arXiv subjects

R. F. Werner

Publications and source records attributed to R. F. Werner.

At least 19 recordsLinked to original sources

arXiv2022

Chiral Floquet systems and quantum walks at half period

We classify chiral symmetric periodically driven quantum systems on a one-dimensional lattice. The driving process is local, can be continuous or discrete in time, and we assume a gap condition for the corresponding Floquet operator. The analysis is in terms of the unitary operator at a half-period, the half-step operator. We give a complete classification of the connected classes of half-step operators in terms of five integer indices. On the basis of these indices it can be decided whether the half-step operator can be obtained from a continuous Hamiltonian driving, or not. The half-step operator determines two Floquet operators, obtained by starting the driving at zero or at half period, respectively. These are called timeframes and are chiral symmetric quantum walks. Conversely, we show under which conditions two chiral symmetric walks determine a common half-step operator. Moreover, we clarify the connection between the classification of half-step operators and the corresponding quantum walks. Within this theory we prove bulk-edge correspondence and show that a second timeframe allows to distinguish between symmetry protected edge states at $+1$ and $-1$ which is not possible for a single timeframe.

math-ph↗

arXiv2021

Numerical optimization of amplitude-modulated pulses in microwave-driven entanglement generation

Microwave control of trapped ions can provide an implementation of high-fidelity two-qubit gates free from errors induced by photon scattering. Furthermore, microwave conductors may be embedded into a scalable trap structure, providing the chip-level integration of control that is desirable for scaling. Recent developments have demonstrated how amplitude modulation of the gate drive can permit a two-qubit entangling operation to become robust against motional mode noise and other experimental imperfections. Here, we discuss a method for the numerical optimization of the microwave pulse envelope to produce gate pulses with improved resilience, faster operation and higher energy efficiency.

quant-ph↗

arXiv2021

Quantum walks: Schur functions meet symmetry protected topological phases

This paper uncovers and exploits a link between a central object in harmonic analysis, the so-called Schur functions, and the very hot topic of symmetry protected topological phases of quantum matter. This connection is found in the setting of quantum walks, i.e. quantum analogs of classical random walks. We prove that topological indices classifying symmetry protected topological phases of quantum walks are encoded by matrix Schur functions built out of the walk. This main result of the paper reduces the calculation of these topological indices to a linear algebra problem: calculating symmetry indices of finite-dimensional unitaries obtained by evaluating such matrix Schur functions at the symmetry protected points $\pm1$. The Schur representation fully covers the complete set of symmetry indices for 1D quantum walks with a group of symmetries realizing any of the symmetry types of the tenfold way. The main advantage of the Schur approach is its validity in the absence of translation invariance, which allows us to go beyond standard Fourier methods, leading to the complete classification of non-translation invariant phases for typical examples.

math-ph↗

arXiv2021

An algorithm to factorize quantum walks into shift and coin operations

We provide an algorithm that factorizes one-dimensional quantum walks into a protocol of two basic operations: A fixed conditional shift that transports particles between cells and suitable coin operators that act locally in each cell. This allows to tailor quantum walk protocols to any experimental setup by rephrasing it on the cell structure determined by the experimental limitations. We give the example of a walk defined on a qutrit chain compiled to run an a qubit chain.

quant-ph↗

arXiv2019

Robust and resource-efficient microwave near-field entangling $^9$Be$^+$ gate

Microwave trapped-ion quantum logic gates avoid spontaneous emission as a fundamental source of decoherence. However, microwave two-qubit gates are still slower than laser-induced gates and hence more sensitive to fluctuations and noise of the motional mode frequency. We propose and implement amplitude-shaped gate drives to obtain resilience to such frequency changes without increasing the pulse energy per gate operation. We demonstrate the resilience by noise injection during a two-qubit entangling gate with $^9$Be$^+$ ion qubits. In absence of injected noise, amplitude modulation gives an operation infidelity in the $10^{-3}$ range.

quant-ph↗

arXiv2018

Complete homotopy invariants for translation invariant symmetric quantum walks on a chain

We provide a classification of translation invariant one-dimensional quantum walks with respect to continuous deformations preserving unitarity, locality, translation invariance, a gap condition, and some symmetry of the tenfold way. The classification largely matches the one recently obtained (arXiv:1611.04439) for a similar setting leaving out translation invariance. However, the translation invariant case has some finer distinctions, because some walks may be connected only by breaking translation invariance along the way, retaining only invariance by an even number of sites. Similarly, if walks are considered equivalent when they differ only by adding a trivial walk, i.e., one that allows no jumps between cells, then the classification collapses also to the general one. The indices of the general classification can be computed in practice only for walks closely related to some translation invariant ones. We prove a completed collection of simple formulas in terms of winding numbers of band structures covering all symmetry types. Furthermore, we determine the strength of the locality conditions, and show that the continuity of the band structure, which is a minimal requirement for topological classifications in terms of winding numbers to make sense, implies the compactness of the commutator of the walk with a half-space projection, a condition which was also the basis of the general theory. In order to apply the theory to the joining of large but finite bulk pieces, one needs to determine the asymptotic behaviour of a stationary Schrödinger equation. We show exponential behaviour, and give a practical method for computing the decay constants.

