Faithful State Research Articles

Eventually entanglement breaking divisible quantum dynamics

Abstract It is shown that a large class of quantum dynamical maps on complex matrix algebras governed by time-local Master Equations tend to become entanglement breaking in the course of time. Such situation seems to be generic for quantum evolution and in particular, completely positive dynamical semigroups with a unique faithful stationary state enjoy this property. Inspired by this observation, we propose a new concept of eventually entanglement breaking divisible (eEB-divisible) dynamics. A dynamical map is eEB-divisible if any propagator becomes entanglement breaking in finite time. It turns out that eEB-divisibility is quite general and holds for a large class of quantum evolutions.

On Haagerup noncommutative quasi [formula omitted] spaces

Let M be a σ-finite von Neumann algebra, equipped with a normal faithful state φ, and let A be a maximal subdiagonal subalgebra of M. We have proved that for 0<p<1, Hp(A) is independent of φ. Furthermore, in the case that A is a type 1 subdiagonal subalgebra, we have obtained an interpolation theorem for Hp(A) in the case where 0<θ<1, 0<p0,p1≤∞ and p=1−θp0+θp1≥1.

OneSC: A computational platform for recapitulating cell state transitions.

Computational modelling of cell state transitions has been a great interest of many in the field of developmental biology, cancer biology and cell fate engineering because it enables performing perturbation experiments in silico more rapidly and cheaply than could be achieved in a wet lab. Recent advancements in single-cell RNA sequencing (scRNA-seq) allow the capture of high-resolution snapshots of cell states as they transition along temporal trajectories. Using these high-throughput datasets, we can train computational models to generate in silico 'synthetic' cells that faithfully mimic the temporal trajectories. Here we present OneSC, a platform that can simulate synthetic cells across developmental trajectories using systems of stochastic differential equations govern by a core transcription factors (TFs) regulatory network. Different from the current network inference methods, OneSC prioritizes on generating Boolean network that produces faithful cell state transitions and steady cell states that mimic real biological systems. Applying OneSC to real data, we inferred a core TF network using a mouse myeloid progenitor scRNA-seq dataset and showed that the dynamical simulations of that network generate synthetic single-cell expression profiles that faithfully recapitulate the four myeloid differentiation trajectories going into differentiated cell states (erythrocytes, megakaryocytes, granulocytes and monocytes). Finally, through the in-silico perturbations of the mouse myeloid progenitor core network, we showed that OneSC can accurately predict cell fate decision biases of TF perturbations that closely match with previous experimental observations.

Pretty good measurement for bosonic Gaussian ensembles

The pretty good measurement is a fundamental analytical tool in quantum information theory, giving a method for inferring the classical label that identifies a quantum state chosen probabilistically from an ensemble. Identifying and constructing the pretty good measurement for the class of bosonic Gaussian states is of immediate practical relevance in quantum information processing tasks. Holevo recently showed that the pretty good measurement for a bosonic Gaussian ensemble is a bosonic Gaussian measurement that attains the accessible information of the ensemble [IEEE Trans. Inf. Theory 66(9) (2020) 5634]. In this paper, we provide an alternate proof of Gaussianity of the pretty good measurement for a Gaussian ensemble of multimode bosonic states, with a focus on establishing an explicit and efficiently computable Gaussian description of the measurement. We also compute an explicit form of the mean square error of the pretty good measurement, which is relevant when using it for parameter estimation. Generalizing the pretty good measurement is a quantum instrument, called the pretty good instrument. We prove that the post-measurement state of the pretty good instrument is a faithful Gaussian state if the input state is a faithful Gaussian state whose covariance matrix satisfies a certain condition. Combined with our previous finding for the pretty good measurement and provided that the same condition holds, it follows that the expected output state is a faithful Gaussian state as well. In this case, we compute an explicit Gaussian description of the post-measurement and expected output states. Our findings imply that the pretty good instrument for bosonic Gaussian ensembles is no longer merely an analytical tool, but that it can also be implemented experimentally in quantum optics laboratories.

Nonassociative Lp-spaces and embeddings in noncommutative Lp-spaces

We define a notion of nonassociative Lp-space associated to a JBW⁎-algebra (Jordan von Neumann algebra) equipped with a normal faithful state φ. In the particular case of JW⁎-algebras underlying von Neumann algebras, we connect these spaces to a complex interpolation theorem of Ricard and Xu on noncommutative Lp-spaces. We also make the link with the nonassociative Lp-spaces of Iochum associated to JBW-algebras and the investigation of contractively complemented subspaces of noncommutative Lp-spaces. More precisely, we show that our nonassociative Lp-spaces contain isometrically the Lp-spaces of Iochum and that all tracial nonassociative Lp-spaces from JW⁎-factors arise as positively contractively complemented subspaces of noncommutative Lp-spaces.

