Faithful State Research Articles

On Unital Full Amalgamated Free Products of Quasidiagonal C*-Algebras

In the paper, we consider the question as to whether a unital full amalgamated free product of quasidiagonal C*-algebras is itself quasidiagonal. We give a sufficient condition for a unital full amalgamated free product of quasidiagonal C*-algebras with amalgamation over a finite-dimensional C*-algebra to be quasidiagonal. By applying this result, we conclude that the unital full free product of two AF algebras with amalgamation over a finite-dimensional C*-algebra is AF if there exists a faithful tracial state on each of the two AF algebras such that the restrictions of these states to the common subalgebra coincide.

Compatibility of observables on effect algebras

We compare different notions of simultaneous measurability (compatibility) of observables on lattice $$\sigma $$ź-effect algebras and more generally, on $$\sigma $$ź-effect algebras that can be covered by $$\sigma $$ź-MV-algebras. We prove that every $$\sigma $$ź-MV-algebra is the range of a $$\sigma $$ź-additive observable, and we compare the following notions of compatibility of observables: joint measurability, coexistence, joint measurability of binarizations, coexistence of binarizations, smearings of the same observable. We prove that if there is a faithful state on the effect algebra, then any two standard observables that are smearings of the same (sharp) observable admit a generalized joint observable.

Closed convex hulls of unitary orbits in C⁎-algebras of real rank zero

In this paper, we study closed convex hulls of unitary orbits in various C$^*$-algebras. For unital C$^*$-algebras with real rank zero and a faithful tracial state determining equivalence of projections, a notion of majorization describes the closed convex hulls of unitary orbits for self-adjoint operators. Other notions of majorization are examined in these C$^*$-algebras. Combining these ideas with the Dixmier property, we demonstrate unital, infinite dimensional C$^*$-algebras of real rank zero and strict comparison of projections with respect to a faithful tracial state must be simple and have a unique tracial state. Also, closed convex hulls of unitary orbits of self-adjoint operators are fully described in unital, simple, purely infinite C$^*$-algebras.

Process tomography via sequential measurements on a single quantum system

We utilize a discrete (sequential) measurement protocol to investigate quantum process tomography of a single two-level quantum system, with an unknown initial state, undergoing Rabi oscillations. The ignorance of the dynamical parameters is encoded into a continuous-variable classical system which is coupled to the two-level quantum system via a generalized Hamiltonian. This combined estimate of the quantum state and dynamical parameters is updated by using the information obtained from sequential measurements on the quantum system and, after a sufficient waiting period, faithful state monitoring and parameter determination is obtained. Numerical evidence is used to demonstrate the convergence of the state estimate to the true state of the hybrid system.

Environment induced decoherence for Markovian evolutions

We study environmental decoherence for a quantum Markov semigroup T acting on an arbitrary von Neumann algebra M. In particular, we analyze the relationships between the decomposition of the algebra induced by decoherence and a sort of “isometric-sweeping decomposition” for the space of states. Moreover, when the semigroup has a faithful normal invariant state, we embed the algebra M in its completion M̂ with respect to the scalar product induced by the faithful state, and we compare the decomposition induced by decoherence with some other kind of isometric-sweeping decompositions of M and M̂.

GROUP ACTION PRESERVING THE HAAGERUP PROPERTY OF -ALGEBRAS

From the viewpoint of $C^{\ast }$-dynamical systems, we define a weak version of the Haagerup property for the group action on a $C^{\ast }$-algebra. We prove that this group action preserves the Haagerup property of $C^{\ast }$-algebras in the sense of Dong [‘Haagerup property for $C^{\ast }$-algebras’, J. Math. Anal. Appl.377 (2011), 631–644], that is, the reduced crossed product $C^{\ast }$-algebra $A\rtimes _{{\it\alpha},\text{r}}{\rm\Gamma}$ has the Haagerup property with respect to the induced faithful tracial state $\widetilde{{\it\tau}}$ if $A$ has the Haagerup property with respect to ${\it\tau}$.

On the properties of some sets of von Neumann algebras under perturbation

Let ℒ be a type II1 factor with separable predual and τ be a normal faithful tracial state of ℒ. We first show that the set of subfactors of ℒ with property Γ, the set of type II1 subfactors of ℒ with similarity property and the set of all McDuff subfactors of ℒ are open and closed in the Hausdorff metric d 2 induced by the trace norm; then we show that the set of all hyperfinite von Neumann subalgebras of ℒ is closed in d 2. We also consider the connection of perturbation of operator algebras under d 2 with the fundamental group and the generator problem of type II1 factors. When Open image in new window is a finite von Neumann algebra with a normal faithful trace, the set of all von Neumann subalgebras Open image in new window of Open image in new window such that Open image in new window is rigid is closed in the Hausdorff metric d 2.

