Relating an entanglement measure with statistical correlators for two-qudit mixed states using only a pair of complementary observables

Abstract

We focus on characterizing entanglement of high dimensional bipartite states using various statistical correlators for two-qudit mixed states. The salient results obtained are as follows: (a) A scheme for determining the entanglement measure given by Negativity is explored by analytically relating it to the widely used statistical correlators viz. mutual predictability, mutual information and Pearson Correlation coefficient for different types of bipartite arbitrary dimensional mixed states. Importantly, this is demonstrated using only a pair of complementary observables pertaining to the mutually unbiased bases. (b) The relations thus derived provide the separability bounds for detecting entanglement obtained for a fixed choice of the complementary observables, while the bounds per se are state-dependent. Such bounds are compared with the earlier suggested separability bounds. (c) We also show how these statistical correlators can enable distinguishing between the separable, distillable and bound entanglement domains of the one-parameter Horodecki two-qutrit states. Further, the relations linking Negativity with the statistical correlators have been derived for such Horodecki states in the domain of distillable entanglement. Thus, this entanglement characterisation scheme based on statistical correlators and harnessing complementarity of the obsevables opens up a potentially rich direction of study which is applicable for both distillable and bound entangled states.

Direct determination of arbitrary dimensional entanglement monotones using statistical correlators and minimal complementary measurements

“…However, Maccone et al justified this conjecture only by showing its applicability for bipartite qubits and the validity of this conjecture has remained uninvestigated for dimensions d > 2. In this work, we justify the validity of this conjecture for pure bipartite qutrits by deriving an analytic relation between PCC and N for any arbitrary dimension d (details of this derivation are provided in a separate paper [22]), and this relation is tested by applying it to quantify the amount of entanglement in our experimentally generated pure bipartite qutrits (d = 3). PCC for a joint observable X A ⊗ X B is given by equation ( 3) where the observable, X can be written as X = i,j: i|j =0 |j i| and is such that it projects each computational basis |i to the superposition of all the remaining basis vectors of the complete set.…”

“…for the state, ψ AB as given by the equation ( 4). Furthermore, it can be shown that as discussed in [22],…”

“…Hence, this constitutes the first direct experimental determination of the standard EMs, and that too, using only one set of joint local complementary measurements in the performed experiment, for the higher dimensional states. Furthermore, in a separate work [22], it has been shown how the scheme formulated here can be extended for different types of empirically relevant mixed bipartite entangled states.…”

Inverse-Design of High-Dimensional Quantum Optical Circuits in a Complex Medium

“…Knowledge of the precise noise present in our experiment can lead to stronger entanglement criteria 85,86 , but since we do not know the relevant noise, we adopt an approach where the state is treated as uncharacterized. In our analysis, we assume that Alice's measurements correspond to the computational basis and the first MUB, respectively.…”