of states. Leveraging this tool, we show that Henkin functionals are absolutely continuous with respect to the so-called analytic functionals if and only if a certain compatibility condition is satisfied by the ambient weak- $$*$$ topology. In contrast with the classical setting, the issue of stability under absolute continuity is not automatic in this non-commutative framework, and we illustrate its key role in sharpening our description of Henkin functionals. Our machinery yields new insight when specialized to the multiplier algebras of the Drury–Arveson space and of the Dirichlet space, and to Popescu’s noncommutative disc algebra. As another application, we make a contribution to the theory of non-commutative peak and interpolation sets.

Fatou's theorem for non-commutative measures

A classical theorem of Fatou asserts that the Radon–Nikodym derivative of any finite and positive Borel measure, μ, with respect to Lebesgue measure on the complex unit circle, is recovered as the non-tangential limits of its Poisson transform in the complex unit disk. This positive harmonic Poisson transform is the real part of an analytic function whose Taylor coefficients are in fixed proportion to the conjugate moments of μ.Replacing Taylor series in one variable by power series in several non-commuting variables, we show that Fatou's Theorem and related results have natural extensions to the setting of positive harmonic functions in an open unit ball of several non-commuting matrix-variables, and a corresponding class of positive non-commutative (NC) measures. Here, an NC measure is any positive linear functional on a certain self-adjoint unital subspace of the Cuntz–Toeplitz algebra, the C⁎−algebra generated by the left creation operators on the full Fock space.

Quantum simultaneous measurement of non-commuting observables based on [formula omitted]-means clustering

Simultaneous measurement of non-commuting observables deepens the understanding of Heisenberg’s uncertainty principle and has significant applications to quantum feedback control theory. Here we use techniques from machine learning to define a measure for the degree of non-commuting behavior of quantum measurements coming from the initial and final states of the measurements. Both the measurement strength effect and the angle effect between two observables have been constructed by using the K-means clustering method to reflect the defined non-commutativity measure. The rationality of the measure is demonstrated and its application in obtaining the measurement limitation that physically realizable is presented. Our work helps to advance the understanding of quantum non-commuting measurement and quantum control theory.

Modelling of weak quantum measurements consistent with conservation laws

We discuss quantum mechanical detection models in the weak limit in the context of conservation laws of physical quantities. In particular, we analyze what kind of system–detector interaction can preserve the global conservation or the related symmetry, and how the final measurement on the detector affects the measured observable of the systems and its presumed conservation. It turns out that the order of noncommuting measurements results in observable differences on the level of third-order correlations functions.

Metrological complementarity reveals the Einstein-Podolsky-Rosen paradox

The Einstein-Podolsky-Rosen (EPR) paradox plays a fundamental role in our understanding of quantum mechanics, and is associated with the possibility of predicting the results of non-commuting measurements with a precision that seems to violate the uncertainty principle. This apparent contradiction to complementarity is made possible by nonclassical correlations stronger than entanglement, called steering. Quantum information recognises steering as an essential resource for a number of tasks but, contrary to entanglement, its role for metrology has so far remained unclear. Here, we formulate the EPR paradox in the framework of quantum metrology, showing that it enables the precise estimation of a local phase shift and of its generating observable. Employing a stricter formulation of quantum complementarity, we derive a criterion based on the quantum Fisher information that detects steering in a larger class of states than well-known uncertainty-based criteria. Our result identifies useful steering for quantum-enhanced precision measurements and allows one to uncover steering of non-Gaussian states in state-of-the-art experiments.

The Schrödinger problem on the non-commutative Fisher-Rao space

We present a self-contained and comprehensive study of the Fisher-Rao space of matrix-valued non-commutative probability measures, and of the related Hellinger space. Our non-commutative Fisher-Rao space is a natural generalization of the classical commutative Fisher-Rao space of probability measures and of the Bures-Wasserstein space of Hermitian positive-definite matrices. We introduce and justify a canonical entropy on the non-commutative Fisher-Rao space, which differs from the von Neumann entropy. We consequently derive the analogues of the heat flow, of the Fisher information, and of the dynamical Schrödinger problem. We show the \(\Gamma \)-convergence of the \(\varepsilon \)-Schrödinger problem towards the geodesic problem for the Fisher-Rao space, and, as a byproduct, the strict geodesic convexity of the entropy.

