Schroedinger’s Principle Resolves EPR-Locality Paradox for Two Spin One-Half States

Scientists have long grappled with the EPR-locality paradox, a cornerstone debate in quantum mechanics concerning the seemingly instantaneous connection between entangled particles. Now, Walter F. Wreszinski from the Instituto de Fisica, Universidade de S ao Paulo (USP), Brasil, alongside colleagues, demonstrates a resolution to this paradox by revisiting a principle embedded within Erwin Schrödinger’s original 1935 paper. Their work proves the non-existence of the paradox within the Copenhagen interpretation, even when considering the complexities of spin measurement highlighted by Araki and Yanase. This research is significant because it clarifies a fundamental conflict between quantum mechanics and classical notions of locality, potentially reshaping our understanding of entanglement and measurement in the quantum realm.

This research is significant because it clarifies a fundamental conflict between quantum mechanics and classical notions of locality, potentially reshaping our understanding of entanglement and measurement in the quantum realm.

EPR Paradox Resolved via Wave Packet Collapse

The team achieved a definitive answer to the question posed by Einstein, Podolsky, and Rosen concerning the completeness of the quantum mechanical description of nature, arriving at the same conclusion as Griffiths through a probabilistic interpretation. Experiments show the EPR-locality paradox arises clearly in the case of entangled states of two spins one-half, allowing for a focused analysis of the underlying principles. Researchers define the Hilbert space for a finite region Λ ⊂Zν as HΛ = ⊗x∈ΛC2x, with the set of observables A(Λ) generated by operators Si x = 1/2σi x, where i = 1, 2, 3 and σi x represent the Pauli matrices at site x. A key aspect of the research involves examining states and observables within a Hilbert space, defining density matrices ρΛ and states ωΛ, and differentiating between pure and separable states.

Notably, the research leverages the Lieb-Robinson bound, which demonstrates that interactions within quantum spin systems are limited by a quantifiable exponential decay dependent on distance and time, ensuring relativistic causality is maintained. The team considered a scenario involving observers, Alice and Bob, measuring the spin of entangled particles prepared by a third party, Charlie, to illustrate the paradox. By applying Schrödinger’s principle, the scientists prove that the apparent non-locality arises from a misunderstanding of the measurement process, and that the observed correlations do not violate relativistic causality.

Hilbert Space Construction and State Definition are fundamental

The study began by defining the Hilbert space HΛ as ⊗x∈ΛC2x, where Λ represents a region in ν-dimensional space and each C2x denotes a two-dimensional complex space at site x. Observables were then constructed as bounded operators on this Hilbert space, specifically Si x = 1/2σi x, utilising the Pauli matrices σi x. Researchers employed density matrices ρΛ, positive Hermitean matrices with unit trace, to represent states, and defined states ωΛ as positive, normed linear functionals on the observable algebra A(Λ), calculated via TrHΛ(ρΛA). To characterise quantum states, the team distinguished between pure states, represented by projection operators PΛ, and mixed states, which are convex combinations of other states.

Separable states, defined as convex combinations of product states, were contrasted with entangled states, establishing a “classical” versus “quantum” dichotomy. Notably, the study referenced Bell states, such as ωB(. ) ≡(ΨB,−, . ΨB,−), where ΨB,−≡ 1/p ( (|+)1 ⊗|−)2 −|−)1 ⊗|+)2), utilising the basis of eigenvectors |±)1,2 of σ3. This precise mathematical formulation allowed for a detailed analysis of correlations between entangled particles. The research pioneered the application of the Lieb-Robinson bound to quantum spin systems, establishing a limit on the spatial and temporal propagation of correlations.

Specifically, the bound, expressed as ||[(τxτt)(A), B]|| ≤2C||A||||B|| exp[−|t|(λ|x|/|t| −D)], quantified the decay of the commutator between observables A and B, demonstrating exponential decay outside the light cone defined by the group velocity vg, where 0 < vg < ∞. This bound served as a crucial tool for assessing the consistency of the proposed resolution to the EPR paradox. Experiments focused on a scenario involving Alice and Bob, measuring σ3 on spatially separated particles prepared in a spin singlet state by Charlie. The team analysed whether Alice’s measurement could instantaneously determine Bob’s, seemingly violating relativistic causality.

EPR Paradox Resolved via Wave-Packet Collapse, suggesting non-locality

The team focused on states within a Hilbert space defined as HΛ = ⊗x∈ΛC2x, utilizing observables generated by the set A(Λ) of bounded operators Si x = 1/2σi x, where i ranges from 1 to 3 and σi x represent the Pauli matrices at each site x. Measurements confirm that a density matrix ρΛ, a positive, Hermitean matrix with a unit trace, accurately describes the state of the system. A state ωΛ on A(Λ) is defined as a positive, normed linear functional, calculated as ωΛ(A) = TrHΛ(ρΛA) for any operator A in A(Λ), ensuring ωΛ(1) equals 1. Results demonstrate that states can be classified as either pure, represented by a projection operator PΛ, or as convex combinations of product states.

Separable states, exhibiting “product correlations”, are distinguished from entangled states, which display “intrinsically quantum” behaviour. Notably, the researchers examined Bell states, such as ωB(. ) ≡(ΨB,−, . ΨB,−), where ΨB,−≡ 1/p ( (|+)1 ⊗|−)2 −|−)1 ⊗|+)2), utilising the basis of orthonormal eigenvectors |±)1,2 of σ3. The study leverages the Lieb-Robinson bound, establishing that the commutator [(τxτt)(A), B] for observables A, B in A0, is bounded by ||[(τxτt)(A), B]|| ≤2C||A||||B|| exp[−|t|(λ|x|/|t| −D)], where C, λ are positive constants and D is a quantity dependent on the interaction. This bound, applicable to finite-range interactions, confirms that the influence of an observation at the origin diminishes exponentially with both time and distance, constrained by the group velocity vg, satisfying 0 < vg < ∞.

Considering a spin singlet prepared by Charlie and distributed to Alice and Bob, the work addresses the EPR-locality paradox. The research clarifies that Alice’s measurement, while seemingly instantaneous, does not violate relativistic causality because it is not a local measurement in the sense of being associated with a projector of finite spatial and temporal support. This analysis, grounded in the principles of algebraic quantum field theory, provides a definitive answer to the question of whether the quantum mechanical description of nature is complete, aligning with conclusions reached by Griffiths.

EPR Paradox Resolved Under Copenhagen Interpretation through wavefunction

The work clarifies that the EPR paradox arises even in basic two-spin-one-half systems, and leverages the properties of density matrices and states within Hilbert spaces to demonstrate this. Furthermore, the research highlights the importance of the Lieb-Robinson bound, which limits the speed at which information can propagate in quantum spin systems, reinforcing the notion of locality despite apparent non-local correlations. While the analysis successfully addresses the EPR paradox in this context, extending these results to more complex systems or alternative interpretations of quantum mechanics remains an open question.