Maximally incompatible quantum observables

Toigo

2017

Entropy

Self Cite

Heisenberg's uncertainty principle has recently led to general measurement uncertainty relations for quantum systems: incompatible observables can be measured jointly or in sequence only with some unavoidable approximation, which can be quantified in various ways. The relative entropy is the natural theoretical quantifier of the information loss when a 'true' probability distribution is replaced by an approximating one. In this paper, we provide a lower bound for the amount of information that is lost by replacing the distributions of the sharp position and momentum observables, as they could be obtained with two separate experiments, by the marginals of any smeared joint measurement. The bound is obtained by introducing an entropic error function, and optimizing it over a suitable class of covariant approximate joint measurements. We fully exploit two cases of target observables: (1) n-dimensional position and momentum vectors; (2) two components of position and momentum along different directions. In (1), we connect the quantum bound to the dimension n; in (2), going from parallel to orthogonal directions, we show the transition from highly incompatible observables to compatible ones. For simplicity, we develop the theory only for Gaussian states and measurements.

Section: Proposition 11mentioning

confidence: 93%

“…A very rich literature on this topic flourished in the last 20 years, and various kinds of MURs have been proposed, based on distances between probability distributions, noise quantifications, conditional entropy, etc. [12,[14][15][16][17][18][19][20][21][27][28][29][30][31][32].…”

Section: Introductionmentioning

confidence: 99%

Measurement Uncertainty Relations for Position and Momentum: Relative Entropy Formulation

Toigo

2017

Entropy

Self Cite

“…A different approach [9,10,30,29,31,32] to the construction of compatible observables is to consider noisy versions of the target observables.…”

Section: Noise and Compatibilitymentioning

confidence: 99%

“…where p j is a classical probability, independent of the system state [9,30,31]. A review of some choices for the noise classes introduced in the literature is given in [32]; the typical choices are: (a) classical noises, (b) noises represented by compatible POVMs, (c) general POVMs.…”

Section: Noise and Compatibilitymentioning

confidence: 99%

“…Our aim in introducing the device information loss (46) and the minimum information loss (54), (55) was to have uncertainty measures, based on information theory, by which MURs could be expressed in a simple way, section 3.3; this construction produced also an incompatibility measure, the minimum information loss. The result above gives a link with the robustness measures [9,10,[29][30][31][32] which quantify the incompatibility by maximizing the visibility; in other terms, these measures are based on the ability of the target observables to maintain incompatibility against noise.…”

Section: Minimum Information Loss and Noisy Versions Of The Target Obmentioning

confidence: 99%

See 1 more Smart Citation

Entropic measurement uncertainty relations for all the infinite components of a spin vector

2020

J. Phys. Commun.

The information-theoretic formulation of quantum measurement uncertainty relations (MURs), based on the notion of relative entropy between measurement probabilities, is extended to the set of all the spin components for a generic spin s. For an approximate measurement of a spin vector, which gives approximate joint measurements of the spin components, we define the device information loss as the maximum loss of information per observable occurring in approximating the ideal incompatible components with the joint measurement at hand. By optimizing on the measuring device, we define the notion of minimum information loss. By using these notions, we show how to give a significant formulation of state independent MURs in the case of infinitely many target observables. The same construction works as well for finitely many observables, and we study the related MURs for two and three orthogonal spin components. The minimum information loss plays also the role of measure of incompatibility and in this respect it allows us to compare quantitatively the incompatibility of various sets of spin observables, with different number of involved components and different values of s.

Measurement Uncertainty Relations for Discrete Observables: Relative Entropy Formulation

Toigo

2018

Commun. Math. Phys.

Self Cite

We introduce a new information-theoretic formulation of quantum measurement uncertainty relations, based on the notion of relative entropy between measurement probabilities. In the case of a finitedimensional system and for any approximate joint measurement of two target discrete observables, we define the entropic divergence as the maximal total loss of information occurring in the approximation at hand. For fixed target observables, we study the joint measurements minimizing the entropic divergence, and we prove the general properties of its minimum value. Such a minimum is our uncertainty lower bound: the total information lost by replacing the target observables with their optimal approximations, evaluated at the worst possible state. The bound turns out to be also an entropic incompatibility degree, that is, a good information-theoretic measure of incompatibility: indeed, it vanishes if and only if the target observables are compatible, it is state-independent, and it enjoys all the invariance properties which are desirable for such a measure. In this context, we point out the difference between general approximate joint measurements and sequential approximate joint measurements; to do this, we introduce a separate index for the tradeoff between the error of the first measurement and the disturbance of the second one. By exploiting the symmetry properties of the target observables, exact values, lower bounds and optimal approximations are evaluated in two different concrete examples: (1) a couple of spin-1/2 components (not necessarily orthogonal); (2) two Fourier conjugate mutually unbiased bases in prime power dimension. Finally, the entropic incompatibility degree straightforwardly generalizes to the case of many observables, still maintaining all its relevant properties; we explicitly compute it for three orthogonal spin-1/2 components. arXiv:1608.01986v3 [math-ph] 9 Jan 2018Trivial and sharp observables An observable A is trivial if A = p1 for some probability p, where 1 is the identity of H. In particular, we will make use of the uniform distribution u X on X, u X (x) = 1/ |X|, and the trivial uniform observable U X = u X 1.An observable A is sharp if A(x) is a projection ∀x ∈ X. Note that we allow A(x) = 0 for some x, which is required when dealing with sets of observables sharing the same outcome space. Of course, for every sharp observable we have |{x : A(x) = 0}| ≤ d.Bi-observables and compatible observables When the outcome set has the product form X × Y, we speak of bi-observables. In this case, given the POVM M ∈ M(X × Y), we can introduce also the marginal observablesIn the same way, for p ∈ P(X×Y), we get the marginal probabilities p [1] ∈ P(X) and p [2] ∈ P(Y). Clearly, (M [i] ) ρ = (M ρ ) [i] ; hence there is no ambiguity in writing M ρ [i] for both probabilities. Two observables A ∈ M(X) and B ∈ M(Y) are jointly measurable or compatible if there exists a bi-observable M ∈ M(X × Y) such that M [1] = A and M [2] = B; then, we call M a joint measurement of A and B.Two classical probabilities p...