Tail algebras of quantum exchangeable random variables

We show that any countably generated von Neumann algebra with specified normal faithful state can arise as the tail algebra of a quantum exchangeable sequence of noncommutative random variables. We also characterize the cases when the state corresponds to a limit of convex combinations of free products states.

Heisenberg picture approach to the stability of quantum Markov systems

Quantum Markovian systems, modeled as unitary dilations in the quantum stochastic calculus of Hudson and Parthasarathy, have become standard in current quantum technological applications. This paper investigates the stability theory of such systems. Lyapunov-type conditions in the Heisenberg picture are derived in order to stabilize the evolution of system operators as well as the underlying dynamics of the quantum states. In particular, using the quantum Markov semigroup associated with this quantum stochastic differential equation, we derive sufficient conditions for the existence and stability of a unique and faithful invariant quantum state. Furthermore, this paper proves the quantum invariance principle, which extends the LaSalle invariance principle to quantum systems in the Heisenberg picture. These results are formulated in terms of algebraic constraints suitable for engineering quantum systems that are used in coherent feedback networks.

Quantum automorphism groups and [formula omitted]-deformations

We show that any compact quantum group having the same fusion rules as the ones of SO(3) is the quantum automorphism group of a pair (A,φ), where A is a finite dimensional C⁎-algebra endowed with a homogeneous faithful state. We also study the representation category of the quantum automorphism group of (A,φ) when φ is not necessarily positive, generalizing some known results, and we discuss the possibility of classifying the cosemisimple (not necessarily compact) Hopf algebras whose corepresentation semi-ring is isomorphic to that of SO(3).

Ultraproducts of von Neumann algebras

We study several notions of ultraproducts of von Neumann algebras from a unified viewpoint. In particular, we show that for a sigma-finite von Neumann algebra M, the ultraproduct Mω introduced by Ocneanu is a corner of the ultraproduct ∏ωM introduced by Groh and Raynaud. Using this connection, we show that the ultraproduct action of the modular automorphism group of a normal faithful state φ of M on the Ocneanu ultraproduct is the modular automorphism group of the ultrapower state (σtφω=(σtφ)ω). Applying these results, we obtain several properties of the Ocneanu ultraproduct of type III factors, which are not present in the tracial ultraproducts. For instance, it turns out that the ultrapower Mω of a Type III0 factor is never a factor. Moreover we settle in the affirmative a recent problem by Ueda about the connection between the relative commutant of M in Mω and Connes' asymptotic centralizer algebra Mω.

Normal states of type III factors

Let M be a factor of type III with separable predual and with normal states phi_1,...,phi_k, omega with omega faithful. Let A be a finite dimensional C*-subalgebra of M. Then it is shown that there is a unitary operator u in M such that phi_i Ad u =omega on A for i=1,...,k. This follows from an embedding result of a finite dimensional C*-algebra with a faithful state into M with finitely many given states. We also give similar embedding results of C*-algebras and von Neumann algebras with faithful states into M. Another similar result for a factor of type II_1 instead of type III holds.

A multievent approach to estimating pair fidelity and heterogeneity in state transitions

Fidelity rates of pair-bonded individuals are of considerable interest to behavioral and population biologists as they can influence population structure, mating rates, population productivity, and gene flow. Estimates of fidelity rates calculated from direct observations of pairs in consecutive breeding seasons may be biased because (i) individuals that are not seen are assumed to be dead, (ii) variation in the detectability of individuals is ignored, and (iii) pair status must be known with certainty. This can lead to a high proportion of observations being ignored. This approach also restricts the way variation in fidelity rates for different types of individuals, or the covariation between fidelity and other vital rates (e.g., survival) can be analyzed. In this study, we develop a probabilistic multievent capture–mark–recapture (MECMR) modeling framework for estimating pair fidelity rates that accounts for imperfect detection rates and capture heterogeneity, explicitly incorporates uncertainty in the assessment of pair status, and allows estimates of state-dependent survival and fidelity rates to be obtained simultaneously. We demonstrate the utility of our approach for investigating patterns of fidelity in pair-bonded individuals, by applying it to 30 years of breeding data from a wild population of great tits Parus major Linnaeus. Results of model selection supported state-dependent recapture, survival, and fidelity rates. Recapture rates were higher for individuals breeding with their previous partner than for those breeding with a different partner. Faithful birds that were breeding with the same partner as in the previous breeding season (i.e., at t − 1) experienced substantially higher survival rates (between t and t + 1) and were also more likely to remain faithful to their current partner (i.e., to remain in the faithful state at t + 1). First year breeders were more likely to change partner than older birds. These findings imply that traditional estimates, which do not account for state-dependent parameters, may be both inaccurate and biased, and hence, inferences based on them may conceal important biological effects. This was demonstrated in the analysis of simulated capture histories, which showed that our MECMR model was able to estimate state-dependant survival and pair fidelity rates in the face of varying state-dependant recapture rates robustly, and more accurately, than the traditional method. In addition, this new modeling approach provides a statistically rigorous framework for testing hypothesis about the causes and consequences of fidelity to a partner for natural populations. The novel modeling approach described here can readily be applied, either in its current form or via extension, to other populations and other types of dyadic interactions (e.g., between nonpaired individuals, such as parent–offspring relationships, or between individuals and locations, such as nest-site fidelity).

