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Modern Physics Letters B

KNOWLEDGE EXCESS DUALITY AND VIOLATION OF BELL INEQUALITIES: THEORY AND EXPERIMENT

Miroslav Gavenda, Antonín Černoch, Jan Soubusta, Miloslav Dušek, Radim Filip

Published Mar 10, 2005• 3 citations
Publisher

Abstract

We analyze correlations between two parties sharing a general mixed two-qubit state ρSM in order to predict measurement outcomes. It is an interesting question as to what can be predicted about the results of two complementary projective measurements on one qubit from the measurement results on the other qubit. To quantify these predictions the complementary knowledge excessesΔK(ΠM→ΠS) and ΔK(Π′M→Π′S) are used, where ΠS, Π′S resp. ΠM, Π′M represent projective measurements on the qubit S resp. M. We derive an inequality ΔK2(ΠM→ΠS) + ΔK2(Π′M→Π′S)≤(B max /2)2, where B max is the maximum violation of Bell inequalities. This inequality restricts our capability to enhance the complementary predictions for any state ρSM and for arbitrary ΠS, Π′S and ΠM, Π′M. This result is experimentally verified on two-photon Werner states prepared by means of spontaneous parametric down-conversion.

DOI
10.1142/s021798490500827x

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We analyze correlations between two parties sharing a general mixed two-qubit state ρSM in order to predict measurement outcomes. It is an interesting question as to what can be predicted about the results of two complementary projective measurements on one qubit from the measurement results on the other qubit. To quantify these predictions the complementary knowledge excessesΔK(ΠM→ΠS) and ΔK(Π′M→Π′S) are used, where ΠS, Π′S resp. ΠM, Π′M represent projective measurements on the qubit S resp. M. We derive an inequality ΔK2(ΠM→ΠS) + ΔK2(Π′M→Π′S)≤(B max /2)2, where B max is the maximum violation of Bell inequalities. This inequality restricts our capability to enhance the complementary predictions for any state ρSM and for arbitrary ΠS, Π′S and ΠM, Π′M. This result is experimentally verified on two-photon Werner states prepared by means of spontaneous parametric down-conversion.
If there are correlations between two qubits, then the results of the measurement on one of them can help to predict measurement results on the other one. It is an interesting question as to what can be predicted about the results of two complementary projective measurements on the first qubit. To quantify these predictions the complementary knowledge excesses are used. A nontrivial constraint restricting them is derived. For any mixed state and for arbitrary measurements the knowledge excesses are bounded by a factor depending only on the maximal violation of Bell's inequalities. This result is experimentally verified on two-photon Werner states prepared by means of spontaneous parametric down-conversion.
A constraint on two complementary knowledge excesses by maximal violation of Bell inequalities for a single copy of any mixed state of two qubits S,MS,MS,M is analyzed. The complementary knowledge excesses ΔK(ΠM→ΠS){\bf ΔK}(Π_{M}\to Π_{S})ΔK(ΠM​→ΠS​) and ΔK(ΠM′→ΠS′){\bf ΔK}(Π'_{M}\to Π'_{S})ΔK(ΠM′​→ΠS′​) quantify an enhancement of ability to predict results of the complementary projective measurements ΠS,ΠS′Π_{S},Π'_{S}ΠS​,ΠS′​ on the qubit SSS from the projective measurements ΠM,ΠM′Π_{M},Π'_{M}ΠM​,ΠM′​ performed on the qubit MMM. For any state ρSMρ_{SM}ρSM​ and for arbitrary ΠS,ΠS′Π_{S},Π'_{S}ΠS​,ΠS′​ and ΠM,ΠM′Π_{M},Π'_{M}ΠM​,ΠM′​, the knowledge excesses satisfy the following inequality ΔK2(ΠM→ΠS)+ΔK2(ΠM′→ΠS′)≤(Bmax/2)2{\bf ΔK}^{2}(Π_{M}\to Π_{S})+{\bf ΔK}^{2} (Π'_{M}\to Π'_{S})\leq (B_{max}/2)^2ΔK2(ΠM​→ΠS​)+ΔK2(ΠM′​→ΠS′​)≤(Bmax​/2)2, where BmaxB_{max}Bmax​ is maximum of violation of Bell inequalities under single-copy local operations (local filtering and unitary transformations). Particularly, for the Bell-diagonal states only an appropriate choice of the measurements ΠS,ΠS′Π_{S},Π'_{S}ΠS​,ΠS′​ and ΠM,ΠM′Π_{M},Π'_{M}ΠM​,ΠM′​ are sufficient to saturate the inequality.
