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arXiv:1701.08331 · Physical Review A

Mermin inequalities for perfect correlations in many-qutrit systems

Jay Lawrence

Published Apr 18, 2017• 19 citations
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Abstract

The existence of Greenberger-Horne-Zeilinger (GHZ) contradictions in many-qutrit systems was a long-standing theoretical question until its (affirmative) resolution in 2013. To enable experimental tests, we derive Mermin inequalities from concurrent observable sets identified in those proofs. These employ a weighted sum of observables, called $\mathcal{M}$, in which every term has the chosen GHZ state as an eigenstate with eigenvalue unity. The quantum prediction for $\mathcal{M}$ is then just the number of concurrent observables, and this grows asymptotically as ${2}^{N}/3$ as the number of qutrits $N\ensuremath{\rightarrow}\ensuremath{\infty}$. The maximum classical value falls short for every $N\ensuremath{\ge}3$, so that the quantum to classical ratio (starting at 1.5 when $N=3$) diverges exponentially ($\ensuremath{\sim}1.{064}^{N}$) as $N\ensuremath{\rightarrow}\ensuremath{\infty}$, where the system is in a Schr\"odinger-cat-like superposition of three macroscopically distinct states.

DOI
10.1103/physreva.95.042123

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The existence of Greenberger-Horne-Zeilinger (GHZ) contradictions in many-qutrit systems was a long-standing theoretical question until its (affirmative) resolution in 2013. To enable experimental tests, we derive Mermin inequalities from concurrent observable sets identified in those proofs. These employ a weighted sum of observables, called M\mathcal{M}M, in which every term has the chosen GHZ state as an eigenstate with eigenvalue unity. The quantum prediction for M\mathcal{M}M is then just the number of concurrent observables, and this grows asymptotically as 2N/3{2}^{N}/32N/3 as the number of qutrits N\ensuremath→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞. The maximum classical value falls short for every N\ensuremath≥3N\ensuremath{\ge}3N\ensuremath≥3, so that the quantum to classical ratio (starting at 1.5 when N=3N=3N=3) diverges exponentially (\ensuremath∼1.064N\ensuremath{\sim}1.{064}^{N}\ensuremath∼1.064N) as N\ensuremath→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞, where the system is in a Schr"odinger-cat-like superposition of three macroscopically distinct states.
Mermin inequalities are derived for systems of three-state particles (qutrits) employing three local measurement bases. These establish perfect correlations which violate local realistic bounds more strongly than those previously reported with two bases. The quantum eigenvalue of the Mermin operator grows as the dimension of the Hilbert space, 3N3^N3N, rather than 2N2^N2N, as obtained with two measurement bases. Unexpectedly, the proof is simplified with three bases because of the increased rotational symmetry of the construction.
We present a generalized Greenberger-Horne-Zeilinger (GHZ) theorem, which involves more than two local measurement settings for some parties, and cannot be reduced to one with less settings. Our results hold for an odd number of parties. We use a set of observables, which are incompatible but share a common eigenstate, here a GHZ state. Such observables are called concurrent. The idea is illustrated with an example of a three-qutrit system and then generalized to systems of higher dimensions, and more parties. The GHZ paradoxes can lead to, e.g., secret sharing protocols.
Multipartite Bell tests provide a correlation-only route to benchmarking quantum processors, but their application at large scales is hindered by the rapid decay of many-body correlators under noise and exponentially many terms in conventional Bell expressions. Here we address these scalability obstacles by introducing a finite-setting generalized Mermin family of state-tailored Bell inequalities with analytic certification bounds, in which the measurement-setting number mmm provides an additional certification dimension complementary to the system size nnn. We show that, for the powers-of-two setting choices considered here, increasing mmm leaves the ideal normalized multipartite quantum value unchanged while lowering the relevant classical bounds, thereby strengthening the Bell-violation ratios and yielding an improved noise-robustness scaling compared to the standard Mermin inequality. We test this construction experimentally on a programmable superconducting processor by preparing Greenberger-Horne-Zeilinger (GHZ) states of up to 80 qubits. Using randomized sampling for direct Bell-operator estimation, we observe Bell ratios that grow exponentially with system size, certify a nonlocality depth of 14, and show that increasing mmm strengthens both the Bell ratio and depth certification. All results are obtained solely from measured correlators and analytical bounds, without readout correction, tomography, or model-based mitigation. Generalized Mermin inequalities therefore provide a sharper Bell benchmark for noisy large-scale GHZ states.
The Greenberger-Horne-Zeilinger (GHZ) paradox provides an all-versus-nothing test for the quantum nonlocality. In most of the GHZ paradoxes known so far each observer is allowed to measure only two alternative observables. Here we present a general construction for GHZ paradoxes in which each observer measures more than two observables given that the system is prepared in the nnn-qudit GHZ state. By doing so we are able to construct a multisetting GHZ paradox for the nnn-qubit GHZ state, with nnn being arbitrary, which is genuine nnn-partite; i.e., no GHZ paradox exists when restricted to a subset of a number of observers for a given set of Mermin observables. Our result fills up the gap of the absence of a genuine GHZ paradox for the GHZ state of an even number of qubits, especially the four-qubit GHZ state as used in GHZ's original proposal.
We generalize Greenberger-Horne-Zeilinger (GHZ) theorem to an arbitrary number of DDD-dimensional systems. Contrary to conventional approaches using compatible composite observables, we employ incompatible and concurrent observables, whose common eigenstate is still a generalized GHZ state. It is these concurrent observables which enable one to prove a genuinely NNN-partite and DDD-dimensional GHZ theorem. Our principal idea is illustrated for a four-partite system with DDD which is an arbitrary multiple of 3. By extending to NNN qudits, we show that GHZ theorem holds as long as NNN is not divisible by all nonunit divisors of DDD, smaller than NNN.
A proof of Bell's theorem without inequalities valid for both inequivalent classes of three-qubit entangled states under local operations assisted by classical communication, namely Greenberger-Horne-Zeilinger (GHZ) and W,W,W, is described. This proof leads to a Bell inequality that allows more conclusive tests of Bell's theorem for three-qubit systems. Another Bell inequality involving both tri- and bipartite correlations is introduced which illustrates the different violations of local realism exhibited by the GHZ and WWW states.