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Incompatibility of Observables as State-Independent Bound of Uncertainty Relations
For a pair of observables, they are called "incompatible", if and only if the
commutator between them does not vanish, which represents one of the key
features in quantum mechanics. The question is, how can we characterize the
incompatibility among three or more observables? Here we explore one possible
route towards this goal through Heisenberg's uncertainty relations, which
impose fundamental constraints on the measurement precisions for incompatible
observables. Specifically, we quantify the incompatibility by the optimal
state-independent bounds of additive variance-based uncertainty relations. In
this way, the degree of incompatibility becomes an intrinsic property among the
operators, but not on the quantum state. To justify our case, we focus on the
incompatibility of spin systems. For an arbitrary setting of two or three
linearly-independent Pauli-spin operators, the incompatibility is analytically
solved, the spins are maximally incompatible if and only if they are orthogonal
to each other. On the other hand, the measure of incompatibility represents a
versatile tool for applications such as testing entanglement of bipartite
states, and EPR-steering criteria.Comment: Comments are welcom
A Framework for Uncertainty Relations
Uncertainty principle, which was first introduced by Werner Heisenberg
in 1927, forms a fundamental component of quantum mechanics.
A graceful aspect of quantum mechanics is that the uncertainty
relations between incompatible observables allow for succinct quan-
titative formulations of this revolutionary idea: it is impossible to
simultaneously measure two complementary variables of a particle in
precision. In particular, information theory offers two basic ways to
express the Heisenberg’s principle: variance-based uncertainty relations
and entropic uncertainty relations.
We first investigate the uncertainty relations based on the sum of
variances and derive a family of weighted uncertainty relations to
provide an optimal lower bound for all situations. Our work indicates
that it seems unreasonable to assume a priori that incompatible
observables have equal contribution to the variance-based sum form
uncertainty relations. We also study the role of mutually exclusive
physical states in the recent work and generalize the variance-based
uncertainty relations to mutually exclusive uncertainty relations.
Next, we develop a new kind of entanglement detection criteria within
the framework of marjorization theory and its matrix representation.
By virtue of majorization uncertainty bounds, we are able to construct
the entanglement criteria which have advantage over the scalar detect-
ing algorithms as they are often stronger and tighter.
Furthermore, we explore various expression of entropic uncertainty
relations, including sum of Shannon entropies, majorization uncer-
tainty relations and uncertainty relations in presence of quantum
memory. For entropic uncertainty relations without quantum side
information, we provide several tighter bounds for multi-measurements,
with some of them also valid for RĂ©nyi and Tsallis entropies besides
the Shannon entropy. We employ majorization theory and actions
of the symmetric group to obtain an admixture bound for entropic
uncertainty relations with multi-measurements. Comparisons among
existing bounds for multi-measurements are also given. However,classical entropic uncertainty relations assume there has only classical
side information. For modern uncertainty relations, those who allowed
for non-trivial amount of quantum side information, their bounds
have been strengthened by our recent result for both two and multi-
measurements.
Finally, we propose an approach which can extend all uncertainty
relations on Shannon entropies to allow for quantum side information
and discuss the applications of our entropic framework. Combined with
our uniform entanglement frames, it is possible to detect entanglement
via entropic uncertainty relations even if there is no quantum side in-
formation. With the rising of quantum information theory, uncertainty
relations have been established as important tools for a wide range of
applications, such as quantum cryptography, quantum key distribution,
entanglement detection, quantum metrology, quantum speed limit and
so on. It is thus necessary to focus on the study of uncertainty relations
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