[Paper Review] Weak measurements of non local variables

Aharon Brodutch|arXiv (Cornell University)|Nov 11, 2008

Quantum Mechanics and Applications24 references3 citations

TL;DR

This paper investigates weak measurements of non-local quantum variables, such as the sum and product of observables on entangled particles at distant locations. It proposes direct non-local weak measurement protocols using entangled measuring devices and analyzes indirect local methods, concluding that direct methods are more efficient and that non-local weak values of products cannot be reliably measured due to fundamental inefficiencies in existing indirect approaches.

ABSTRACT

Methods for measuring the weak value of non local variables are investigated. We analyze local (indirect) measurement methods for obtaining the weak values. We also describe some new (direct) methods (Non local weak measurements) for measuring the weak values of some non local variables.

Motivation & Objective

• To investigate the feasibility and efficiency of measuring weak values of non-local quantum variables, such as the sum and product of observables on entangled particles at spatially separated locations.
• To develop and analyze direct non-local weak measurement protocols using specially prepared entangled measuring devices for non-local variables.
• To compare indirect local measurement methods (using separate local devices) with direct non-local methods in terms of resource efficiency and practicality.
• To assess the physical status of non-local weak values, particularly for non-local products, and determine whether they can be considered 'elements of reality' based on measurement feasibility.
• To evaluate the limitations of existing indirect methods, such as 'joint weak values,' and demonstrate their impractical inefficiency for measuring non-local products.

Proposed method

• Proposes a direct non-local weak measurement method for non-local sums using an entangled measuring device prepared in a specific superposition state to couple to the sum of observables on distant particles.
• Describes an indirect local method using two separate local measuring devices—one at each location—based on the relation between local weak values and the non-local weak value of the sum.
• Analyzes the 'joint weak values' method for non-local products, which relies on sequential local weak measurements and classical post-processing of results.
• Introduces a sequential weak measurement protocol for non-local products, requiring prior knowledge of the system to be measured.
• Uses the Two-State Vector Formalism (TSVF) to describe pre- and post-selected systems and derives the weak value as the effective coupling strength between the measuring device pointer and the weak value.
• Compares measurement efficiency by evaluating the required ensemble size for a given precision, using the uncertainty in the pointer shift as a proxy for resource cost.

Experimental results

Research questions

• RQ1 Can non-local weak values of the sum of two observables on entangled particles be measured directly using a non-local measuring device?
• RQ2 How does the efficiency of indirect local weak measurement methods compare to direct non-local methods for measuring non-local sums?
• RQ3 Is there a practical and efficient method for measuring non-local weak values of the product of two observables on distant particles?
• RQ4 Why are existing indirect methods like 'joint weak values' for non-local products considered highly inefficient?
• RQ5 Can non-local weak values of products be considered 'elements of reality' if they cannot be measured via effective coupling to a measuring device?

Key findings

• Direct non-local weak measurement of the sum of two observables is feasible and significantly more efficient than indirect local methods, especially when the non-local uncertainty is small and local uncertainty is large.
• The indirect local method for measuring non-local sums requires a larger ensemble than the direct method, making it inefficient, particularly when the local uncertainties are large.
• No direct method exists for measuring non-local weak values of products; all current approaches rely on indirect or sequential local measurements.
• The 'joint weak values' method for non-local products is shown to be even more inefficient than the indirect local method for sums, due to the extremely small deviations in pointer shifts relative to the uncertainty in the observables.
• For non-local products, the lack of effective coupling between the measuring device and the weak value means that such weak values cannot be reliably interpreted as 'elements of reality'.
• The paper concludes that weak values of non-local products have a lesser physical status than those of non-local sums due to the absence of an efficient measurement procedure.