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arXiv:quant-ph/0503047 · Physical Review A

Optimal classical-communication-assisted local model of n -qubit Greenberger-Horne-Zeilinger correlations

Tracey E. Tessier, Carlton M. Caves, Ivan Deutsch, Bryan Eastin, Dave Bacon

Published Sep 6, 2005• 16 citations
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Abstract

We present a model, motivated by the criterion of reality put forward by Einstein, Podolsky, and Rosen and supplemented by classical communication, which correctly reproduces the quantum-mechanical predictions for measurements of all products of Pauli operators on an $n$-qubit GHZ state (or ``cat state''). The $n\ensuremath{-}2\phantom{\rule{0.3em}{0ex}}\text{bits}$ employed by our model are shown to be optimal for the allowed set of measurements, demonstrating that the required communication overhead scales linearly with $n$. We formulate a connection between the generation of the local values utilized by our model and the stabilizer formalism, which leads us to conjecture that a generalization of this method will shed light on the content of the Gottesman-Knill theorem.

DOI
10.1103/physreva.72.032305

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•Apr 22, 2022•0 citations
We present a model, motivated by the criterion of reality put forward by Einstein, Podolsky, and Rosen and supplemented by classical communication, which correctly reproduces the quantum-mechanical predictions for measurements of all products of Pauli operators on an nnn-qubit GHZ state (or ``cat state''). The n\ensuremath−2bitsn\ensuremath{-}2\phantom{\rule{0.3em}{0ex}}\text{bits}n\ensuremath−2bits employed by our model are shown to be optimal for the allowed set of measurements, demonstrating that the required communication overhead scales linearly with nnn. We formulate a connection between the generation of the local values utilized by our model and the stabilizer formalism, which leads us to conjecture that a generalization of this method will shed light on the content of the Gottesman-Knill theorem.
We present a local hidden-variable model supplemented by classical communication that reproduces the quantum-mechanical predictions for measurements of all products of Pauli operators on an n-qubit GHZ state (or "cat state"). The simulation is efficient since the required amount of communication scales linearly with the number of qubits, even though there are Bell-type inequalities for these states for which the amount of violation grows exponentially with n. The structure of our model yields insight into the Gottesman-Knill theorem by demonstrating that, at least in this limited case, the correlations in the set of nonlocal hidden variables represented by the stabilizer generators are captured by an appropriate set of local hidden variables augmented by n-2 bits of classical communication.
Despite the exponential overhead to describe general multiqubit quantum states and processes, efficient methods for certain state families and operations have been developed and utilized. The stabilizer formalism and the Gottesman-Knill theorem, where pure stabilizer or graph states are manipulated by Clifford operations and Pauli measurements, are prominent examples, and these states play a major role in many applications in quantum technologies. Here we develop a noisy stabilizer formalism, i.e., a method that allows one to efficiently describe and follow not only pure states under Clifford operations and Pauli measurements but also Pauli noise processes acting on such stabilizer states, including uncorrelated and correlated dephasing and single- or multiple-qubit depolarizing noise. The method scales linearly in the number of qubits of the initial state, but exponentially in the size of the target state. Thus, whenever a noisy stabilizer state is manipulated by means of local Pauli measurements such that a multipartite entangled state of a few qubits is generated, one can efficiently describe the resulting state.
We use techniques for lower bounds on communication to derive necessary conditions (in terms of detector efficiency or amount of superluminal communication) for being able to reproduce the quantum correlations occurring in Einstein-Podolsky-Rosen-type experiments with classical local hidden-variable theories. As an application, we consider n parties sharing a Greenberger-Horne-Zeilinger-type state and show that the amount of superluminal classical communication required to reproduce the correlations is at least n(log((2)n-3) bits and the maximum detector efficiency eta(*) for which the resulting correlations can still be reproduced by a local hidden-variable theory is upper bounded by eta(*)</=8/n and thus decreases with n.