Faithful Solid State Optical Memory with Dynamically Decoupled Spin Wave Storage

We report a high fidelity optical memory in which dynamical decoupling is used to extend the storage time. This is demonstrated in a rare-earth doped crystal in which optical coherences were transferred to nuclear spin coherences and then protected against environmental noise by dynamical decoupling, leading to storage times of up to 4.2 ms. An interference experiment shows that relative phases of input pulses are preserved through the whole storage and retrieval process with a visibility ≈1, demonstrating the usefulness of dynamical decoupling for extending the storage time of quantum memories. We also show that dynamical decoupling sequences insensitive to initial spin coherence increase retrieval efficiency.

Haagerup property for [formula omitted]-algebras and rigidity of [formula omitted]-algebras with property (T)

We study the Haagerup property for C⁎-algebras. We first give new examples of C⁎-algebras with the Haagerup property. A nuclear C⁎-algebra with a faithful tracial state always has the Haagerup property, and the permanence of the Haagerup property for C⁎-algebras is established. As a consequence, the class of all C⁎-algebras with the Haagerup property turns out to be quite large. We then apply Popaʼs results and show the C⁎-algebras with property (T) have a certain rigidity property. Unlike the case of von Neumann algebras, for the reduced group C⁎-algebras of groups with relative property (T), the rigidity property strongly fails in general. Nevertheless, for some groups without nontrivial property (T) subgroups, we show a rigidity property in some cases. As examples, we prove the reduced group C⁎-algebras of the (non-amenable) affine groups of the affine planes have a rigidity property.

TWISTED COMMUTING SQUARES

We consider generalizations of commuting squares, called twisted commuting squares, obtained by having the commuting square orthogonality condition hold with respect to the inner product given by a faithful state on a finite-dimensional *-algebra. We present various examples of twisted commuting squares, most of which are computationally easy to work with, and we prove an isolation result. We also show how parametric families of (not necessarily) twisted commuting squares yield associative deformations of the matrix multiplication.

Free products and the lack of state-preserving approximations of nuclear $C^*$-algebras

Let $A$ be a homogeneous C*-algebra and $\phi$ a state on $A.$ We show that if $\phi$ satisfies a certain faithfulness condition, then there is a net of finite-rank, unital completely positive, $\phi$-preserving maps on $A$ that tend to the identity pointwise. This combined with results of Ricard and Xu show that the reduced free product of homogeneous C*-algebras with respect to these states have the completely contractive approximation property. We also give an example of a faithful state on $M_2\otimes C[0,1]$ for which no such state-preserving approximation of the identity map exists, thus answering a question of Ricard and Xu.

Controlling and exploiting phases in multi-spin systems using electron spin resonance and nuclear magnetic resonance

The phase of a superposition state is a quintessential characteristic that differentiates a quantum bit of information from a classical one. This phase can be manipulated dynamically or geometrically, and can be exploited to sensitively estimate Hamiltonian parameters, perform faithful quantum state tomography and encode quantum information into multiple modes of an ensemble. Here we discuss the methods that we have employed to manipulate and exploit the phase information of single-, two-, multi-qubit and multi-mode spin systems.

Decoherence for Quantum Markov Semi-Groups on Matrix Algebras

In this work we describe a necessary and sufficient condition for decoherence of quantum Markov evolutions acting on matrix spaces (according to the definition introduced by Blanchard and Olkiewicz). This condition is related to the spectral analysis of the generator $${\mathcal{L}}$$ of the semigroup and is easily stated: the evolution displays decoherence if and only if the maximal algebra $${\mathcal{N}(\mathcal{T})}$$ on which the semigroup is *-automorphic contains all the eigenvalues of $${\mathcal{L}}$$ associated with eigenvectors with null real part. Moreover, this condition is surely verified when the semigroup admits a faithful invariant state.

Purely infinite simple reduced [formula omitted]-algebras of one-relator separated graphs

Given a separated graph (E,C), there are two different C∗-algebras associated to it: the full graph C∗-algebra C∗(E,C) and the reduced one Cred∗(E,C). For a large class of separated graphs (E,C), we prove that Cred∗(E,C) either is purely infinite simple or admits a faithful tracial state. The main tool we use to show pure infiniteness of reduced graph C∗-algebras is a generalization to the amalgamated case of a result on purely infinite simple free products due to Dykema.