28,667 research outputs found
Quantum Proofs for Classical Theorems
Alongside the development of quantum algorithms and quantum complexity theory in recent years, quantum techniques have also proved instrumental in obtaining results in diverse classical (non-quantum) areas, such as coding theory, communication complexity, and polynomial approximations. In this paper we survey these results and the quantum toolbox they use
Gromov-Witten invariants on Grassmannians
We prove that any three-point genus zero Gromov-Witten invariant on a type A
Grassmannian is equal to a classical intersection number on a two-step flag
variety. We also give symplectic and orthogonal analogues of this result; in
these cases the two-step flag variety is replaced by a sub-maximal isotropic
Grassmannian. Our theorems are applied, in type A, to formulate a conjectural
quantum Littlewood-Richardson rule, and in the other classical Lie types, to
obtain new proofs of the main structure theorems for the quantum cohomology of
Lagrangian and orthogonal Grassmannians.Comment: 15 pages, LaTeX2e, to appear in J. Amer. Math. So
A Resource Framework for Quantum Shannon Theory
Quantum Shannon theory is loosely defined as a collection of coding theorems,
such as classical and quantum source compression, noisy channel coding
theorems, entanglement distillation, etc., which characterize asymptotic
properties of quantum and classical channels and states. In this paper we
advocate a unified approach to an important class of problems in quantum
Shannon theory, consisting of those that are bipartite, unidirectional and
memoryless.
We formalize two principles that have long been tacitly understood. First, we
describe how the Church of the larger Hilbert space allows us to move flexibly
between states, channels, ensembles and their purifications. Second, we
introduce finite and asymptotic (quantum) information processing resources as
the basic objects of quantum Shannon theory and recast the protocols used in
direct coding theorems as inequalities between resources. We develop the rules
of a resource calculus which allows us to manipulate and combine resource
inequalities. This framework simplifies many coding theorem proofs and provides
structural insights into the logical dependencies among coding theorems.
We review the above-mentioned basic coding results and show how a subset of
them can be unified into a family of related resource inequalities. Finally, we
use this family to find optimal trade-off curves for all protocols involving
one noisy quantum resource and two noiseless ones.Comment: 60 page
Exponentially Accurate Semiclassical Dynamics: Propagation, Localization, Ehrenfest Times, Scattering and More General States
We prove six theorems concerning exponentially accurate semiclassical quantum
mechanics. Two of these theorems are known results, but have new proofs. Under
appropriate hypotheses, they conclude that the exact and approximate dynamics
of an initially localized wave packet agree up to exponentially small errors in
for finite times and for Ehrenfest times. Two other theorems state that
for such times the wave packets are localized near a classical orbit up to
exponentially small errors. The fifth theorem deals with infinite times and
states an exponentially accurate scattering result. The sixth theorem provides
extensions of the other five by allowing more general initial conditions
Commuting Quantum Circuits with Few Outputs are Unlikely to be Classically Simulatable
We study the classical simulatability of commuting quantum circuits with n
input qubits and O(log n) output qubits, where a quantum circuit is classically
simulatable if its output probability distribution can be sampled up to an
exponentially small additive error in classical polynomial time. First, we show
that there exists a commuting quantum circuit that is not classically
simulatable unless the polynomial hierarchy collapses to the third level. This
is the first formal evidence that a commuting quantum circuit is not
classically simulatable even when the number of output qubits is exponentially
small. Then, we consider a generalized version of the circuit and clarify the
condition under which it is classically simulatable. Lastly, we apply the
argument for the above evidence to Clifford circuits in a similar setting and
provide evidence that such a circuit augmented by a depth-1 non-Clifford layer
is not classically simulatable. These results reveal subtle differences between
quantum and classical computation.Comment: 19 pages, 6 figures; v2: Theorems 1 and 3 improved, proofs modifie
Strong converse theorems using R\'enyi entropies
We use a R\'enyi entropy method to prove strong converse theorems for certain
information-theoretic tasks which involve local operations and quantum or
classical communication between two parties. These include state
redistribution, coherent state merging, quantum state splitting, measurement
compression with quantum side information, randomness extraction against
quantum side information, and data compression with quantum side information.
The method we employ in proving these results extends ideas developed by Sharma
[arXiv:1404.5940], which he used to give a new proof of the strong converse
theorem for state merging. For state redistribution, we prove the strong
converse property for the boundary of the entire achievable rate region in the
-plane, where and denote the entanglement cost and quantum
communication cost, respectively. In the case of measurement compression with
quantum side information, we prove a strong converse theorem for the classical
communication cost, which is a new result extending the previously known weak
converse. For the remaining tasks, we provide new proofs for strong converse
theorems previously established using smooth entropies. For each task, we
obtain the strong converse theorem from explicit bounds on the figure of merit
of the task in terms of a R\'enyi generalization of the optimal rate. Hence, we
identify candidates for the strong converse exponents for each task discussed
in this paper. To prove our results, we establish various new entropic
inequalities, which might be of independent interest. These involve conditional
entropies and mutual information derived from the sandwiched R\'enyi
divergence. In particular, we obtain novel bounds relating these quantities, as
well as the R\'enyi conditional mutual information, to the fidelity of two
quantum states.Comment: 40 pages, 5 figures; v4: Accepted for publication in Journal of
Mathematical Physic
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