Quantum teleportation is a cornerstone protocol for future quantum networks and distributed quantum com- puting. However, scaling teleportation to an arbitrary N- qubit state using conventional methods, which require N separate Bell pairs, presents a significant challenge in terms of both quantum and classical resource overhead. This paper introduces a novel and resource-efficient protocol for tele- porting an arbitrary N-qubit state using a single, pre-shared (N+1)-qubit Greenberger-Horne-Zeilinger (GHZ) state. Our approach reduces the required entangled channel qubits and classical communication bits from O(2N) to N+1, offering a substantial improvement in communication efficiency. The protocol mechanism involves a sequence of local and non-local CNOT gates performed by the sender (Alice), followed by local measurements on her N+1 qubits. The N+1 classical measurement outcomes are then transmitted to the receiver (Bob), who recovers the original quantum state by applying a specific corrective unitary transformation U-f, followed by a set of conditional Pauli gates. We provide a high-level proof of the protocol’s correctness and show that the corrective trans- formation U-f can be implemented efficiently using only local operations on the receiver’s side. Recognizing the practical difficulty of implementing non-local quantum gates between spatially separated nodes, we also propose a modified scheme to enhance near-term feasibility. This practical approach re- locates the non-local operations to a co-located preparation phase, thereby trading the formidable challenge of non-local interactions for a high-fidelity quantum memory requirement. This work analyzes the critical trade-offs between communication efficiency and operational complexity, presenting a vi- able alternative for designing scalable and resource-optimized quantum communication systems.
We introduce a real-valued functional I(n⃗1,n⃗2)I(\vec{n}_1,\vec{n}_2)I(n1​,n2​) of four three-qubit correlations that turns the Greenberger-Horne-Zeilinger (GHZ) algebraic paradox into a quantitative witness of genuine tripartite entanglement. For every three-qubit state ρρρ and every pair of directions, ∣I(n⃗1,n⃗2;ρ)∣≤2|I(\vec{n}_1,\vec{n}_2;ρ)|\le 2∣I(n1​,n2​;ρ)∣≤2, with equality if and only if n⃗1⊥n⃗2\vec{n}_1\perp\vec{n}_2n1​⊥n2​ and ρρρ is locally unitarily equivalent to the GHZ state. On the five-parameter Acín family, I(x^,y^)I(\hat{x},\hat{y})I(x^,y^​) depends only on λ0λ4λ_0λ_4λ0​λ4​ and is maximised by the GHZ state; for the W state I(x^,y^)=0I(\hat{x},\hat{y})=0I(x^,y^​)=0 and max⁡∣IW∣=35/27\max|I_{\mathrm{W}}|=35/27max∣IW​∣=35/27. Maximising over independent local orthonormal frames yields a local-unitary invariant measure EGHZ∈[0,1]\mathcal{E}_{\mathrm{GHZ}}\in[0,1]EGHZ​∈[0,1], computable from four three-body Pauli correlators, that equals one exactly on the GHZ orbit and takes the values 35/5435/5435/54 on the W state and 1/21/21/2 on product states. We show that EGHZ\mathcal{E}_{\mathrm{GHZ}}EGHZ​ is not convex: an equal mixture of three biseparable states has EGHZ=14/27\mathcal{E}_{\mathrm{GHZ}}=14/27EGHZ​=14/27, larger than the average 1/21/21/2 of its components. We also relate EGHZ\mathcal{E}_{\mathrm{GHZ}}EGHZ​ to GHZ fidelity witnesses, discuss its behaviour under local operations and classical communication, and outline a generalisation to three qudits via Heisenberg-Weyl operators.
The Greenberger-Horne-Zeilinger (GHZ) version of the Einstein-Podolsky-Rosen (EPR) paradox is widely regarded as a conclusive logical argument that rules out the possibility of describing quantum phenomena within the framework of any model of local hidden variables in which the observers are free to choose their own experimental settings. In this paper, we show, however, that the GHZ argument implicitly relies on a subtle, though crucial, assumption that is not required by fundamental physical principles, and which had gone unnoticed. Namely, we note that the argument implicitly assumes the existence of an absolute angular frame of reference with respect to which the polarization properties of the hypothetical hidden configurations of the entangled particles, as well as the orientations of the measurement devices that test them, can be defined. We further note that such an absolute frame of reference may not exist if the hidden configurations of the entangled particles spontaneously break the gauge rotational symmetry. Indeed, by skipping the disputed requirement we are able to build a model of local hidden variables for the GHZ state, which complies with the 'free-will' hypothesis and reproduces the quantum mechanical predictions. Our model, thus, completes-in the EPR sense-the quantum description of the GHZ state.