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arXiv:2001.08821 · Quantum Information Processing

Absolutely maximally entangled states in tripartite heterogeneous systems

Yi Shen, Lin Chen

Published Mar 1, 2021• 8 citations
arXivHTMLPDF
DOI
10.1007/s11128-021-03034-y

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As emerging quantum architectures evolve into heterogeneous networks combining different physical substrates, such as qubits for logic and higher-dimensional qudits for robust communication, the traditional scalar metrics of quantum error correction become insufficient. To address this, we introduce a mathematical framework based on dimension multisets to characterize quantum error-correcting codes (QECC) and absolutely maximally entangled (AME) states in mixed-dimensional Hilbert spaces. By replacing scalar weights with multisets, we accurately capture the exact physical composition of error supports across these diverse systems. Our central result is the mixed-dimensional quantum MacWilliams identity, which establishes the formal algebraic relationship between Shor-Laflamme enumerators and unitary weight enumerators. From this foundation, we deduce the mixed-dimensional shadow identity and derive rigorous, generalized constraints on code parameters, explicitly formulating the mixed-dimensional quantum Hamming, Singleton and Scott bounds, and developing a linear program to systematically evaluate code viability. For the Singleton bound, a tighter bound that has no homogeneous analogue is derived for pure mixed-dimensional codes. Finally, we deploy this enumerator machinery to thoroughly analyze AME states, utilizing shadow inequalities to constrain their existence and introducing a combinatorial grid method for the explicit construction of mixed-dimensional tripartite AME states.
The Kochen Specker theorem revealed contextuality as a fundamental nonclassical feature of nature. Nonlocality arises as a special case of contextuality, where entangled states shared by space like separated parties exhibit nonlocal correlations. The notion of maximality in correlations, analogous to maximal entanglement, is less explored in multipartite systems. In our work, we have defined maximal correlations in terms of contextual models, which are analogous to absolutely maximally entangled (AME) states. Employing the sheaf theoretic framework, we introduce maximal contextual correlations associated with the corresponding maximal contextual model. The formalism introduces the contextual fraction CF as a measure of contextuality, taking values from 0 (noncontextual) to 1 (fully contextual). This enables the formulation of a new class of correlations termed absolutely maximal contextual correlations (AMCC), which are both maximally contextual and maximal marginals. In the bipartite setting, the canonical example is the Popescu Rohrlich (PR) box, while in the tripartite case, it includes Greenberger Horne Zeilinger (GHZ) correlations and three way nonlocal correlations. In this work, we extend these findings to four party correlations. Notably, no AME state exists for four qubits, which introduces a subtle difference between AMCC and AME. The construction follows the constraint satisfaction problem (CSP) and parity check methods. In particular, the explicit realization of a non AMCC correlation that is maximally contextual yet not maximal marginal is obtained within the CSP framework.