83,530 research outputs found
Noise and Disturbance of Qubit Measurements: An Information-Theoretic Characterisation
Information-theoretic definitions for the noise associated with a quantum
measurement and the corresponding disturbance to the state of the system have
recently been introduced [F. Buscemi et al., Phys. Rev. Lett. 112, 050401
(2014)]. These definitions are invariant under relabelling of measurement
outcomes, and lend themselves readily to the formulation of state-independent
uncertainty relations both for the joint estimate of observables (noise-noise
relations) and the noise-disturbance tradeoff. Here we derive such relations
for incompatible qubit observables, which we prove to be tight in the case of
joint estimates, and present progress towards fully characterising the
noise-disturbance tradeoff. In doing so, we show that the set of obtainable
noise-noise values for such observables is convex, whereas the conjectured form
for the set of obtainable noise-disturbance values is not. Furthermore,
projective measurements are not optimal with respect to the joint-measurement
noise or noise-disturbance tradeoffs. Interestingly, it seems that four-outcome
measurements are needed in the former case, whereas three-outcome measurements
are optimal in the latter.Comment: Minor changes, corresponds to final published version. 14 pages, 5
figure
Experimental test of an entropic measurement uncertainty relation for arbitrary qubit observables
A tight information-theoretic measurement uncertainty relation is
experimentally tested with neutron spin-1/2 qubits. The noise associated to the
measurement of an observable is defined via conditional Shannon entropies and a
tradeoff relation between the noises for two arbitrary spin observables is
demonstrated. The optimal bound of this tradeoff is experimentally obtained for
various non-commuting spin observables. For some of these observables this
lower bound can be reached with projective measurements, but we observe that,
in other cases, the tradeoff is only saturated by general quantum measurements
(i.e., positive-operator valued measures), as predicted theoretically.Comment: 6 pages, 3 figure
Measurement uncertainty relations
Measurement uncertainty relations are quantitative bounds on the errors in an
approximate joint measurement of two observables. They can be seen as a
generalization of the error/disturbance tradeoff first discussed heuristically
by Heisenberg. Here we prove such relations for the case of two canonically
conjugate observables like position and momentum, and establish a close
connection with the more familiar preparation uncertainty relations
constraining the sharpness of the distributions of the two observables in the
same state. Both sets of relations are generalized to means of order
rather than the usual quadratic means, and we show that the optimal constants
are the same for preparation and for measurement uncertainty. The constants are
determined numerically and compared with some bounds in the literature. In both
cases the near-saturation of the inequalities entails that the state (resp.
observable) is uniformly close to a minimizing one.Comment: This version 2 contains minor corrections and reformulation
A transform of complementary aspects with applications to entropic uncertainty relations
Even though mutually unbiased bases and entropic uncertainty relations play
an important role in quantum cryptographic protocols they remain ill
understood. Here, we construct special sets of up to 2n+1 mutually unbiased
bases (MUBs) in dimension d=2^n which have particularly beautiful symmetry
properties derived from the Clifford algebra. More precisely, we show that
there exists a unitary transformation that cyclically permutes such bases. This
unitary can be understood as a generalization of the Fourier transform, which
exchanges two MUBs, to multiple complementary aspects. We proceed to prove a
lower bound for min-entropic entropic uncertainty relations for any set of
MUBs, and show that symmetry plays a central role in obtaining tight bounds.
For example, we obtain for the first time a tight bound for four MUBs in
dimension d=4, which is attained by an eigenstate of our complementarity
transform. Finally, we discuss the relation to other symmetries obtained by
transformations in discrete phase space, and note that the extrema of discrete
Wigner functions are directly related to min-entropic uncertainty relations for
MUBs.Comment: 16 pages, 2 figures, v2: published version, clarified ref [30
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