High-spin measurements in an arbitrary two-qudit state

TL;DR

This paper derives an explicit analytical formula for the maximal CHSH expectation in two-qudit states with arbitrary spin measurements, revealing that high-spin measurements do not violate the CHSH inequality and show different entanglement behavior compared to spin-1/2 cases.

Contribution

It introduces the spin-$s$ correlation matrix and derives a general expression for the maximal CHSH expectation for two-qudit states with arbitrary spin measurements.

Findings

High-spin measurements do not violate the CHSH inequality in certain nonseparable states.

Maximal CHSH expectation decreases with increased entanglement in high-spin cases.

Contrasts the behavior of entanglement and CHSH violation between spin-1/2 and higher-spin systems.

Abstract

Violation of the CHSH inequality by a bipartite quantum state is now used in many quantum applications. However, the explicit analytical expression for the maximal value of the CHSH expectation under local Alice and Bob spin-$s$ measurements is still known only for $s=1/2$. In the present article, for an arbitrary state of two spin-$s$ qudits, each of dimension $d=2s+1\geq 2$, we introduce the notion of the spin-$s$ correlation matrix, which has dimension $3\times 3$ for all $s\geq \frac{1}{2}$; establish its relation to the general correlation $(d^{2}-1)\times (d^{2}-1)$ matrix of this state within the generalized Pauli representation and derive in terms of the spin-$s$ correlation matrix the explicit analytical expression for the maximal value of the CHSH expectation under local Alice and Bob spin-$s$ measurements in this state. Specifying this general expression for the two-qudit GHZ state, the nonlocal two-qudit Werner state, and some nonseparable pure two-qudit states, we find that, under local Alice and Bob high-spin ($s\geq1$) measurements in each of these nonseparable states, including the maximally entangled one, the CHSH inequality is not violated. Moreover, unlike the case of spin-$1/2$ measurements, where each pure nonseparable two-qubit state violates the CHSH inequality and the maximal value of its CHSH expectation increases monotonically with a growth of its entanglement, the situation under high-spin measurements is quite different -- for a pure two-qudit state with a higher degree of entanglement, the maximal value of the CHSH expectation turns out to be less than for a pure two-qudit state with lower entanglement and even for a separable one.