Contextuality in Phase Space

Asadian, Ali; Budroni, Costantino; Steinhoff, F. E. S.; Rabl, Peter; Gühne, Otfried

doi:10.1103/physrevlett.114.250403

Cited by 31 publications

(46 citation statements)

References 42 publications

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“…1(b), the three displacements implementing the logical Pauli operations (19), (20) and (23), form a triangular in phase space that encloses an area of π/2. The latter is closely related to the fact that the anti-commutators between the displacements (20), (19) and (23), vanish [25], yielding the anti-commutation relations of our logical Pauli operators:…”

Section: Pauli and Weyl-heisenberg Operatorsmentioning

confidence: 94%

“…We can also define spatial and temporal correlators among observables of the kind of Eq. (28), which have already been proven to be useful in the context of testing quantum mechanical properties in CV in [20,22,25,26,29,30]. All these works involve measurements of particularly chosen modular variables that can be expressed in the form of Eq.…”

Section: State Readout With Modular Variablesmentioning

confidence: 99%

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Quantum information processing in phase space: A modular variables approach

et al. 2016

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Binary quantum information can be fault tolerantly encoded in states defined in infinite dimensional Hilbert spaces. Such states define a computational basis, and permit a perfect equivalence between continuous and discrete universal operations. The drawback of this encoding is that the corresponding logical states are unphysical, meaning infinitely localized in phase space. We use the modular variables formalism to show that, in a number of protocols relevant for quantum information and for the realization of fundamental tests of quantum mechanics, it is possible to loosen the requirements on the logical subspace without jeopardizing their usefulness or their successful implementation. Such protocols involve measurements of appropriately chosen modular variables that permit the readout of the encoded discrete quantum information from the corresponding logical states. Finally, we demonstrate the experimental feasibility of our approach by applying it to the transverse degrees of freedom of single photons.

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Section: Pauli and Weyl-heisenberg Operatorsmentioning

confidence: 94%

Section: State Readout With Modular Variablesmentioning

confidence: 99%

Quantum information processing in phase space: A modular variables approach

et al. 2016

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“…For example, they exhibit nonlocal Heisenberg equations of motion [7]. Modular operators have been proposed for testing macrorealism via Leggett-Garg inequalities (LGI) [8] as well as contextuality with continuous-variable systems [9]. The commutation of modular position and momentum operators allows their use as stabilizers for fault-tolerant continuous-variable computation, as proposed by Gottesman, Kitaev, and Preskill (GKP) [10].…”

Section: Introductionmentioning

confidence: 99%

Sequential Modular Position and Momentum Measurements of a Trapped Ion Mechanical Oscillator

et al. 2018

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The noncommutativity of position and momentum observables is a hallmark feature of quantum physics. However, this incompatibility does not extend to observables that are periodic in these base variables. Such modular-variable observables have been suggested as tools for fault-tolerant quantum computing and enhanced quantum sensing. Here, we implement sequential measurements of modular variables in the oscillatory motion of a single trapped ion, using state-dependent displacements and a heralded nondestructive readout. We investigate the commutative nature of modular variable observables by demonstrating no-signaling in time between successive measurements, using a variety of input states. Employing a different periodicity, we observe signaling in time. This also requires wave-packet overlap, resulting in quantum interference that we enhance using squeezed input states. The sequential measurements allow us to extract two-time correlators for modular variables, which we use to violate a Leggett-Garg inequality. Signaling in time and Leggett-Garg inequalities serve as efficient quantum witnesses, which we probe here with a mechanical oscillator, a system that has a natural crossover from the quantum to the classical regime.

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“…Together with a theorem bounding the expectation value of general anti-commuting observables we furthermore demonstrate the usefulness of the HW-basis in entanglement detection by deriving a general method to obtain linear entanglement witnesses. Interesting roads for the future would be to investigate whether the set of observables presented in this work can simplify the analysis stabilizer systems defined in terms of HW operators [38]. And since our construction of Bloch vectors is closely related to the phase-space picture it would be interesting to investigate applications for significant problems where phase space formulations are exploited, such as finding magic states [39], determining the Wigner function [40,41] or the problem of finding symmetric informationally complete POVM [42].…”

mentioning

confidence: 99%

“…Being a natural choice for higher-dimensional spin representations they have been extensively used in parametrizations of corresponding density matrices [4,13] and in entanglement detection. Other choices, such as the HeisenbergWeyl (HW) operators, and the non-Hermitian generalization of the 1/2-spin Pauli operators, have also been explored [3,[14][15][16][17][18]. While having some convenient properties, they are unitary, but not Hermitian matrices.…”

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confidence: 99%

Heisenberg-Weyl Observables: Bloch vectors in phase space

et al. 2016

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We introduce a Hermitian generalization of Pauli matrices to higher dimensions which is based on Heisenberg-Weyl operators. The complete set of Heisenberg-Weyl observables allows us to identify a real-valued Bloch vector for an arbitrary density operator in discrete phase space, with a smooth transition to infinite dimensions. Furthermore, we derive bounds on the sum of expectation values of any set of anti-commuting observables. Such bounds can be used in entanglement detection and we show that Heisenberg-Weyl observables provide a first non-trivial example beyond the dichotomic case.PACS numbers: 07.10. Cm, 03.65.Ta, 03.65.Ud Introduction. The Bloch representation is a cornerstone of analyzing the characteristics of quantum systems. It was first introduced for two-level systems by Bloch [1] and has since been used in a wide variety of settings (for comprehensive reviews consult [2][3][4]). It is usually defined via a decomposition of the density matrix into a complete operator basis. Defining quantum states via the expectation values of a complete set of measurements gives a very practical account of their properties. Apart from being intuitive this approach gives convenient solutions for Hamiltonian evolutions (see, e.g., [5]) and has found many applications in entanglement theory [6][7][8][9][10][11][12].However, there is not a unique Bloch decomposition of a given quantum state; this fact may favor a particular representation over another for certain tasks. The canonical choice for a complete basis of observables is usually given by the so-called generalized Gell-Mann matrices, generators of the special unitary group [SU(d)]. Being a natural choice for higher-dimensional spin representations they have been extensively used in parametrizations of corresponding density matrices [4,13] and in entanglement detection. Other choices, such as the HeisenbergWeyl (HW) operators, and the non-Hermitian generalization of the 1/2-spin Pauli operators, have also been explored [3,[14][15][16][17][18]. While having some convenient properties, they are unitary, but not Hermitian matrices. Thus the associated Bloch vector itself has imaginary entries that do not correspond to expectation values of physical observables. This makes both the theoretical description and experimental realization more cumbersome and therefore requires more effort to identify the relevant parameters.In this article we introduce a Hermitian Bloch-basis derived from HW-operators. It conveniently combines multiple desirable properties of Bloch vector parametrizations, allowing a smooth transition to the infinite dimensional limit. We first explore properties such as (anti-) commutativity. We then proceed with the derivation of an inequality which bounds sums of anti-commuting observables with which we show in exemplary cases how one can construct powerful criteria for entanglement detection in this new basis. Finally we present a scheme for practical experimental acquisition through a Ramseytype measurement.Phase-space displacements. We start our a...

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Contextuality in Phase Space

Cited by 31 publications

References 42 publications

Quantum information processing in phase space: A modular variables approach

Quantum information processing in phase space: A modular variables approach

Sequential Modular Position and Momentum Measurements of a Trapped Ion Mechanical Oscillator

Heisenberg-Weyl Observables: Bloch vectors in phase space

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