quant-ph↗

arXiv2018

The topological classification of one-dimensional symmetric quantum walks

We give a topological classification of quantum walks on an infinite 1D lattice, which obey one of the discrete symmetry groups of the tenfold way, have a gap around some eigenvalues at symmetry protected points, and satisfy a mild locality condition. No translation invariance is assumed. The classification is parameterized by three indices, taking values in a group, which is either trivial, the group of integers, or the group of integers modulo 2, depending on the type of symmetry. The classification is complete in the sense that two walks have the same indices if and only if they can be connected by a norm continuous path along which all the mentioned properties remain valid. Of the three indices, two are related to the asymptotic behaviour far to the right and far to the left, respectively. These are also stable under compact perturbations. The third index is sensitive to those compact perturbations which cannot be contracted to a trivial one. The results apply to the Hamiltonian case as well. In this case all compact perturbations can be contracted, so the third index is not defined. Our classification extends the one known in the translation invariant case, where the asymptotic right and left indices add up to zero, and the third one vanishes, leaving effectively only one independent index. When two translationally invariant bulks with distinct indices are joined, the left and right asymptotic indices of the joined walk are thereby fixed, and there must be eigenvalues at $1$ or $-1$ (bulk-boundary correspondence). Their location is governed by the third index. We also discuss how the theory applies to finite lattices, with suitable homogeneity assumptions.

quant-ph↗

arXiv2018

Quantum walks in external gauge fields

Describing a particle in an external electromagnetic field is a basic task of quantum mechanics. The standard scheme for this is known as "minimal coupling", and consists of replacing the momentum operators in the Hamiltonian by modified ones with an added vector potential. In lattice systems it is not so clear how to do this, because there is no continuous translation symmetry, and hence there are no momenta. Moreover, when time is also discrete, as in quantum walk systems, there is no Hamiltonian, only a unitary step operator. We present a unified framework of gauge theory for such discrete systems, keeping a close analogy to the continuum case. In particular, we show how to implement minimal coupling in a way that automatically guarantees unitary dynamics. The scheme works in any lattice dimension, for any number of internal degree of freedom, for walks that allow jumps to a finite neighborhood rather than to nearest neighbours, is naturally gauge invariant, and prepares possible extensions to non-abelian gauge groups.

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arXiv2016

A quantum dynamical approach to matrix Khrushchev's formulas

Khrushchev's formula is the cornerstone of the so called Khrushchev theory, a body of results which has revolutionized the theory of orthogonal polynomials on the unit circle. This formula can be understood as a factorization of the Schur function for an orthogonal polynomial modification of a measure on the unit circle. No such formula is known in the case of matrix-valued measures. This constitutes the main obstacle to generalize Khrushchev theory to the matrix-valued setting which we overcome in this paper. It was recently discovered that orthogonal polynomials on the unit circle and their matrix-valued versions play a significant role in the study of quantum walks, the quantum mechanical analogue of random walks. In particular, Schur functions turn out to be the mathematical tool which best codify the return properties of a discrete time quantum system, a topic in which Khrushchev's formula has profound and surprising implications. We will show that this connection between Schur functions and quantum walks is behind a simple proof of Khrushchev's formula via `quantum' diagrammatic techniques for CMV matrices. This does not merely give a quantum meaning to a known mathematical result, since the diagrammatic proof also works for matrix-valued measures. Actually, this path counting approach is so fruitful that it provides different matrix generalizations of Khrushchev's formula, some of them new even in the case of scalar measures. Furthermore, the path counting approach allows us to identify the properties of CMV matrices which are responsible for Khrushchev's formula. On the one hand, this helps to formalize and unify the diagrammatic proofs using simple operator theory tools. On the other hand, this is the origin of our main result which extends Khrushchev's formula beyond the CMV case, as a factorization rule for Schur functions related to general unitary operators.

math.CA↗

arXiv2016

Revivals in Quantum Walks with quasi-periodically time-dependent coin

We provide an explanation of recent experimental results of Xue et al., where full revivals in a time-dependent quantum walk model with a periodically changing coin are found. Using methods originally developed for "electric" walks with a space-dependent, rather than a time-dependent coin, we provide a full explanation of the observations of Xue et al. We extend the analysis from periodic time-dependence to quasi-periodic behaviour with periods incommensurate to the step size. Spectral analysis, one of the principal tools for the study of electric walks, fails for time-dependent systems, but we find qualitative propagation behaviour of the time-dependent system in close analogy to the electric case.

quant-ph↗

arXiv2016

Bulk-edge correspondence of one-dimensional quantum walks

We outline a theory of symmetry protected topological phases of one-dimensional quantum walks. We assume spectral gaps around the symmetry-distinguished points +1 and -1, in which only discrete eigenvalues are allowed. The phase classification by integer or binary indices extends the classification known for translation invariant systems in terms of their band structure. However, our theory requires no translation invariance whatsoever, and the indices we define in this general setting are invariant under arbitrary symmetric local perturbations, even those that cannot be continuously contracted to the identity. More precisely we define two indices for every walk, characterizing the behavior far to the right and far to the left, respectively. Their sum is a lower bound on the number of eigenstates at +1 and -1. For a translation invariant system the indices add up to zero, so one of them already characterizes the phase. By joining two bulk phases with different indices we get a walk in which the right and left indices no longer cancel, so the theory predicts bound states at +1 or -1. This is a rigorous statement of bulk-edge correspondence. The results also apply to the Hamiltonian case with a single gap at zero.

