464 research outputs found
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 ± 1. 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. © 2021, The Author(s)
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 Schrodinger equation. We show exponential behaviour, and give a practical method for computing the decay constants
Propagation of Quantum Walks in Electric Fields
We study one-dimensional quantum walks in a homogenous 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
Parameter estimation with mixed quantum states
We consider quantum enhanced measurements with initially mixed states. We
show very generally that for any linear propagation of the initial state that
depends smoothly on the parameter to be estimated, the sensitivity is bound by
the maximal sensitivity that can be achieved for any of the pure states from
which the initial density matrix is mixed. This provides a very general proof
that purely classical correlations cannot improve the sensitivity of parameter
estimation schemes in quantum enhanced measurement schemes.Comment: 6 page
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 ± 1. 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
Applying causality principles to the axiomatization of probabilistic cellular automata
Cellular automata (CA) consist of an array of identical cells, each of which
may take one of a finite number of possible states. The entire array evolves in
discrete time steps by iterating a global evolution G. Further, this global
evolution G is required to be shift-invariant (it acts the same everywhere) and
causal (information cannot be transmitted faster than some fixed number of
cells per time step). At least in the classical, reversible and quantum cases,
these two top-down axiomatic conditions are sufficient to entail more
bottom-up, operational descriptions of G. We investigate whether the same is
true in the probabilistic case. Keywords: Characterization, noise, Markov
process, stochastic Einstein locality, screening-off, common cause principle,
non-signalling, Multi-party non-local box.Comment: 13 pages, 6 figures, LaTeX, v2: refs adde
The quantum world is not built up from correlations
It is known that the global state of a composite quantum system can be
completely determined by specifying correlations between measurements performed
on subsystems only. Despite the fact that the quantum correlations thus suffice
to reconstruct the quantum state, we show, using a Bell inequality argument,
that they cannot be regarded as objective local properties of the composite
system in question. It is well known since the work of J.S. Bell, that one
cannot have locally preexistent values for all physical quantities, whether
they are deterministic or stochastic. The Bell inequality argument we present
here shows this is also impossible for correlations among subsystems of an
individual isolated composite system. Neither of them can be used to build up a
world consisting of some local realistic structure. As a corrolary to the
result we argue that entanglement cannot be considered ontologically robust.
The argument has an important advantage over others because it does not need
perfect correlations but only statistical correlations. It can therefore easily
be tested in currently feasible experiments using four particle entanglement.Comment: Published version. Title change
Quantum Key Distribution between N partners: optimal eavesdropping and Bell's inequalities
Quantum secret-sharing protocols involving N partners (NQSS) are key
distribution protocols in which Alice encodes her key into qubits, in
such a way that all the other partners must cooperate in order to retrieve the
key. On these protocols, several eavesdropping scenarios are possible: some
partners may want to reconstruct the key without the help of the other ones,
and consequently collaborate with an Eve that eavesdrops on the other partners'
channels. For each of these scenarios, we give the optimal individual attack
that the Eve can perform. In case of such an optimal attack, the authorized
partners have a higher information on the key than the unauthorized ones if and
only if they can violate a Bell's inequality.Comment: 14 pages, 1 figur
Transfer of quantum states using finite resources
We discuss the problem of transfering a qubit from Alice to Bob using a noisy
quantum channel and only finite resources. As the basic protocol for the
transfer we apply quantum teleportation. It turns out that for a certain
quality of the channel direct teleportation combined with qubit purification is
superior to entanglement purification of the channel. If, however, the quality
of the channel is rather low one should simply apply an estimation-preparation
scheme.Comment: 9 pages RevTeX including 5 figures, replaced with revised version, to
appear in Phys. Rev.
Index theory of one dimensional quantum walks and cellular automata
If a one-dimensional quantum lattice system is subject to one step of a
reversible discrete-time dynamics, it is intuitive that as much "quantum
information" as moves into any given block of cells from the left, has to exit
that block to the right. For two types of such systems - namely quantum walks
and cellular automata - we make this intuition precise by defining an index, a
quantity that measures the "net flow of quantum information" through the
system. The index supplies a complete characterization of two properties of the
discrete dynamics. First, two systems S_1, S_2 can be pieced together, in the
sense that there is a system S which locally acts like S_1 in one region and
like S_2 in some other region, if and only if S_1 and S_2 have the same index.
Second, the index labels connected components of such systems: equality of the
index is necessary and sufficient for the existence of a continuous deformation
of S_1 into S_2. In the case of quantum walks, the index is integer-valued,
whereas for cellular automata, it takes values in the group of positive
rationals. In both cases, the map S -> ind S is a group homomorphism if
composition of the discrete dynamics is taken as the group law of the quantum
systems. Systems with trivial index are precisely those which can be realized
by partitioned unitaries, and the prototypes of systems with non-trivial index
are shifts.Comment: 38 pages. v2: added examples, terminology clarifie
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