1,210 research outputs found
Geometric derivation of the quantum speed limit
The Mandelstam-Tamm and Margolus-Levitin inequalities play an important role
in the study of quantum mechanical processes in Nature, since they provide
general limits on the speed of dynamical evolution. However, to date there has
been only one derivation of the Margolus-Levitin inequality. In this paper,
alternative geometric derivations for both inequalities are obtained from the
statistical distance between quantum states. The inequalities are shown to hold
for unitary evolution of pure and mixed states, and a counterexample to the
inequalities is given for evolution described by completely positive
trace-preserving maps. The counterexample shows that there is no quantum speed
limit for non-unitary evolution.Comment: 8 pages, 1 figure
Graph-theoretic strengths of contextuality
Cabello-Severini-Winter and Abramsky-Hardy (building on the framework of
Abramsky-Brandenburger) both provide classes of Bell and contextuality
inequalities for very general experimental scenarios using vastly different
mathematical techniques. We review both approaches, carefully detail the links
between them, and give simple, graph-theoretic methods for finding
inequality-free proofs of nonlocality and contextuality and for finding states
exhibiting strong nonlocality and/or contextuality. Finally, we apply these
methods to concrete examples in stabilizer quantum mechanics relevant to
understanding contextuality as a resource in quantum computation.Comment: 13 pages; significantly rewritte
Negative Quasi-Probability as a Resource for Quantum Computation
A central problem in quantum information is to determine the minimal physical
resources that are required for quantum computational speedup and, in
particular, for fault-tolerant quantum computation. We establish a remarkable
connection between the potential for quantum speed-up and the onset of negative
values in a distinguished quasi-probability representation, a discrete analog
of the Wigner function for quantum systems of odd dimension. This connection
allows us to resolve an open question on the existence of bound states for
magic-state distillation: we prove that there exist mixed states outside the
convex hull of stabilizer states that cannot be distilled to non-stabilizer
target states using stabilizer operations. We also provide an efficient
simulation protocol for Clifford circuits that extends to a large class of
mixed states, including bound universal states.Comment: 15 pages v4: This is a major revision. In particular, we have added a
new section detailing an explicit extension of the Gottesman-Knill simulation
protocol to deal with positively represented states and measurement (even
when these are non-stabilizer). This paper also includes significant
elaboration on the two main results of the previous versio
Quantum Error Correction with the Toric-GKP Code
We examine the performance of the single-mode GKP code and its concatenation
with the toric code for a noise model of Gaussian shifts, or displacement
errors. We show how one can optimize the tracking of errors in repeated noisy
error correction for the GKP code. We do this by examining the
maximum-likelihood problem for this setting and its mapping onto a 1D Euclidean
path-integral modeling a particle in a random cosine potential. We demonstrate
the efficiency of a minimum-energy decoding strategy as a proxy for the path
integral evaluation. In the second part of this paper, we analyze and
numerically assess the concatenation of the GKP code with the toric code. When
toric code measurements and GKP error correction measurements are perfect, we
find that by using GKP error information the toric code threshold improves from
to . When only the GKP error correction measurements are perfect
we observe a threshold at . In the more realistic setting when all error
information is noisy, we show how to represent the maximum likelihood decoding
problem for the toric-GKP code as a 3D compact QED model in the presence of a
quenched random gauge field, an extension of the random-plaquette gauge model
for the toric code. We present a new decoder for this problem which shows the
existence of a noise threshold at shift-error standard deviation for toric code measurements, data errors and GKP ancilla errors.
If the errors only come from having imperfect GKP states, this corresponds to
states with just 4 photons or more. Our last result is a no-go result for
linear oscillator codes, encoding oscillators into oscillators. For the
Gaussian displacement error model, we prove that encoding corresponds to
squeezing the shift errors. This shows that linear oscillator codes are useless
for quantum information protection against Gaussian shift errors.Comment: 50 pages, 14 figure
Holographic entropy inequalities and gapped phases of matter
We extend our studies of holographic entropy inequalities to gapped phases of
matter. For any number of regions, we determine the linear entropy inequalities
satisfied by systems in which the entanglement entropy satisfies an exact area
law. In particular, we find that all holographic entropy inequalities are valid
in such systems. In gapped systems with topological order, the "cyclic
inequalities" derived recently for the holographic entanglement entropy
generalize the Kitaev-Preskill formula for the topological entanglement
entropy. Finally, we propose a candidate linear inequality for general 4-party
quantum states.Comment: 20 pages, 4 figures. v2: section 4 rewritten, where all linear
entropy (in)equalities satisfied by area-law systems are derived and an error
in their relations to graph theory is correcte
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