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arXiv2014

What Maudlin replied to

A recent post by Tim Maudlin to this archive (arXiv:1408.1828) was entitled "Reply to Werner". However, it was not clear to what text this was supposed to be a reply. Here I briefly provide this context, and show that Maudlin's post is as ill-conceived as the original paper (arXiv:1408.1826).

quant-ph↗

arXiv2014

Comment on "Experimental Test of Error-Disturbance Uncertainty Relations by Weak Measurement"

In this comment on the paper by F. Kaneda, S.-Y. Baek, M. Ozawa and K. Edamatsu [Phys. Rev. Lett. 112, 020402, 2014, arXiv:1308.5868], we point out that the claim of having refuted Heisenberg's error-disturbance relation is unfounded since it is based on the choice of unsuitable and operationally problematical quantifications of measurement error and disturbance. As we have shown elsewhere [PRL 111, 160405, 2013], for appropriate choices of operational measures of error and disturbance, Heisenberg's heuristic relation can be turned into a precise inequality which is a rigorous consequence of quantum mechanics.

quant-ph↗

arXiv2014

Comment on "Noise and Disturbance in Quantum Measurements: An Information-Theoretic Approach"

In this comment on the work of F. Buscemi, M.J.W. Hall, M. Ozawa and M.M. Wilde [PRL 112, 050401, 2014, arXiv:1310.6603], we point out a misrepresentation of measures of error and disturbance introduced in our recent work [PRL 111, 160405, 2013, arXiv:1306.1565] as being "purely formal, with no operational counterparts". We also exhibit an tension in the authors' message, in that their main result is an error-disturbance relation for state-independent measures, but its importance is declared to be limited to discrete variables. In contrast, we point out the separate roles played by such relations for either state-dependent or state-independent measures of error and disturbance.

quant-ph↗

arXiv2014

Measurement Uncertainty: Reply to Critics

In a recent publication [PRL 111, 160405 (2013)] we proved a version of Heisenberg's error-disturbance tradeoff. This result was in apparent contradiction to claims by Ozawa of having refuted these ideas of Heisenberg. In a direct reaction [arXiv:1308.3540] Ozawa has called our work groundless, and has claimed to have found both a counterexample and an error in our proof. Here we answer to these allegations. We also comment on the submission [arXiv:1307.3604] by Rozema et al, in which our approach is unfavourably compared to that of Ozawa.

quant-ph↗

arXiv2013

Propagation and spectral properties of quantum walks in electric fields

We study one-dimensional quantum walks in a homogeneous electric field. The field is given by a phase which depends linearly on position and is applied after each step. The long time propagation properties of this system, such as revivals, ballistic expansion and Anderson localization, depend very sensitively on the value of the electric field $Φ$, e.g., on whether $Φ/(2π)$ is rational or irrational. We relate these properties to the continued fraction expansion of the field. When the field is given only with finite accuracy, the beginning of the expansion allows analogous conclusions about the behavior on finite time scales.

quant-ph↗

arXiv2012

Quantum Walks with Non-Orthogonal Position States

Quantum walks have by now been realized in a large variety of different physical settings. In some of these, particularly with trapped ions, the walk is implemented in phase space, where the corresponding position states are not orthogonal. We develop a general description of such a quantum walk and show how to map it into a standard one with orthogonal states, thereby making available all the tools developed for the latter. This enables a variety of experiments, which can be implemented with smaller step sizes and more steps. Tuning the non-orthogonality allows for an easy preparation of extended states such as momentum eigenstates, which travel at a well-defined speed with low dispersion. We introduce a method to adjust their velocity by momentum shifts, which allows to investigate intriguing effects such as the analog of Bloch oscillations.

quant-ph↗

arXiv2012

Recurrence for discrete time unitary evolutions

We consider quantum dynamical systems specified by a unitary operator U and an initial state vector ϕ. In each step the unitary is followed by a projective measurement checking whether the system has returned to the initial state. We call the system recurrent if this eventually happens with probability one. We show that recurrence is equivalent to the absence of an absolutely continuous part from the spectral measure of U with respect to ϕ. We also show that in the recurrent case the expected first return time is an integer or infinite, for which we give a topological interpretation. A key role in our theory is played by the first arrival amplitudes, which turn out to be the (complex conjugated) Taylor coefficients of the Schur function of the spectral measure. On the one hand, this provides a direct dynamical interpretation of these coefficients; on the other hand it links our definition of first return times to a large body of mathematical literature.

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