Entanglement and Local Hidden Variables (LHV)

  1. Entangled States and Wigner Functions:
    • Entangled states with non-negative Wigner functions can be interpreted using local hidden variables (LHV)​​.
    • Bell’s theorem indicates that quantum mechanics cannot generally be underpinned by LHV theories, especially when Bell’s inequalities are violated​​.
  2. Bell’s Inequality Violation (BIQV):
    • Bell’s inequality can be violated even for non-maximally entangled states like the two-mode squeezed state (TMSS)​​.
    • A non-negative Wigner function might imply a classical model for the correlations, but this is not always the case for all observables​​.
  3. Non-Dispersive Observables:
    • For LHV interpretation of quantum mechanics, both the wavefunction and the observables must be non-dispersive (i.e., their Wigner functions take on eigenvalues as possible values)​​.

Quantum Teleportation

  1. Teleportation Protocols:
    • Teleportation relies on entangled states and classical communication to transmit quantum states from one location to another​​.
    • When entangled states are represented by non-negative Wigner functions, teleportation can theoretically be explained by LHV without invoking non-classicality​​.
  2. Teleportation Fidelity:
    • Teleportation protocols achieving fidelity greater than 50% imply the presence of entanglement, typically considered a quantum feature. A fidelity of 66% or higher is required for secure quantum communication​​.
  3. Classical Interpretation:
    • The document explores whether teleportation always indicates quantum mechanics’ non-classicality. It shows that with non-negative Wigner functions and non-dispersive observables, teleportation can be interpreted classically​​.

Wigner Functions

  1. Properties and Significance:
    • Wigner functions are used to represent quantum states in phase space and can be equivalent to the density operator formulation in quantum mechanics​​.
    • Non-negative Wigner functions for pure states are necessarily Gaussian and thus may be used to support LHV interpretations​​.
  2. Examples:
    • The TMSS, when maximally squeezed, approximates the EPR state and has a non-negative Wigner function, suggesting a classical underpinning for certain aspects of its entanglement​​.
  3. Teleportation with Gaussian States:
    • The protocol for teleportation is analyzed for Gaussian states, both physically realizable and non-realizable, showing that teleportation efficiency can be maintained within classical bounds under certain conditions​​.

This post delves into the interplay between classical interpretations (via LHV) and quantum phenomena like entanglement and teleportation, highlighting conditions under which seemingly quantum behaviors might be understood through classical theories.