Dimension required for nonlocal correlations
For specified finite-dimensional tensor-product correlation families and a fixed metric, determine tight lower and upper bounds on the minimal local dimension achieving accuracy , with explicit parameter dependence.
Problem statement
For specified finite-dimensional tensor-product correlation families and a fixed metric, determine tight lower and upper bounds on the minimal local dimension achieving accuracy , with explicit parameter dependence.
Lean 4
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Background
Dimension witnesses certify lower bounds on resources needed to reproduce observed Bell statistics. For a target in , define approximation in a stated metric such as the maximum total-variation distance over input pairs. A commuting-operator target outside need not have arbitrarily accurate finite-dimensional realizations at all.
Known examples force dimension to grow as the requested error decreases, sometimes extremely rapidly. There can be no uniform computable dimension cutoff sufficient to approximate every unrestricted finite Bell-game value: together with bounded-dimensional optimization this would contradict MIP*=RE. Thus a general computable dimension bound is not an open deliverable.
Self-testing certifies a reference subsystem up to local isometries and unused auxiliary systems, not the exact total dimension of an unknown physical device. Conversely, specified noisy-resource models can restore computable bounds; Qin–Yao supply one such result for fully quantum games using noisy maximally entangled resources.
The useful quantitative frontier is restricted families, optimal dimension witnesses and explicit dependence on accuracy, inputs and outputs.
Progress & source assessment
Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.
- [1]Qin & Yao, 'Decidability of fully quantum nonlocal games with noisy maximally entangled states' (2023 revision) link
- [2]Ji et al., 'MIP*=RE' link
- [3]Brunner, Cavalcanti, Pironio, Scarani, Wehner, 'Bell nonlocality', Rev. Mod. Phys. 86, 419 (2014) link
- [4]Brunner, Pironio, Acín, Gisin, Méthot, Scarani, 'Testing the dimension of Hilbert spaces', PRL 100, 210503 (2008) link
Recorded progress
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