4.2 Bell states and EPR paradox

Bell states are the building blocks of quantum entanglement, showcasing the bizarre nature of quantum mechanics. These four special two-qubit states exhibit correlations that defy classical intuition, leading to phenomena like quantum teleportation and superdense coding.

The EPR paradox, proposed by Einstein and colleagues, challenges the completeness of quantum mechanics using entangled particles. It highlights the tension between quantum theory and local realism, sparking decades of research into the foundations of quantum mechanics and the nature of reality.

Bell States and the EPR Paradox

Four Bell states

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• Maximally entangled two-qubit states exhibit quantum phenomenon where particles are correlated and cannot be described independently
  • Entanglement occurs when two or more particles are linked in a way that measuring one instantly determines the state of the others, regardless of distance (quantum teleportation, superdense coding)
• Denoted as $|\Phi^+\rangle$, $|\Phi^-\rangle$, $|\Psi^+\rangle$, and $|\Psi^-\rangle$
  • $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ is a superposition of $|00\rangle$ and $|11\rangle$ with equal amplitudes
  • $|\Phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle)$ has a minus sign between the basis states
  • $|\Psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)$ and $|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle)$ are superpositions of $|01\rangle$ and $|10\rangle$
• Each Bell state is a superposition of two basis states with specific phase relationships between them
• Measuring one qubit of a Bell state collapses the wavefunction and instantaneously determines the state of the other qubit, showcasing the non-local nature of quantum mechanics (Einstein's "spooky action at a distance")

EPR paradox and significance

• Thought experiment proposed by Einstein, Podolsky, and Rosen in 1935 highlights apparent contradiction between quantum mechanics and local realism
  • Local realism assumes particle properties are determined independently of measurement and effects cannot propagate faster than light (causality, special relativity)
• Considers a pair of entangled particles, such as those in a Bell state, where measuring one instantly determines the state of the other, even at large distances
  • Seemingly instantaneous correlation appears to violate local realism and suggests "hidden variables" determine measurement outcomes
• Challenges the completeness of quantum mechanics and implies the existence of a deeper, more fundamental theory
  • Sparked decades of research into the foundations of quantum mechanics and the nature of reality (Bohm's interpretation, many-worlds interpretation)

Bell states vs EPR paradox

• Bell states are prime examples of entangled particles considered in the EPR paradox
  • Correlation between measurements of two qubits in a Bell state is precisely the "spooky action at a distance" Einstein found problematic
• Measuring one qubit of a Bell state collapses the wavefunction of the entire system, determining the state of the other qubit regardless of spatial separation
  • Demonstrates the non-local nature of quantum mechanics and challenges the concept of local realism
• EPR paradox, using Bell states as an example, highlights the apparent conflict between quantum mechanics and classical intuitions about locality and causality

Implications of Bell's theorem

• Bell's theorem (1964) provides a mathematical framework for testing the EPR paradox and local realism
  • States that any theory based on local hidden variables cannot reproduce all predictions of quantum mechanics
• Bell derived inequalities that must be satisfied by any local hidden variable theory
  • Quantum mechanics predicts the violation of these inequalities for certain entangled states, such as Bell states
• Experimental tests consistently show that quantum mechanics is correct and local hidden variable theories are incompatible with observed results (Aspect's experiments, loophole-free tests)
  • Violation of Bell's inequalities implies that quantum mechanics is fundamentally non-local and the state of a particle can be instantaneously influenced by measurements on another, even at large distances
• Far-reaching implications for understanding quantum mechanics and the nature of reality
  • Local realism does not hold in the quantum world and entanglement is a fundamental feature of quantum systems
  • Challenges our classical intuitions about causality, locality, and the objective nature of reality (Bohr's complementarity principle, Heisenberg's uncertainty principle)