Experimental Test of Residual Error-Disturbance Uncertainty Relations for Mixed Spin-½States

Abstract

The indeterminacy inherent in quantum measurements is an outstanding character of quantum theory, which manifests itself typically in the uncertainty principle. In the last decade, several universally valid forms of error-disturbance uncertainty relations were derived for completely general quantum measurements for arbitrary states. Subsequently, Branciard established a form that is optimal for spin measurements for some pure states. However, the bound in his inequality is not stringent for mixed states. One of the present authors recently derived a new bound tight in the corresponding mixed state case. Here, a neutron-optical experiment is carried out to investigate this new relation: it is tested whether error and disturbance of quantum measurements disappear or persist in mixing up the measured ensemble. The attainability of the new bound is experimentally observed, falsifying the tightness of Branciard's bound for mixed spin states.

Soundness and completeness of quantum root-mean-square errors

“…Universally valid uncertainty relations. In what follows, we shall show that all the universally valid measurement uncertainty relations obtained so far [8,10,11,[13][14][15][16] for the noise-operator based q-rms error are maintained with the same forms by property (V) and that their experimental confirmations reported so far [18][19][20][21][22][23][24] for dichotomic measurements are also reinterpreted to confirm the relations for the new error measure by property (VI). Moreover, the state-independent formulation based on this notion maintains Heisenberg's original form for the measurement uncertainty relation, whereas the state-dependent formulation violates it.…”

Error-Disturbance Trade-off in Sequential Quantum Measurements

“…Heisenberg's intuition, (Â)η(B) ≥ 1 2 | [Â,B] | [3], was negated in the experiments [9][10][11][12][13][14]. Ozawa recently showed that the error and disturbance quantified by the root-meansquare (rms) deviations, (Â) = (NÂ −Â ⊗Î) 2 (1/2) and η(B) = (NB −B ⊗Î) 2 (1/2) , satisfy the following relation [15][16][17], (1) was soon verified in a number of experiments [9][10][11][12], which thereafter has stimulated a great deal of interest in the investigation of joint measurability of two observables with fruitful outcomes [1,2,13,14,[18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. Quantum mechanics is fundamentally probabilistic.…”

“…It is not physically reasonable that (Â) can vanish for noisy measurements, while η(B) needs not to vanish for nondisturbing measurements. Note that in [31] the right hand side of relation (1) is replaced by a stronger constant given by D AB = T r(| √ ρ[Â,B] √ ρ|)/2, which does not vanish for a maximally-mixed state. In addition, relation (1) varies with the relabelings of eigenvalues and measurement outcomes.…”

Dynamically unpolarized single-photon source in diamond with intrinsic randomness

“…The new ‘dynamically unpolarized’ property, i.e. the property of no polarization correlation between time adjacent single photons, constitutes a new criterion of randomness for unpolarized single photons which is useful for true random number generation789 as well as for tests of fundamental quantum mechanics pertaining to mixed states3456.…”

“…If we measure the polarization of a single photon in a certain basis, the measurement outcome will be either of the two orthogonal polarizations with even probability, regardless of the choice of basis. The unpolarized mixed state of a single photon is crucial to explore fundamental problems for mixed states such as testing error-disturbance relations of measurements34 and exploring the nature of mixed states themselves56, as well as to realize genuine random number generators789. For these applications, in addition to the statical statistics predicted by the density matrix, we must examine the dynamical statistics of the measurement outcomes.…”