The Fell topology and the modular Gromov-Hausdorff propinquity

We show that any (norm-closed two-sided) ideal of a unital C*-algebra that comes equipped with certain natural quantum metric structure is a metrized quantum vector bundle, when canonically viewed as a module over A A . Next, given a unital AF-algebra A A equipped with a faithful tracial state, we equip each ideal of A A with a metrized quantum vector bundle structure using previous work of the first author and Latrémolière. Moreover, we show that convergence of ideals in the Fell topology implies convergence of the associated metrized quantum vector bundles in the modular Gromov-Hausdorff propinquity of Latrémolière. In a similar vein but requiring a different approach, given a compact metric space ( X , d ) (X,d) , we equip each ideal of C ( X ) C(X) with a metrized quantum vector bundle structure, and show that convergence in the Fell topology implies convergence in the modular Gromov-Hausdorff propinquity.

A tracial characterization of Furstenberg’s conjecture

AbstractWe investigate almost minimal actions of abelian groups and their crossed products. As an application, given multiplicatively independent integers p and q, we show that Furstenberg’s $\times p,\times q$ conjecture holds if and only if the canonical trace is the only faithful extreme tracial state on the $C^*$ -algebra of the group $\mathbb {Z}[\frac {1}{pq}]\rtimes \mathbb {Z}^2$ . We also compute the primitive ideal space and K-theory of $C^*(\mathbb {Z}[\frac {1}{pq}]\rtimes \mathbb {Z}^2)$ .

Challenges for Inline Observation Error Estimation in the Presence of Misspecified Background Uncertainty

Abstract For data assimilation to provide faithful state estimates for dynamical models, specifications of observation uncertainty need to be as accurate as possible. Innovation-based methods based on Desroziers diagnostics, are commonly used to estimate observation uncertainty, but such methods can depend greatly on the prescribed background uncertainty. For ensemble data assimilation, this uncertainty comes from statistics calculated from ensemble forecasts, which require inflation and localization to address under sampling. In this work, we use an ensemble Kalman filter (EnKF) with a low-dimensional Lorenz model to investigate the interplay between the Desroziers method and inflation. Two inflation techniques are used for this purpose: 1) a rigorously tuned fixed multiplicative scheme and 2) an adaptive state-space scheme. We document how inaccuracies in observation uncertainty affect errors in EnKF posteriors and study the combined impacts of misspecified initial observation uncertainty, sampling error, and model error on Desroziers estimates. We find that whether observation uncertainty is over- or underestimated greatly affects the stability of data assimilation and the accuracy of Desroziers estimates and that preference should be given to initial overestimates. Inline estimates of Desroziers tend to remove the dependence between ensemble spread–skill and the initially prescribed observation error. In addition, we find that the inclusion of model error introduces spurious correlations in observation uncertainty estimates. Further, we note that the adaptive inflation scheme is less robust than fixed inflation at mitigating multiple sources of error. Last, sampling error strongly exacerbates existing sources of error and greatly degrades EnKF estimates, which translates into biased Desroziers estimates of observation error covariance. Significance Statement To generate accurate predictions of various components of the Earth system, numerical models require an accurate specification of state variables at our current time. This step adopts a probabilistic consideration of our current state estimate versus information provided from environmental measurements of the true state. Various strategies exist for estimating uncertainty in observations within this framework, but are sensitive to a host of assumptions, which are investigated in this study.

Quantum Inverse Algorithm via Adaptive Variational Quantum Linear Solver: Applications to General Eigenstates.

We propose a quantum inverse algorithm (QInverse) to directly determine general eigenstates by repeatedly applying the inverse power of a shifted Hamiltonian to an arbitrary initial state. To properly deal with the strongly entangled inverse power states and the resultant excited states, we solved the underlying linear equation, both variationally and adaptively, to obtain a faithful inverse power state with a shallow quantum circuit. QInverse is singularity-free and successfully obtains the target excited states with an energy closest to the shift ω, which is difficult to reach using variational methods. We also propose a subspace expansion approach to accelerate convergence and show that it is helpful to determine the two nearest eigenvalues when they are equally close to ω. These approaches were compared with the folded-spectrum method, which aims to generate excited states through variational optimization. It is shown that, whereas the folded-spectrum approach often fails to predict the target state by falling into a local minimum owing to its variational features, the success rate and accuracy of our algorithms are systematically improvable.

