137 research outputs found
Symmetric Informationally Complete Measurements of Arbitrary Rank
There has been much interest in so-called SIC-POVMs: rank 1 symmetric
informationally complete positive operator valued measures. In this paper we
discuss the larger class of POVMs which are symmetric and informationally
complete but not necessarily rank 1. This class of POVMs is of some independent
interest. In particular it includes a POVM which is closely related to the
discrete Wigner function. However, it is interesting mainly because of the
light it casts on the problem of constructing rank 1 symmetric informationally
complete POVMs. In this connection we derive an extremal condition alternative
to the one derived by Renes et al.Comment: Contribution to proceedings of International Conference on Quantum
Optics, Minsk, 200
Quantum Theory is a Quasi-stochastic Process Theory
There is a long history of representing a quantum state using a
quasi-probability distribution: a distribution allowing negative values. In
this paper we extend such representations to deal with quantum channels. The
result is a convex, strongly monoidal, functorial embedding of the category of
trace preserving completely positive maps into the category of quasi-stochastic
matrices. This establishes quantum theory as a subcategory of quasi-stochastic
processes. Such an embedding is induced by a choice of minimal informationally
complete POVM's. We show that any two such embeddings are naturally isomorphic.
The embedding preserves the dagger structure of the categories if and only if
the POVM's are symmetric, giving a new use of SIC-POVM's, objects that are of
foundational interest in the QBism community. We also study general convex
embeddings of quantum theory and prove a dichotomy that such an embedding is
either trivial or faithful.Comment: In Proceedings QPL 2017, arXiv:1802.0973
Verifying the quantumness of bipartite correlations
Entanglement is at the heart of most quantum information tasks, and therefore
considerable effort has been made to find methods of deciding the entanglement
content of a given bipartite quantum state. Here, we prove a fundamental
limitation to deciding if an unknown state is entangled or not: we show that
any quantum measurement which can answer this question necessarily gives enough
information to identify the state completely. Therefore, only prior information
regarding the state can make entanglement detection less expensive than full
state tomography in terms of the demanded quantum resources. We also extend our
treatment to other classes of correlated states by considering the problem of
deciding if a state is NPT, discordant, or fully classically correlated.
Remarkably, only the question related to quantum discord can be answered
without resorting to full state tomography
Unknown Quantum States and Operations, a Bayesian View
The classical de Finetti theorem provides an operational definition of the
concept of an unknown probability in Bayesian probability theory, where
probabilities are taken to be degrees of belief instead of objective states of
nature. In this paper, we motivate and review two results that generalize de
Finetti's theorem to the quantum mechanical setting: Namely a de Finetti
theorem for quantum states and a de Finetti theorem for quantum operations. The
quantum-state theorem, in a closely analogous fashion to the original de
Finetti theorem, deals with exchangeable density-operator assignments and
provides an operational definition of the concept of an "unknown quantum state"
in quantum-state tomography. Similarly, the quantum-operation theorem gives an
operational definition of an "unknown quantum operation" in quantum-process
tomography. These results are especially important for a Bayesian
interpretation of quantum mechanics, where quantum states and (at least some)
quantum operations are taken to be states of belief rather than states of
nature.Comment: 37 pages, 3 figures, to appear in "Quantum Estimation Theory," edited
by M.G.A. Paris and J. Rehacek (Springer-Verlag, Berlin, 2004
Connections of geometric measure of entanglement of pure symmetric states to quantum state estimation
We study the geometric measure of entanglement (GM) of pure symmetric states
related to rank-one positive-operator-valued measures (POVMs) and establish a
general connection with quantum state estimation theory, especially the maximum
likelihood principle. Based on this connection, we provide a method for
computing the GM of these states and demonstrate its additivity property under
certain conditions. In particular, we prove the additivity of the GM of pure
symmetric multiqubit states whose Majorana points under Majorana representation
are distributed within a half sphere, including all pure symmetric three-qubit
states. We then introduce a family of symmetric states that are generated from
mutually unbiased bases (MUBs), and derive an analytical formula for their GM.
These states include Dicke states as special cases, which have already been
realized in experiments. We also derive the GM of symmetric states generated
from symmetric informationally complete POVMs (SIC~POVMs) and use it to
characterize all inequivalent SIC~POVMs in three-dimensional Hilbert space that
are covariant with respect to the Heisenberg--Weyl group. Finally, we describe
an experimental scheme for creating the symmetric multiqubit states studied in
this article and a possible scheme for measuring the permanent of the related
Gram matrix.Comment: 11 pages, 1 figure, published versio
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