7 research outputs found
Sufficiency of R\'enyi divergences
A set of classical or quantum states is equivalent to another one if there
exists a pair of classical or quantum channels mapping either set to the other
one. For dichotomies (pairs of states) this is closely connected to (classical
or quantum) R\'enyi divergences (RD) and the data-processing inequality: If a
RD remains unchanged when a channel is applied to the dichotomy, then there is
a recovery channel mapping the image back to the initial dichotomy. Here, we
prove for classical dichotomies that equality of the RDs alone is already
sufficient for the existence of a channel in any of the two directions and
discuss some applications. We conjecture that equality of the minimal quantum
RDs is sufficient in the quantum case and prove it for special cases. We also
show that neither the Petz quantum nor the maximal quantum RDs are sufficient.
As a side-result of our techniques we obtain an infinite list of inequalities
fulfilled by the classical, the Petz quantum, and the maximal quantum RDs.
These inequalities are not true for the minimal quantum RDs.Comment: Comments welcome, 31 pages, v2: Removed insignifcant error; v3:
Misupload; v4: Significantly improved presentatio
Convergence of Dynamics on Inductive Systems of Banach Spaces
Many features of physical systems, both qualitative and quantitative, become
sharply defined or tractable only in some limiting situation. Examples are
phase transitions in the thermodynamic limit, the emergence of classical
mechanics from quantum theory at large action, and continuum quantum field
theory arising from renormalization group fixed points. It would seem that few
methods can be useful in such diverse applications. However, we here present a
flexible modeling tool for the limit of theories: soft inductive limits
constituting a generalization of inductive limits of Banach spaces. In this
context, general criteria for the convergence of dynamics will be formulated,
and these criteria will be shown to apply in the situations mentioned and more.Comment: Comments welcom
Exact and lower bounds for the quantum speed limit in finite dimensional systems
A fundamental problem in quantum engineering is determining the lowest time
required to ensure that all possible unitaries can be generated with the tools
available, which is one of a number of possible quantum speed limits. We
examine this problem from the perspective of quantum control, where the system
of interest is described by a drift Hamiltonian and set of control
Hamiltonians. Our approach uses a combination of Lie algebra theory, Lie groups
and differential geometry, and formulates the problem in terms of geodesics on
a differentiable manifold. We provide explicit lower bounds on the quantum
speed limit for the case of an arbitrary drift, requiring only that the control
Hamiltonians generate a topologically closed subgroup of the full unitary
group, and formulate criteria as to when our expression for the speed limit is
exact and not merely a lower bound. These analytic results are then tested and
confirmed using a numerical optimization scheme. Finally we extend the analysis
to find a lower bound on the quantum speed limit in the common case where the
system is described by a drift Hamiltonian and a single control Hamiltonian.Comment: 13 page
State-dependent Trotter Limits and their approximations
The Trotter product formula is a key instrument in numerical simulations of
quantum systems. However, computers cannot deal with continuous degrees of
freedom, such as the position of particles in molecules, or the amplitude of
electromagnetic fields. It is therefore necessary to discretize these variables
to make them amenable to digital simulations. Here, we give sufficient
conditions to conclude the validity of this approximate discretized physics.
Essentially, it depends on the state-dependent Trotter error, for which we
establish explicit bounds that are also of independent interest.Comment: 16 pages, 5 figure
The Schmidt rank for the commuting operator framework
In quantum information theory, the Schmidt rank is a fundamental measure for
the entanglement dimension of a pure bipartite state. Its natural definition
uses the Schmidt decomposition of vectors on bipartite Hilbert spaces, which
does not exist (or at least is not canonically given) if the observable
algebras of the local systems are allowed to be general C*-algebras. In this
work, we generalize the Schmidt rank to the commuting operator framework where
the joint system is not necessarily described by the minimal tensor product but
by a general bipartite algebra. We give algebraic and operational definitions
for the Schmidt rank and show their equivalence. We analyze bipartite states
and compute the Schmidt rank in several examples: The vacuum in quantum field
theory, Araki-Woods-Powers states, as well as ground states and translation
invariant states on spin chains which are viewed as bipartite systems for the
left and right half chains. We conclude with a list of open problems for the
commuting operator framework.Comment: 44 pages, 3 figure
Covariant catalysis requires correlations and good quantum reference frames degrade little
Catalysts are quantum systems that open up dynamical pathways between quantum states which are otherwise inaccessible under a given set of operational restrictions while, at the same time, they do not change their quantum state. We here consider the restrictions imposed by symmetries and conservation laws, where any quantum channel has to be covariant with respect to the unitary representation of a symmetry group, and present two results. First, for an exact catalyst to be useful, it has to build up correlations to either the system of interest or the degrees of freedom dilating the given process to covariant unitary dynamics. This explains why catalysts in pure states are useless. Second, if a quantum system ("reference frame") is used to simulate to high precision unitary dynamics (which possibly violates the conservation law) on another system via a global, covariant quantum channel, then this channel can be chosen so that the reference frame is approximately catalytic. In other words, a reference frame that simulates unitary dynamics to high precision degrades only very little
Self-Adjointness of Toeplitz Operators in the Segal-Bargmann Space
We prove a new criterion that guarantees self-adjointness of Toeplitz
operator with unbounded operator-valued symbols. Our criterion applies, in
particular, to symbols with Lipschitz continuous derivatives, which is the
natural class of Hamiltonian functions for classical mechanics. For this we
extend the Berger-Coburn estimate to the case of vector-valued Segal-Bargmann
spaces. Finally, we apply our result to prove self-adjointness for a class of
(operator-valued) quadratic forms on the space of Schwartz functions in the
Schr\"odinger representation.Comment: 17 page