Constructing Extremal Compatible Quantum Observables by Means of Two Mutually Unbiased Bases

Carmeli, Claudio; Cassinelli, Gianni; Toigo, Alessandro

doi:10.1007/s10701-019-00274-y

Cited by 4 publications

(14 citation statements)

References 31 publications

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“…In even dimensions, by decomposing the space into many qubit subspaces and by having MUBs on each of them, we can reach again the value of equation (80). For instance in dimension d 4 = this means (75)), the lowest value we found for p h (that is, equation (80) for even dimensions and numerical results based on an analytical construction described in the main text for odd dimensions), the lowest value we found for d h (equation (80)), and the lower bound (28).…”

Section: Higher Dimensions 431 Dimension D 3 =mentioning

confidence: 71%

“…For non-commuting measurement operators, however, the anticommutator might have some negative eigenvalues for which the remaining terms are supposed to compensate. Note that the same construction for parent POVMs has recently been used in [28].…”

Section: Lower Boundsmentioning

confidence: 99%

“…The noise set depends on the specific measurements, which makes this measure different than all the other measures considered in this work. It has been investigated in many works [12,28,30,[32][33][34][35][36], often in relation with Einstein-Podolsky-Rosen steering. This specific type of noise has also been considered in scenarios different from incompatibility [37].…”

Section: Incompatibility Depolarising Robustness 311 Definition Anmentioning

confidence: 99%

“…the measurement generating a uniform distribution of outcomes regardless of the state. It has been investigated in many works [7,28,31,34,35,40], and also in the framework of general probabilistic theories [41,42].…”

Section: Incompatibility Random Robustness 321 Definition and Propmentioning

confidence: 99%

“…the feasible point presented in equation(28) of the main text. It is easy to check that this choice of x and y saturates equation (C5) for a particular value of c ab , which we refer to as the critical overlap Note that this coincides with the MUB overlap only in dimension d 2 = .…”

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confidence: 99%

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Incompatibility robustness of quantum measurements: a unified framework

2019

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In quantum mechanics performing a measurement is an invasive process which generally disturbs the system. Due to this phenomenon, there exist incompatible quantum measurements, i.e. measurements that cannot be simultaneously performed on a single copy of the system. It is then natural to ask what the most incompatible quantum measurements are. To answer this question, several measures have been proposed to quantify how incompatible a set of measurements is, however their properties are not well-understood. In this work, we develop a general framework that encompasses all the commonly used measures of incompatibility based on robustness to noise. Moreover, we propose several conditions that a measure of incompatibility should satisfy, and investigate whether the existing measures comply with them. We find that some of the widely used measures do not fulfil these basic requirements. We also show that when looking for the most incompatible pairs of measurements, we obtain different answers depending on the exact measure. For one of the measures, we analytically prove that projective measurements onto two mutually unbiased bases are among the most incompatible pairs in every dimension. However, for some of the remaining measures we find that some peculiar measurements turn out to be even more incompatible.performed simultaneously by performing the parent measurement. If such a parent measurement does not exist, we say that the measurements are incompatible (or not jointly measurable). We remark here that other notions of compatibility, such as commutativity, non-disturbance and coexistence, are also used in the literature [1,3]; let us for completeness briefly explain how they are related. Commutativity of a measurement pair implies nondisturbance, which in turn implies joint measurability, which then implies coexistence. Moreover, it is known that none of the converse implications hold in general, therefore these notions are strictly distinct [4]. In this work we focus solely on the notion of joint measurability, because the existence (or not) of a parent measurement has a clear operational meaning. Therefore, throughout the present paper we use the terms '(in) compatibility' and '(non-)joint measurability' interchangeably. It is important to notice that whenever two measurements are compatible, they cannot be used to produce quantum advantage in tasks like Bell nonlocality [5] or Einstein-Podolsky-Rosen steering [6,7]. Moreover, it was recently shown that joint measurability is equivalent to a specific notion of classicality, namely, preparation non-contextuality [8,9]. Hence, one may think of compatible measurements as 'classical' , and incompatible measurements as a resource for the above tasks. Therefore, it is of fundamental importance to characterise and understand the structure of incompatible measurements.What is particularly important is to go beyond the dichotomy of compatible and incompatible measurements, and quantify to what extent a pair of measurements is incompatible. A natural framework for this qua...

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Section: Higher Dimensions 431 Dimension D 3 =mentioning

confidence: 71%

Section: Lower Boundsmentioning

confidence: 99%

Section: Incompatibility Depolarising Robustness 311 Definition Anmentioning

confidence: 99%

Section: Incompatibility Random Robustness 321 Definition and Propmentioning

confidence: 99%

mentioning

confidence: 99%

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Incompatibility robustness of quantum measurements: a unified framework

2019

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Geometry of skew information-based quantum coherence

Huang

Fei

et al. 2020

Commun. Theor. Phys.

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We study the skew information-based coherence of quantum states and derive explicit formulas for Werner states and isotropic states in a set of autotensor of mutually unbiased bases (AMUBs). We also give surfaces of skew information-based coherence for Bell-diagonal states and a special class of X states in both computational basis and in mutually unbiased bases. Moreover, we depict the surfaces of the skew information-based coherence for Bell-diagonal states under various types of local nondissipative quantum channels. The results show similar as well as different features compared with relative entropy of coherence and l 1 norm of coherence.

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Amount of quantum coherence needed for measurement incompatibility

2022

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A pair of quantum observables diagonal in the same "incoherent" basis can be measured jointly, so some coherence is obviously required for measurement incompatibility. Here we first observe that coherence in a single observable is linked to the diagonal elements of any observable jointly measurable with it, leading to a general criterion for the coherence needed for incompatibility. Specialising to the case where the second observable is incoherent (diagonal), we develop a concrete method for solving incompatibility problems, tractable even in large systems by analytical bounds, without resorting to numerical optimisation. We verify the consistency of our method by a quick proof of the known noise bound for mutually unbiased bases, and apply it to study emergent classicality in the spin-boson model of an N -qubit open quantum system. Finally, we formulate our theory in an operational resource-theoretic setting involving "genuinely incoherent operations" used previously in the literature, and show that if the coherence is insufficient to sustain incompatibility, the associated joint measurements have sequential implementations via incoherent instruments. with! j G(i, j) = M(i) for all i, and ! i G(i, j) = F(j) for all j; otherwise M and F are incompatible [45]. A. Joint measurability criteria III. INCOMPATIBILITY DUE TO A COHERENCE PATTERNHere we specialise to the physically relevant class of observables, introducing first their general structure, and then focusing on the spin-boson model. A. General consideration of coherence matricesStarting with a brief motivation, we consider again the above qubit example: when m 3 = 0, we have M = C * Q 0 where C is a PSD matrix with c 00 = c 11 = 1, c 01 = m 1 − im 2 , and Q 0 (0) = 1 2 ( + σ x ). Note that

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Constructing Extremal Compatible Quantum Observables by Means of Two Mutually Unbiased Bases

Cited by 4 publications

References 31 publications

Incompatibility robustness of quantum measurements: a unified framework

Incompatibility robustness of quantum measurements: a unified framework

Geometry of skew information-based quantum coherence

Amount of quantum coherence needed for measurement incompatibility

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