High-dimensional quantum entanglement and the advancements in their experimental realization provide a playground for fundamental research and eventually lead to quantum technological developments. The Horodecki criterion determines whether a state violates the Clauser-Horne-Shimony-Holt (CHSH) inequality for a two-qubit entangled state, solely from the state parameters. However, it remains a challenging task to formulate similar necessary and sufficient criteria for a high-dimensional entangled state for the violation of a suitable Bell inequality. Here we develop a Horodecki-like criterion based on the state parameters of arbitrary two-qudit states to violate a two-outcome Bell inequality involving ${2}^{n\ensuremath{-}1}$ and $n$ measurement settings for Alice and Bob, respectively. This inequality reduces to the well-known CHSH and Gisin elegant Bell inequalities for $n=2$ and 3, respectively. While the proposed criterion is sufficient to violate the Bell inequality, it becomes necessary as well for the following cases: (i) $m$ copies of Bell-diagonal states for arbitrary $n$, (ii) nondecomposable states whose correlation matrix is diagonalized by local unitaries, and (iii) for any arbitrary two-qubit state when $n=3$, where the maximal value of the Bell functional is achieved with Bob's measurements being pairwise anticommuting. For any states, we derive the constraints on Alice's measurements in achieving the maximum quantum violation for this inequality.
High-dimensional quantum entanglement and the advancements in their experimental realization provide a playground for fundamental research and eventually lead to quantum technological developments. The Horodecki criterion determines whether a state violates the Clauser-Horne-Shimony-Holt (CHSH) inequality for a two-qubit entangled state, solely from the state parameters. However, it remains a challenging task to formulate similar necessary and sufficient criteria for a high-dimensional entangled state for the violation of a suitable Bell inequality. Here we develop a Horodecki-like criterion based on the state parameters of arbitrary two-qudit states to violate a two-outcome Bell inequality involving 2n\ensuremath−1 and n measurement settings for Alice and Bob, respectively. This inequality reduces to the well-known CHSH and Gisin elegant Bell inequalities for n=2 and 3, respectively. While the proposed criterion is sufficient to violate the Bell inequality, it becomes necessary as well for the following cases: (i) m copies of Bell-diagonal states for arbitrary n, (ii) nondecomposable states whose correlation matrix is diagonalized by local unitaries, and (iii) for any arbitrary two-qubit state when n=3, where the maximal value of the Bell functional is achieved with Bob's measurements being pairwise anticommuting. For any states, we derive the constraints on Alice's measurements in achieving the maximum quantum violation for this inequality.
Abstract The Horodecki criterion provides a necessary and sufficient condition for a two-qubit state to be able to manifest Bell nonlocality via violation of the Clauser–Horne–Shimony–Holt (CHSH) inequality. It requires, however, the assumption that suitable projective measurements can be made on each qubit, and is not sufficient for scenarios in which noisy or weak measurements are either desirable or unavoidable. By characterising two-valued qubit observables in terms of strength, bias, and directional parameters, we address such scenarios by providing necessary and sufficient conditions for arbitrary qubit measurements having fixed strengths and relative angles for each observer. In particular, we find the achievable maximal values of the CHSH parameter for unbiased measurements on arbitrary states, and, alternatively, for arbitrary measurements on states with maximally-mixed marginals, and determine the optimal angles in some cases. We also show that for certain ranges of measurement strengths it is only possible to violate the CHSH inequality via biased measurements. Finally, we use the CHSH inequality to obtain a simple necessary condition for the compatibility of two qubit observables.
The Horodecki criterion provides a necessary and sufficient condition for a two-qubit state to be able to manifest Bell nonlocality via violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality. It requires, however, the assumption that suitable projective measurements can be made on each qubit, and is not sufficient for scenarios in which noisy or weak measurements are either desirable or unavoidable. By characterising general two-valued qubit observables in terms of strength, bias, and directional parameters, we address such scenarios by providing generalised necessary and sufficient conditions for noisy qubit measurements having fixed strengths and relative angles for each observer. In particular, we find the achievable maximal values of the CHSH parameter for unbiased measurements on arbitrary states, and, alternatively, for arbitrary measurements on states with maximally-mixed marginals, and determine the optimal angles in some cases. We also give a strong upper bound for the CHSH parameter in the general case, and show that for certain ranges of measurement strengths it is only possible to violate the CHSH inequality via biased measurements.
Abstract Violation of the Clauser–Horne–Shimony–Holt (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 = 2 s + 1 ⩾ 2 , we introduce the notion of the spin-s correlation matrix, which has dimension 3 × 3 for all s ⩾ 1 2 ; establish its relation to the general correlation ( d 2 − 1 ) × ( 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 Greenberger–Horne–Zeilinger 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 ⩾ 1 ) 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.
Alice and Bob each have half of a pair of entangled qubits. Bob measures his half and then passes his qubit to a second Bob who measures again and so on. The goal is to maximize the number of Bobs that can have an expected violation of the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality with the single Alice. This scenario was introduced in [R. Silva et al., Phys. Rev. Lett. 114, 250401 (2015)PRLTAO0031-900710.1103/PhysRevLett.114.250401] where the authors mentioned evidence that when the Bobs act independently and with unbiased inputs then at most two of them can expect to violate the CHSH inequality with Alice. Here we show that, contrary to this evidence, arbitrarily many independent Bobs can have an expected CHSH violation with the single Alice. Our proof is constructive and our measurement strategies can be generalized to work with a larger class of two-qubit states that includes all pure entangled two-qubit states. Since violation of a Bell inequality is necessary for device-independent tasks, our work represents a step towards an eventual understanding of the limitations on how much device-independent randomness can be robustly generated from a single pair of qubits.
Bell inequalities are an important tool in device-independent quantum information processing because their violation can serve as a certificate of relevant quantum properties. Probably the best known example of a Bell inequality is due to Clauser, Horne, Shimony and Holt (CHSH), which is defined in the simplest scenario involving two dichotomic measurements and whose all key properties are well understood. There have been many attempts to generalise the CHSH Bell inequality to higher-dimensional quantum systems, however, for most of them the maximal quantum violation---the key quantity for most device-independent applications---remains unknown. On the other hand, the constructions for which the maximal quantum violation can be computed, do not preserve the natural property of the CHSH inequality, namely, that the maximal quantum violation is achieved by the maximally entangled state and measurements corresponding to mutually unbiased bases. In this work we propose a novel family of Bell inequalities which exhibit precisely these properties, and whose maximal quantum violation can be computed analytically. In the simplest scenario it recovers the CHSH Bell inequality. These inequalities involve d measurements settings, each having d outcomes for an arbitrary prime number d ≥ 3 . We then show that in the three-outcome case our Bell inequality can be used to self-test the maximally entangled state of two-qutrits and three mutually unbiased bases at each site. Yet, we demonstrate that in the case of more outcomes, their maximal violation does not allow for self-testing in the standard sense, which motivates the definition of a new weak form of self-testing. The ability to certify high-dimensional MUBs makes these inequalities attractive from the device-independent cryptography point of view.
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 = 2s + 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- 1 2 measurements, where the spin- 1 2 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 8 9 , 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 4 2 / 21 < 1 regardless of the entanglement type of this mixed state.