[Paper Review] Structural unification of space and time correlations in quantum theory

Shmuel Marcovitch, Benni Reznik|arXiv (Cornell University)|Mar 13, 2011

Quantum Mechanics and Applications3 citations

TL;DR

This paper proposes a structural unification of spatial and temporal quantum correlations via a Jamioó´lowski isomorphism that maps bipartite quantum states to time-evolution maps. It demonstrates that weak measurement correlations in space exactly match those in time, revealing that entanglement in space corresponds to temporal coherence in evolution, with maximally entangled states mapping to unitary dynamics and non-maximally entangled states to selective measurements.

ABSTRACT

We suggest a natural mapping between bipartite states and quantum evolutions of local states, which is a Jamiolkowski map. It is shown that spatial correlations of weak measurements in bipartite systems precisely coincide with temporal correlations of local systems. This mapping has several practical and conceptual implications on the correspondence between Bell and Leggett-Garg inequalities, the statistical properties of evolutions in large systems, temporal decoherence and computational gain, in evaluation of spatial correlations of large systems.

Motivation & Objective

• To establish a formal mapping between bipartite quantum states and time-evolution maps in quantum mechanics.
• To demonstrate that spatial correlations of weak measurements in entangled systems are isomorphic to temporal correlations of weak measurements in single systems.
• To clarify the conceptual and practical correspondence between Bell inequalities (spatial) and Leggett-Garg inequalities (temporal).
• To explore implications for decoherence, computational efficiency, and the statistical behavior of large quantum systems.
• To show that the structural isomorphism implies that non-relativistic quantum mechanics inherently unifies space and time correlations.

Proposed method

• Constructs a Jamioó´lowski isomorphism between the space of bipartite density matrices and the space of quantum time evolutions described by normalized Kraus operators.
• Applies weak measurements with minimal disturbance to probe correlations in both spatial (bipartite) and temporal (single system evolution) settings.
• Uses the Hilbert-Schmidt scalar product to define the isomorphism, ensuring that spatial and temporal correlation functions match exactly.
• Derives the temporal correlation function as $ E(q_1 q_2) = \frac{1}{2} \text{Tr}\left[ O_2 \sum_z M'_z \{O_1, \rho_S^{\text{in}}\} M_z^{\prime\dagger} \right] $, where $ M'_z $ are normalized Kraus operators.
• Demonstrates that the mapping preserves statistical properties: $ D_T = D_S $ and $ N_T = N_S $, confirming the isomorphism.
• Extends the mapping to mixed states and convex combinations of Kraus operators, preserving the correspondence.

Experimental results

Research questions

• RQ1 Can spatial correlations of weak measurements in bipartite systems be exactly mapped to temporal correlations in single systems?
• RQ2 How does the isomorphism between bipartite states and time evolutions relate to the violation of Bell and Leggett-Garg inequalities?
• RQ3 What is the statistical behavior of large quantum systems under this mapping, particularly regarding the likelihood of unitary evolution?
• RQ4 How does this mapping distinguish between decoherence of states and decohering dynamics in time evolution?
• RQ5 Can this isomorphism lead to computational advantages in evaluating correlation functions and Bell inequality bounds?

Key findings

• The mapping establishes an exact isomorphism between spatial and temporal quantum correlations, confirmed by equality of $ D_T = D_S $ and $ N_T = N_S $, proving the Jamioó´lowski correspondence.
• Maximally entangled bipartite states map precisely to unitary time evolutions, while non-maximally entangled states correspond to selective measurements via non-trace-preserving Kraus operators.
• Pure product states map to projector measurements, and mixed bipartite states map to mixtures of time evolutions.
• The correspondence explains why Leggett-Garg inequalities have the same bounds as Bell inequalities, and why maximal violation depends only on observables in the temporal case—due to unitary evolution mapping.
• In large systems, the mapping implies that most pure evolutions are nearly unitary, as almost all pure states in high-dimensional Hilbert spaces are close to maximally entangled.
• The isomorphism enables computational gain: evaluating two-point correlation functions for $ N \times N $ bipartite systems reduces to $ N \times N $ matrix operations on single systems, bypassing $ N^2 \times N^2 $ matrices.