[Paper Review] Uncertainty and Complementarity Relations in Weak Measurement

Arun Kumar Pati, Junde Wu|arXiv (Cornell University)|Nov 26, 2014

Quantum Information and Cryptography1 references3 citations

TL;DR

This paper establishes new uncertainty relations for weak values of non-commuting observables by defining non-Hermitian operators whose expectation values yield weak values, enabling non-trivial variance-based uncertainty bounds. It further proves a complementarity relation showing that the product of weak values for two non-commuting projectors is always bounded by one, even when individual weak values are large or complex.

ABSTRACT

We prove uncertainty relations that quantitatively express the impossibility of jointly sharp preparation of pre- and post-selected quantum states for measuring incompatible observables during the weak measurement. By defining a suitable operator whose average in the pre-selected quantum state gives the weak value, we show that one can have new uncertainty relations for variances of two such operators corresponding to two non-commuting observables. These generalize the recent stronger uncertainty relations that give non-trivial lower bounds for the sum of variances of two observables which fully capture the concept of incompatible observables. Furthermore, we show that weak values for two non-commuting projection operators obey a complementarity relation. Specifically, we show that for a pre-selected state if we measure a projector corresponding to an observable $A$ weakly followed by the strong measurement of another observable $B$ (for the post-selection) and, for the same pre-selected state we measure a projector corresponding to an observable $B$ weakly followed by the strong measurement of the observable $A$ (for the post-selection), then the product of these two weak values is always less than one. This shows that even though individually they are complex and can be large, their product is always bounded.

Motivation & Objective

• To address the lack of uncertainty relations for joint sharp preparation in pre- and post-selected quantum ensembles during weak measurements.
• To define a variance for non-Hermitian operators whose expectation values give weak values, enabling meaningful uncertainty quantification.
• To generalize stronger uncertainty relations beyond the Robertson form to capture incompatibility in weak measurement contexts.
• To establish a complementarity relation between weak values of non-commuting projectors, showing their product is universally bounded.

Proposed method

• Define a non-Hermitian operator whose expectation value in the pre-selected state yields the weak value of a given observable.
• Formulate the variance of such non-Hermitian operators to construct uncertainty relations based on sum of variances.
• Derive new uncertainty relations that provide non-trivial lower bounds for incompatible observables, generalizing recent stronger uncertainty relations.
• Analyze weak values of projection operators for non-commuting observables A and B in pre-selected states.
• Prove that the product of weak values ⟨Π_A⟩_w and ⟨Π_B⟩_w (with reversed measurement order) is always ≤ 1.
• Verify the complementarity relation in both finite and infinite-dimensional Hilbert spaces using explicit wavefunction integrals.

Experimental results

Research questions

• RQ1 Can non-trivial uncertainty relations be formulated for weak values of two incompatible observables in pre- and post-selected ensembles?
• RQ2 How can variance be meaningfully defined for non-Hermitian operators that yield weak values upon state averaging?
• RQ3 Do weak values of non-commuting projectors obey a universal bound despite their potential for large or complex values?
• RQ4 Is there a complementarity relation between weak values of mutually exclusive quantum features in weak measurement?
• RQ5 Can the new uncertainty relations capture the incompatibility of observables more effectively than the Robertson or Heisenberg forms?

Key findings

• The authors construct non-Hermitian operators whose expectation values in the pre-selected state yield weak values, enabling variance-based uncertainty relations.
• The new uncertainty relations provide non-trivial lower bounds for the sum of variances of weak values, generalizing stronger uncertainty relations beyond the Robertson form.
• For two non-commuting projection operators, the product of their weak values—measured in reversed order—satisfies |⟨Π_A⟩_w⟨Π_B⟩_w| ≤ 1, establishing a universal complementarity relation.
• This complementarity relation holds in both finite and infinite-dimensional Hilbert spaces, demonstrating a fundamental bound on weak values of incompatible projectors.
• The bound is saturated in specific cases, such as when post-selection is in momentum or position eigenstates, yielding ⟨Π_Δx⟩_w⟨Π_Δp⟩_w = 1.
• The results show that while individual weak values can be large or complex, their product remains constrained, revealing a deep structural feature of weak measurements.