Invertibility preserving mappings onto finite $C^*$-algebras

We prove that every surjective unital linear mapping which preserves invertible elements from a Banach algebra onto a $C^*$-algebra carrying a faithful tracial state is a Jordan homomorphism, thus generalising Aupetit’s 1998 result for finite von Neumann

Complete Gradient Estimates of Quantum Markov Semigroups

In this article we introduce a complete gradient estimate for symmetric quantum Markov semigroups on von Neumann algebras equipped with a normal faithful tracial state, which implies semi-convexity of the entropy with respect to the recently introduced noncommutative 2-Wasserstein distance. We show that this complete gradient estimate is stable under tensor products and free products and establish its validity for a number of examples. As an application we prove a complete modified logarithmic Sobolev inequality with optimal constant for Poisson-type semigroups on free group factors.

Property T C⁎-algebras with amenable tracial states

As a non-unital analogue of the main result in [6], we show in this article that if A is a separable quasi-central C⁎-algebra with property T and is nuclear, then there is a sequence of positive integers {nk}k∈N such that the c0-direct sum of the family {Mnk}k∈N of matrix algebras is an ideal JA of A and that A/JA has no tracial state. In particular, a separable C⁎-algebra A is a c0-direct sum of matrix algebras if and only if A is quasi-central, is nuclear, has property T and every unital simple quotient of A has a tracial state. We also show that a separable property TC⁎-algebra A is residually finite dimensional if and only if A admits an amenable faithful tracial state and every ideal of A has non-zero center.

INDUCTIVE LIMITS OF C*-ALGEBRAS AND COMPACT QUANTUM METRIC SPACES

AbstractGiven a unital inductive limit of C*-algebras for which each C*-algebra of the inductive sequence comes equipped with a Rieffel compact quantum metric, we produce sufficient conditions to build a compact quantum metric on the inductive limit from the quantum metrics on the inductive sequence by utilizing the completeness of the dual Gromov–Hausdorff propinquity of Latrémolière on compact quantum metric spaces. This allows us to place new quantum metrics on all unital approximately finite-dimensional (AF) algebras that extend our previous work with Latrémolière on unital AF algebras with faithful tracial state. As a consequence, we produce a continuous image of the entire Fell topology on the ideal space of any unital AF algebra in the dual Gromov–Hausdorff propinquity topology.

Emergent gauge symmetries and quantum operations

The algebraic approach to quantum physics emphasizes the role played by the structure of the algebra of observables and its relation to the space of states. An important feature of this point of view is that subsystems can be described by subalgebras, with partial trace being replaced by the more general notion of restriction to a subalgebra. This, in turn, has recently led to applications to the study of entanglement in systems of identical particles. In the course of those investigations on entanglement and particle identity, an emergent gauge symmetry has been found by Balachandran, de Queiroz and Vaidya. In this letter we establish a novel connection between that gauge symmetry, entropy production and quantum operations. Thus, let A be a system described by a finite dimensional observable algebra and a mixed faithful state. Using the Gelfand–Naimark–Segal (GNS) representation we construct a canonical purification of , allowing us to embed A into a larger system C. Using Tomita–Takesaki theory, we obtain a subsystem decomposition of C into subsystems A and B, without making use of any tensor product structure. We identify a group of transformations that acts as a gauge group on A while at the same time giving rise to entropy increasing quantum operations on C. We provide physical means to simulate this gauge symmetry/quantum operation duality.

The Role of Sufism in Spreading Islam and Fighting Atonement

This paper delivered the need for Islamic thought to confront the Takfiri thought, through the return of Sufi thought. Sufism does not have fatwa to kill others, Sufism talks about tolerance, the love of the Prophet, and the non-compulsion of others tojoin either Islam or the way the murid belongs to. Sufism represents the spiritual and faithful state of Islam, the core of Islam. This paper find out the role of Sufism in the spread of Islam; in the past, and the contemporary in the fight against atonement. The aim of this paper is to describe the culture of Sufism, this paper studies the intellectual of Sufism, Sufim cultural, and politics as well. This paper studied about three main axis of Sufism; the culture and history of Sufism, the role of Sufism in spreading Islam, and the role of Sufism in addressing the Takfiri thought. Based on this study, the Islamic institution and government should adopt and adapt the Sufi thought, and to address the Takfiri culture, to prevent Muslim from joining the Takfiri groups

Transmitting the quantum state of electrons across a metallic island with Coulomb interaction.

