N-term pairwise-correlation inequalities, steering, and joint measurability

Abstract

Chained inequalities involving pairwise correlations of qubit observables in the equatorial plane are constructed based on the positivity of a sequence of moment matrices. When a jointly measurable set of fuzzy POVMs is employed in first measurement of every pair of sequential measurements, the chained pairwise correlations do not violate the classical bound imposed by the moment matrix positivity. We identify that incompatibility of the set of POVMs employed in first measurements is only necessary, but not sufficient, in general, for the violation of the inequality. On the other hand, there exists a one-to-one equivalence between the degree of incompatibility (which quantifies the joint measurability) of the equatorial qubit POVMs and the optimal violation of a non-local steering inequality, proposed by Jones and Wiseman (Phys. Rev. A, 84, 012110 (2011)). To this end, we construct a local analogue of this steering inequality in a single qubit system and show that its violation is a mere reflection of measurement incompatibility of equatorial qubit POVMs, employed in first measurements in the sequential unsharp-sharp scheme.

Numerical determination of monopole scaling dimension in parity-invariant three-dimensional noncompact QED

“…However, such a method is not practical since it requires computations of multiple values of τ per Monte Carlo sample point in the parameter space, and further introduces unwanted systematic errors from the → ∞ extrapolations at fixed τ . As was demonstrated in the case of monopole correlators [31], a better method is to make use of scaling of correlators near the infrared fixed point. That is, one expects the scaling…”

“…Complementary to such bootstrap computations, it was demonstrated [31] that a direct way to compute monopole scaling dimensions using lattice computation is to couple such theories to a background field A QQ (x; τ ) = A Q (x; x 0 ) − A Q (x; x 0 +tτ ) that gives rise to a monopole at x 0 and an anti-monopole at x 0 +tτ , which are separated by a distance τ and compute the scaling of the partition function…”

“…Understanding such infra-red quantum phases obtained by tuning parameters of the underlying QFT is an ongoing field of research (c.f., [6]). Similar studies of the critical number of flavors, N NC c , in non-compact QED 3 (referred to as nc-QED 3 ) is continued to be investigated through ab initio lattice simulations [7][8][9][10][11] as well as through other approximation methods [12][13][14][15][16][17]. Unlike non-compact QED 3 , the presence of monopoles in the compact version even as the continuum limit is approached, is a technical challenge to numerical studies due to the presence of many small eigenvalues of the threedimension Dirac operator [18].…”

“…The stronger assumption is that this continues to remain so until N = N C c . The second assumption is based more on numerical works [10,11] that strongly indicate that N NC c < 2. This also means that only the dressed, gauge-invariant monopole operators become relevant at N = N C c and other U(N ) symmetry breaking operator, such as the four-fermi operators, remain irrelevant [21].…”

“…(11) on toroidal lattice is to take care of both the lattice discretization as well as the periodicity correctly. We checked through a full fledged computation in the case of N = 2 QED 3 that the difference in Z Q /Z 0 between the minimum on the torus as defined above and the discretized field of Dirac monopole-antimonopole pair as defined in [31] is, however, marginal.…”