If there are correlations between two qubits then the results of the measurement on one of them can help to predict measurement results on the other one. It is an interesting question what can be predicted about the results of two complementary projective measurements on the first qubit. To quantify these predictions the complementary knowledge excesses are used. A non-trivial constraint restricting them is derived. For any mixed state and for arbitrary measurements the knowledge excesses are bounded by a factor depending only on the maximal violation of Bell's inequalities. This result is experimentally verified on two-photon Werner states prepared by means of spontaneous parametric down-conversion.
If there are correlations between two qubits then the results of the measurement on one of them can help to predict measurement results on the other one. It is an interesting question what can be predicted about the results of two complementary projective measurements on the first qubit. To quantify these predictions the complementary knowledge excesses are used. A non‐trivial constraint restricting them is derived. For any mixed state and for arbitrary measurements the knowledge excesses are bounded by a factor depending only on the maximal violation of Bell’s inequalities. This result is experimentally verified on two‐photon Werner states prepared by means of spontaneous parametric down‐conversion.
Nonlocality and quantum entanglement constitute two special aspects of the quantum correlations existing in quantum systems, which are of paramount importance in quantum-information theory. Traditionally, they have been regarded as identical (equivalent, in fact, for pure two qubit states, that is, Gisin's Theorem ), yet they constitute different resources. Describing nonlocality by means of the violation of several Bell inequalities, we obtain by direct optimization those states of two qubits that maximally violate a Bell inequality, in terms of their degree of mixture as measured by either their participation ratio R = 1/Tr(ρ 2 ) or their maximum eigenvalue λ max . This optimum value is obtained as well, which coincides with previous results. Comparison with entanglement is performed too. An example of an application is given in the XY model. In this novel approximation, we also concentrate on the nonlocality for linear combinations of pure states of two qubits, providing a closed form for their maximal nonlocality measure. The case of Bell diagonal mixed states of two qubits is also extensively studied. Special attention concerning the connection between nonlocality and entanglement for mixed states of two qubits is paid to the so-called maximally entangled mixed states. Additional aspects for the case of two qubits are also described in detail. Since we deal with qubit systems, we will perform an analogous study for three qubits, employing similar tools. Relation between distillability and nonlocality is explored quantitatively for the whole space of states of three qubits. We finally extend our analysis to four-qubit systems, where nonlocality for generalized Greenberger–Horne–Zeilinger states of arbitrary number of parties is computed.
Abstract In the standard Bell experiment, two parties perform local projective measurements on a shared pair of entangled qubits to generate nonlocal correlations. However, these measurements completely destroy the entanglement, rendering the post-measurement state unable for subsequent use. For a long time, it was believed that only unsharp measurements can be used to share quantum correlations. Remarkably, recent research has shown that classical randomness assisted projective measurements are sufficient for sharing nonlocality (Steffinlongo and Tavakoli 2022 Phys. Rev. Lett. 129 230402). Here, by stochastically combining no more than two different projective measurement strategies, we report an experimental observation of double Clauser–Horne–Shimony–Holt inequality violations with two measurements in a sequence made on each pair of maximally and partially entangled polarization photons. Our results reveal that the double violation achieved by partially entangled states can be 11 standard deviations larger than that achieved by maximally entangled ones. Our scheme eliminates the requirement for entanglement assistance in previous unsharp-measurement-based sharing schemes, making it experimentally easier. Our work provides possibilities for sharing other types of quantum correlations in various physical systems with projective measurements.