60 research outputs found
Probability representation and quantumness tests for qudits and two-mode light states
Using tomographic-probability representation of spin states, quantum behavior
of qudits is examined. For a general j-qudit state we propose an explicit
formula of quantumness witnetness whose negative average value is incompatible
with classical statistical model. Probability representations of quantum and
classical (2j+1)-level systems are compared within the framework of quantumness
tests. Trough employing Jordan-Schwinger map the method is extended to check
quantumness of two-mode light states.Comment: 5 pages, 2 figures, PDFLaTeX, Contribution to the 11th International
Conference on Squeezed States and Uncertainty Relations (ICSSUR'09), June
22-26, 2009, Olomouc, Czech Republi
Symmetric informationally complete positive operator valued measure and probability representation of quantum mechanics
Symmetric informationally complete positive operator valued measures
(SIC-POVMs) are studied within the framework of the probability representation
of quantum mechanics. A SIC-POVM is shown to be a special case of the
probability representation. The problem of SIC-POVM existence is formulated in
terms of symbols of operators associated with a star-product quantization
scheme. We show that SIC-POVMs (if they do exist) must obey general rules of
the star product, and, starting from this fact, we derive new relations on
SIC-projectors. The case of qubits is considered in detail, in particular, the
relation between the SIC probability representation and other probability
representations is established, the connection with mutually unbiased bases is
discussed, and comments to the Lie algebraic structure of SIC-POVMs are
presented.Comment: 22 pages, 1 figure, LaTeX, partially presented at the Workshop
"Nonlinearity and Coherence in Classical and Quantum Systems" held at the
University "Federico II" in Naples, Italy on December 4, 2009 in honor of
Prof. Margarita A. Man'ko in connection with her 70th birthday, minor
misprints are corrected in the second versio
MuSR method and tomographic probability representation of spin states
Muon spin rotation/relaxation/resonance (MuSR) technique for studying matter
structures is considered by means of a recently introduced probability
representation of quantum spin states. A relation between experimental MuSR
histograms and muon spin tomograms is established. Time evolution of muonium,
anomalous muonium, and a muonium-like system is studied in the tomographic
representation. Entanglement phenomenon of a bipartite muon-electron system is
investigated via tomographic analogues of Bell number and positive partial
transpose (PPT) criterion. Reconstruction of the muon-electron spin state as
well as the total spin tomography of composed system is discussed.Comment: 20 pages, 4 figures, LaTeX, submitted to Journal of Russian Laser
Researc
Unitary and Non-Unitary Matrices as a Source of Different Bases of Operators Acting on Hilbert Spaces
Columns of d^2 x N matrices are shown to create different sets of N operators
acting on -dimensional Hilbert space. This construction corresponds to a
formalism of the star-product of operator symbols. The known bases are shown to
be partial cases of generic formulas derived by using d^2 x N matrices as a
source for constructing arbitrary bases. The known examples of the SIC-POVM,
MUBs, and the phase-space description of qubit states are considered from the
viewpoint of the developed unified approach. Star-product schemes are
classified with respect to associated d^2 x N matrices. In particular, unitary
matrices correspond to self-dual schemes. Such self-dual star-product schemes
are shown to be determined by dequantizers which do not form POVM.Comment: 12 pages, 1 figure, 1 table, to appear in Journal of Russian Laser
Researc
Simulability of observables in general probabilistic theories
The existence of incompatibility is one of the most fundamental features of quantum theory and can be found at the core of many of the theory's distinguishing features, such as Bell inequality violations and the no-broadcasting theorem. A scheme for obtaining new observables from existing ones via classical operations, the so-called simulation of observables, has led to an extension of the notion of compatibility for measurements. We consider the simulation of observables within the operational framework of general probabilistic theories and introduce the concept of simulation irreducibility. While a simulation irreducible observable can only be simulated by itself, we show that any observable can be simulated by simulation irreducible observables, which in the quantum case correspond to extreme rank-1 positive-operator-valued measures. We also consider cases where the set of simulators is restricted in one of two ways: in terms of either the number of simulating observables or their number of outcomes. The former is seen to be closely connected to compatibility and k compatibility, whereas the latter leads to a partial characterization for dichotomic observables. In addition to the quantum case, we further demonstrate these concepts in state spaces described by regular polygons
Necessary condition for incompatibility of observables in general probabilistic theories
We quantify the intrinsic noise content of an observable in a general probabilistic theory and derive a noise content inequality for incompatible observables. We apply the derived inequality to standard quantum theory, the quantum theory of processes, and polytope state spaces. The noise content for positive operator-valued measures takes a particularly simple form and equals the sum of minimal eigenvalues of all the effects. We illustrate our findings with a number of examples including the introduced notion of reverse observables
Operational Restrictions in General Probabilistic Theories
The formalism of general probabilistic theories provides a universal paradigm that is suitable for describing various physical systems including classical and quantum ones as particular cases. Contrary to the usual no-restriction hypothesis, the set of accessible meters within a given theory can be limited for different reasons, and this raises a question of what restrictions on meters are operationally relevant. We argue that all operational restrictions must be closed under simulation, where the simulation scheme involves mixing and classical post-processing of meters. We distinguish three classes of such operational restrictions: restrictions on meters originating from restrictions on effects; restrictions on meters that do not restrict the set of effects in any way; and all other restrictions. We fully characterize the first class of restrictions and discuss its connection to convex effect subalgebras. We show that the restrictions belonging to the second class can impose severe physical limitations despite the fact that all effects are accessible, which takes place, e.g., in the unambiguous discrimination of pure quantum states via effectively dichotomic meters. We further demonstrate that there are physically meaningful restrictions that fall into the third class. The presented study of operational restrictions provides a better understanding on how accessible measurements modify general probabilistic theories and quantum theory in particular
Inverse spin-s portrait and representation of qudit states by single probability vectors
Using the tomographic probability representation of qudit states and the
inverse spin-portrait method, we suggest a bijective map of the qudit density
operator onto a single probability distribution. Within the framework of the
approach proposed, any quantum spin-j state is associated with the
(2j+1)(4j+1)-dimensional probability vector whose components are labeled by
spin projections and points on the sphere. Such a vector has a clear physical
meaning and can be relatively easily measured. Quantum states form a convex
subset of the 2j(4j+3) simplex, with the boundary being illustrated for qubits
(j=1/2) and qutrits (j=1). A relation to the (2j+1)^2- and
(2j+1)(2j+2)-dimensional probability vectors is established in terms of spin-s
portraits. We also address an auxiliary problem of the optimum reconstruction
of qudit states, where the optimality implies a minimum relative error of the
density matrix due to the errors in measured probabilities.Comment: 23 pages, 4 figures, PDF LaTeX, submitted to the Journal of Russian
Laser Researc
Qubit portrait of the photon-number tomogram and separability of two-mode light states
In view of the photon-number tomograms of two-mode light states, using the
qubit-portrait method for studying the probability distributions with infinite
outputs, the separability and entanglement detection of the states are studied.
Examples of entangled Gaussian state and Schr\"{o}dinger cat state are
discussed.Comment: 20 pages, 6 figures, TeX file, to appear in Journal of Russian Laser
Researc
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