The Coulomb interaction generally limits the quantum propagation of electrons. However, it can also provide a mechanism to transfer their quantum state over larger distances. Here, we demonstrate such a form of electron teleportation across a metallic island. This effect originates from the low-temperature freezing of the island's charge Q which, in the presence of a single connected electronic channel, enforces a one-to-one correspondence between incoming and outgoing electrons. Such faithful quantum state imprinting is established between well-separated injection and emission locations and evidenced through two-path interferences in the integer quantum Hall regime. The additional quantum phase of 2πQ/e, where e is the electron charge, may allow for decoherence-free entanglement of propagating electrons, and notably of flying qubits.

Structure of bicentralizer algebras and inclusions of type $$\mathrm{III}$$ factors

We investigate the structure of the relative bicentralizer algebra $$\mathrm{B}(N \subset M, \varphi )$$ for inclusions of von Neumann algebras with normal expectation where N is a type $$\mathrm{III}_1$$ subfactor and $$\varphi \in N_*$$ is a faithful state. We first construct a canonical flow $$\beta ^\varphi : \mathbf {R}^*_+ \curvearrowright \mathrm{B}(N \subset M, \varphi )$$ on the relative bicentralizer algebra and we show that the W$$^*$$-dynamical system $$(\mathrm{B}(N \subset M, \varphi ), \beta ^\varphi )$$ is independent of the choice of $$\varphi $$ up to a canonical isomorphism. In the case when $$N=M$$, we deduce new results on the structure of the automorphism group of $$\mathrm{B}(M,\varphi )$$ and we relate the period of the flow $$\beta ^\varphi $$ to the tensorial absorption of Powers factors. For general irreducible inclusions $$N \subset M$$, we relate the ergodicity of the flow $$\beta ^\varphi $$ to the existence of irreducible AFD subfactors in M that sit with normal expectation in N. When the inclusion $$N \subset M$$ is discrete, we prove a relative bicentralizer theorem and we use it to solve Kadison’s problem when N is amenable.

On Period, Cycles and Fixed Points of a Quantum Channel

We consider a quantum channel acting on an infinite dimensional von Neumann algebra of operators on a separable Hilbert space. When there exists an invariant normal faithful state, the cyclic properties of such channels are investigated passing through the decoherence free algebra and the fixed points domain. Both these spaces are proved to be images of a normal conditional expectation so that their consequent atomic structure are analyzed in order to give a better description of the action of the channel and, for instance, of its Kraus form and invariant densities.

Operator-Algebraic Construction of Gauge Theories and Jones’ Actions of Thompson’s Groups

Using ideas from Jones, lattice gauge theory and loop quantum gravity, we construct 1+1-dimensional gauge theories on a spacetime cylinder. Given a separable compact group $G$, we construct localized time-zero fields on the spatial torus as a net of C*-algebras together with an action of the gauge group that is an infinite product of $G$ over the dyadic rationals and, using a recent machinery of Jones, an action of Thompson's group $T$ as a replacement of the spatial diffeomorphism group. Adding a family of probability measures on the unitary dual of $G$ we construct a state and obtain a net of von Neumann algebras carrying a state-preserving gauge group action. For abelian $G$, we provide a very explicit description of our algebras. For a single measure on the dual of $G$, we have a state-preserving action of Thompson's group and semi-finite von Neumann algebras. For $G=\mathbf{S}$ the circle group together with a certain family of heat-kernel states providing the measures, we obtain hyperfinite type III factors with a normal faithful state providing a nontrivial time evolution via Tomita-Takesaki theory (KMS condition). In the latter case, we additionally have a non-singular action of the group of rotations with dyadic angles, as a subgroup of Thompson's group $T$, for geometrically motivated choices of families of heat-kernel states.

Manifolds of classical probability distributions and quantum density operators in infinite dimensions

The manifold structure of subsets of classical probability distributions and quantum density operators in infinite dimensions is investigated in the context of C^{*}-algebras and actions of Banach-Lie groups. Specificaly, classical probability distributions and quantum density operators may be both described as states (in the functional analytic sense) on a given C^{*}-algebra mathscr {A} which is Abelian for Classical states, and non-Abelian for Quantum states. In this contribution, the space of states mathscr {S} of a possibly infinite-dimensional, unital C^{*}-algebra mathscr {A} is partitioned into the disjoint union of the orbits of an action of the group mathscr {G} of invertible elements of mathscr {A}. Then, we prove that the orbits through density operators on an infinite-dimensional, separable Hilbert space mathcal {H} are smooth, homogeneous Banach manifolds of mathscr {G}=mathcal {GL}(mathcal {H}), and, when mathscr {A} admits a faithful tracial state tau like it happens in the Classical case when we consider probability distributions with full support, we prove that the orbit through tau is a smooth, homogeneous Banach manifold for mathscr {G}.