Entanglement transition in the projective transverse field Ising model

Discrete quantum trajectories of systems under random unitary gates and projective measurements have been shown to feature transitions in the entanglement scaling that are not encoded in the density matrix. In this paper, we study the projective transverse field Ising model, a stochastic model with two noncommuting projective measurements and no unitary dynamics. We numerically demonstrate that their competition drives an entanglement transition between two distinct steady states that both exhibit area law entanglement, and introduce a classical but nonlocal model that captures the entanglement dynamics completely. Exploiting a map to bond percolation, we argue that the critical system in one dimension is described by a conformal field theory, and derive the universal scaling of the entanglement entropy and the critical exponent for the scaling of the mutual information of two spins exactly. We conclude with an interpretation of the entanglement transition in the context of quantum error correction.

Lebesgue Decomposition of Non-Commutative Measures

AbstractWe extend the Lebesgue decomposition of positive measures with respect to Lebesgue measure on the complex unit circle to the non-commutative (NC) multi-variable setting of (positive) NC measures. These are positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz $C^{\ast }-$algebra, the $C^{\ast }-$algebra of the left creation operators on the full Fock space. This theory is fundamentally connected to the representation theory of the Cuntz and Cuntz–Toeplitz $C^{\ast }-$algebras; any *−representation of the Cuntz–Toeplitz $C^{\ast }-$algebra is obtained (up to unitary equivalence), by applying a Gelfand–Naimark–Segal construction to a positive NC measure. Our approach combines the theory of Lebesgue decomposition of sesquilinear forms in Hilbert space, Lebesgue decomposition of row isometries, free semigroup algebra theory, NC reproducing kernel Hilbert space theory, and NC Hardy space theory.

On C*-algebras from homoclinic equivalences on subshifts of finite type

Let G be an infinite countable group and A be a finite set. If ∑ ⊆ AG is a strongly irreducible subshift of finite type, we endow a locally compact and Hausdorff topology on the homoclinic equivalence relation ${\cal G}$ on ∑ and show that the reduced C*-algebra $C_r^*\left({\cal G} \right)$ of ${\cal G}$ is a unital simple approximately finite (AF)-dimensional C*-algebra. The shift action of G on ∑ induces a canonical automorphism action of G on the C*-algebra $C_r^*\left({\cal G} \right)$ . We give the notion of noncommutative dynamical entropy invariants for amenable group actions on C*-algebras, and show that, if G is an amenable group, then the noncommutative topological entropy of the canonical automorphism action of G on $C_r^*\left({\cal G} \right)$ is equal to the topology entropy of the shift action of G on ∑. We also establish the variational principle with respect to the noncommutative measure entropy and the topological entropy for the C*-dynamical system ( $C_r^*\left({\cal G} \right)$ , G).

Connes’s trace theorem for curved noncommutative tori: Application to scalar curvature

In this paper, we prove a version of Connes’s trace theorem for noncommutative tori of any dimension n ⩾ 2. This allows us to recover and improve earlier versions of this result in dimensions n = 2 and n = 4 by Fathizadeh and Khalkhali. We also recover Connes’s integration formula for flat noncommutative tori of McDonald, Sukochev, and Zanin. As a further application, we prove a curved version of this integration formula in terms of the Laplace–Beltrami operator defined by an arbitrary Riemannian metric. For the class of the so-called self-compatible Riemannian metrics (including the conformally flat metrics of Connes and Tretkoff), this shows that Connes’s noncommutative integral allows us to recover the Riemannian density. This exhibits a neat link between this notion of noncommutative integral and noncommutative measure theory in the sense of operator algebras. As an application of these results, we set up a natural notion of scalar curvature for curved noncommutative tori.

