Nonviolation of the CHSH inequality under local spin-1 measurements on two spin qutrits

Abstract In the present paper, based on the general analytical expression [arXiv:2412.03470] for the maximum of the CHSH expectation under local Alice and Bob spin- s measurements in a two-qudit state of dimension d = 2 s + 1, s ≥1/2, we analyze whether or not, under spin-1 measurements in an arbitrary two-qutrit state, the CHSH inequality is violated. We find analytically for a variety of pure nonseparable two-qutrit states and also, numerically for 1, 000, 000 randomly generated pure nonseparable two-qutrit states, that, under local Alice and Bob spin-1 measurements in each of these nonseparable states, including maximally entangled, the CHSH inequality is not violated. These results together with the spectral decomposition of a mixed state lead us to the Conjecture that, under local Alice and Bob spin-1 measurements, every nonseparable two-qutrit state, pure or mixed, does not violate the CHSH inequality. For a variety of pure two-qutrit states, we further find the values of their concurrence and compare them with the values of their spin-1 CHSH parameter, which determines violation or nonviolation by a two-qutrit state of the CHSH inequality under spin-1 measurements. This comparison indicates that, in contrast to spin- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mfrac> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:mfrac> </mml:math> measurements, where the spin- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mfrac> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:mfrac> </mml:math> CHSH parameter of a pure two-qubit state is increasing monotonically with a growth of its entanglement, for a pure two-qutrit state, this is not the case. In particular, for the two-qutrit GHZ state, which is maximally entangled, the spin-1 CHSH parameter is equal to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msqrt> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:mn>8</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>9</mml:mn> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:msqrt> </mml:math> , while, for some separable pure two-qutrit states, this parameter can be equal to unity. Moreover, for the two-qutrit Horodecki state, the spin-1 CHSH parameter is equal to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>4</mml:mn> <mml:msqrt> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msqrt> <mml:mo>/</mml:mo> <mml:mn>21</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> </mml:math> regardless of the entanglement type of this mixed state.