Quantum Errors and Disturbances: Response to Busch, Lahti and Werner

Appleby, D. M.

doi:10.3390/e18050174

Cited by 13 publications

(21 citation statements)

References 60 publications

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“…Recall that in our parametrization θ i is the deviation of σ i /2 from its true value, if we were to parametrize the state as ρ(θ) = ρ 0 +θ·σ/2 the lower bounds in Eqs (30),(34). and(35) would be 4 times bigger.…”

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confidence: 99%

Uncertainty and trade-offs in quantum multiparameter estimation

Kull¹,

Guérin²,

Verstraete³

2020

J. Phys. A: Math. Theor.

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Uncertainty relations in quantum mechanics express bounds on our ability to simultaneously obtain knowledge about expectation values of non-commuting observables of a quantum system. They quantify trade-offs in accuracy between complementary pieces of information about the system. In Quantum multiparameter estimation, such trade-offs occur for the precision achievable for different parameters characterizing a density matrix: an uncertainty relation emerges between the achievable variances of the different estimators. This is in contrast to classical multiparameter estimation, where simultaneous optimal precision is attainable in the asymptotic limit. We study trade-off relations that follow from known tight bounds in quantum multiparameter estimation. We compute trade-off curves and surfaces from Cramér-Rao type bounds which provide a compelling graphical representation of the information encoded in such bounds, and argue that bounds on simultaneously achievable precision in quantum multiparameter estimation should be regarded as measurement uncertainty relations. From the state-dependent bounds on the expected cost in parameter estimation, we derive a state independent uncertainty relation between the parameters of a qubit system.Ever since its first formulation, the uncertainty principle has seen many refinements and clarifications. As quantum theory developed, its state-of-the-art concepts and mathematical tools were used to formulate in precise terms the ideas which were put forward in Heisenberg's 1927 paper [1]. As a result, our current understanding of the uncertainties inherent in quantum mechanics is spelled out in a collection of theorems pertaining to well defined operational tasks.Soon after Heisenberg's paper, rigorous proofs of his uncertainty relations were formulated [2][3][4]. Those are usually referred to as preparation uncertainty relations. Most well known is the relation due to Weyl and Robertson arXiv:2002.05961v1 [quant-ph]

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confidence: 99%

Uncertainty and trade-offs in quantum multiparameter estimation

Kull¹,

Guérin²,

Verstraete³

2020

J. Phys. A: Math. Theor.

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“…Theorem 11. Let X and Y be the two orthogonal spin-1/2 components (30). Then, there is a unique optimal approximate joint measurement of X and Y, that is the bi-observable M 0 (x, y) = 1 4…”

Section: Entropic Incompatibility Degree and Optimal Measurementsmentioning

confidence: 99%

“…Let the target observables A 1 , A 2 , A 3 be three mutually orthogonal spin-1/2 components, that is, the sharp observables X, Y, Z ∈ M({+1, −1}) associated with the three Pauli matrices; the observables X, Y are given in (30), and…”

Section: Three Orthogonal Spin-1/2 Componentsmentioning

confidence: 99%

“…When the target observables are the orthogonal spin-1/2 components X and Y in (30), the symmetries of our system increase from D 2 to the enlarged dihedral group D 4 . Here we recall that D 4 ⊂ SO(3) is the order 8 group of the 90 • rotations around the k-axis, together with the 180 • rotations around i, j, n and m; clearly, D 2 ⊂ D 4 .…”

Section: B2 the Case Of Two Orthogonal Componentsmentioning

confidence: 99%

“…Example 1 (Two orthogonal spin-1/2 components). Let us consider as target observables the two sharp spin-1/2 components X, Y ∈ M({+1, −1}) associated with the first two Pauli matrices, defined in (30). This is the easiest example of two Fourier conjugate MUBs.…”

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Measurement Uncertainty Relations for Discrete Observables: Relative Entropy Formulation

Barchielli

Gregoratti

Toigo

2018

Commun. Math. Phys.

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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...

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Quantum Uncertainty and Unruh Temperature

Bı́ró

Jakovác

2019

SpringerBriefs in Physics

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Quantum Errors and Disturbances: Response to Busch, Lahti and Werner

Cited by 13 publications

References 60 publications

Uncertainty and trade-offs in quantum multiparameter estimation

Uncertainty and trade-offs in quantum multiparameter estimation

Measurement Uncertainty Relations for Discrete Observables: Relative Entropy Formulation

Quantum Uncertainty and Unruh Temperature

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