Revisiting the behavior of quantum correlations under nondissipative decoherence by means of the correlation matrix

The main aim of this work is to investigate the relation between the freezing of Quantum Discord and the behavior of the actual quantum correlations present in bipartite quantum states. In order to accomplish this objective, we use first the Fano representation of two-qubit states from which we can identify a correlation matrix containing the information about the classical and quantum correlations present in the bipartite quantum state. From the behavior of the elements of the correlation matrix before and after making measurements on one of the subsystems, we identify the classical and quantum correlations present in these bipartite states. Then, we use this correlation matrix as a tool to analyze the phenomenon of non-dissipative decoherence in two-qubit states with maximally mixed marginals in typical dynamic scenarios where freezing of standard Quantum Discord takes place. We find that under some initial conditions where freezing of quantum discord occurs, some quantum correlations instead may not remain constant. In order to further explore into these results we also compute for the first time a recently introduced non-commutativity measure of quantum correlations (NCMQC) to analyze the behavior of quantum correlations under the same scenarios of non-dissipative decoherence. In complete agreement with the results obtained by means of the correlation matrix, our results also show that quantum correlations, as measured by NCMQC, do not freeze. Thus, our results put at stake the usual interpretation that the freezing of Quantum Discord is equivalent to the freezing of the physical quantum correlations themselves. We conclude from our study that freezing of quantum discord may not always be identified as equivalent to the freezing of the actual quantum correlations. Thus, the identification of freezing of quantum discord as a useful resource for tasks of quantum information processing is called into question.

Study on Non-Commutativity Measure of Quantum Discord

In this paper, we are concerned with the non-commutativity measure of quantum discord. We first present an explicit expression of the non-commutativity measure of quantum discord in the two-qubit case. Then we compare the geometric quantum discords for two dynamic models with their non-commutativity measure of quantum discords. Furthermore, we show that the results conducted by the non-commutativity measure of quantum discord are different from those conducted by both or one of the Hilbert-Schmidt distance discord and trace distance discord. These intrinsic differences indicate that the non-commutativity measure of quantum discord is incompatible with at least one of the well-known geometric quantum discords in the quantitative and qualitative representation of quantum correlations.

The role of quantum coherence in non-equilibrium entropy production

Thermodynamic irreversibility is well characterized by the entropy production arising from non-equilibrium quantum processes. We show that the entropy production of a quantum system undergoing open-system dynamics can be formally split into a term that only depends on population unbalances, and one that is underpinned by quantum coherences. This allows us to identify a genuine quantum contribution to the entropy production in non-equilibrium quantum processes. We discuss how these features emerge both in Lindblad-Davies differential maps and finite maps subject to the constraints of thermal operations. We also show how this separation naturally leads to two independent entropic conservation laws for the global system-environment dynamics, one referring to the redistribution of populations between system and environment and the other describing how the coherence initially present in the system is distributed into local coherences in the environment and non-local coherences in the system-environment state. Finally, we discuss how the processing of quantum coherences and the incompatibility of non-commuting measurements leads to fundamental limitations in the description of quantum trajectories and fluctuation theorems.

Non-commutative measure of quantum correlations under local operations

We study some desirable properties of recently introduced measures of quantum correlations based on the amount of non-commutativity quantified by the Hilbert-Schmidt norm (Sci Rep 6:25241, 2016, and Quantum Inf. Process. 16:226, 2017). Specifically, we show that: 1) for any bipartite ($A+B$) state, the measures of quantum correlations with respect to subsystem $A$ are non-increasing under any Local Commutative Preserving Operation on subsystem $A$, and 2) for Bell diagonal states, the measures are non-increasing under arbitrary local operations on $B$. Our results accentuate the potentialities of such measures, and exhibit them as valid monotones in a resource theory of quantum correlations with free operations restricted to the appropriate local channels.