Three weak measurements of projections below minus one eighth

Abstract

For three sequential weak measurements of projection observables on a pure state, without post-selection, the mean product of the pointer positions can be -1/6, below -1/8. This refutes the conjecture of A. A. Abbott, R. Silva, J. Wechs, N. Brunner and C. Branciard (arXiv:1805.09364) that -1/8 bounds this mean for every number of projection observables.

Definition 1.1 (Nested anticommutator).

$$\mathrm{nestedAnti}\left(0,A\right)=A\left(0\right),\mathrm{nestedAnti}\left(n+1,A\right)=A\left(0\right)\cdot \mathrm{nestedAnti}\left(n,A\circ \mathrm{succ}\right)+\mathrm{nestedAnti}\left(n,A\circ \mathrm{succ}\right)\cdot A\left(0\right)$$

Formalization. D5/S3/Quantum/Measurement/SequentialWeakPointerRefutation.nestedAnti (✓ std3).

Citation. Alastair A. Abbott; Ralph Silva; Julian Wechs; Nicolas Brunner; Cyril Branciard (2019). Anomalous Weak Values Without Post-Selection. DOI: 10.22331/q-2019-10-14-194. URL: https://arxiv.org/abs/1805.09364v3.

Commentary.

The nested anticommutator of n + 1 matrices A_0, …, A_n is A_0 for n = 0 and {A_0, {A_1, …, {A_(n-1), A_n}…}} in general, defined by recursion on n; A o succ denotes the family A_1, …, A_n.

Definition 1.2 (Mean product of the pointer positions).

$$\mathrm{pointerMean}\left(A,\psi \right)=2^{-n}\cdot \mathrm{Re}⟨\psi ,\mathrm{nestedAnti}\left(n,A\right)\psi ⟩$$

Formalization. D5/S3/Quantum/Measurement/SequentialWeakPointerRefutation.pointerMean (✓ std3).

Citation. Alastair A. Abbott; Ralph Silva; Julian Wechs; Nicolas Brunner; Cyril Branciard (2019). Anomalous Weak Values Without Post-Selection. DOI: 10.22331/q-2019-10-14-194. URL: https://arxiv.org/abs/1805.09364v3.

Commentary.

In the weak regime without post-selection, the mean product of the pointer positions of n + 1 sequential measurements of A_0, …, A_n on the pure state psi is 2^(-n) times the real part of <psi, N psi>, where N is the nested anticommutator and <psi, phi> is the sum over i of the conjugate of psi_i times phi_i.

Definition 1.3 (The conjectured bound minus one eighth).

$$claim\iff (\forall d:\mathbb{N},\forall n:\mathbb{N},\forall A:\mathrm{Fin}\left(n+1\right)\to {\mathbb{C}}^{d\times d},(\forall i:\mathrm{Fin}\left(n+1\right),\mathrm{IsStarProjection}\left(A\left(i\right)\right))\Rightarrow \forall \psi :{\mathbb{C}}^d,(⟨\psi ,\psi ⟩=1)\Rightarrow -\frac{1}{8}\le \mathrm{pointerMean}\left(A,\psi \right))$$

Formalization. D5/S3/Quantum/Measurement/SequentialWeakPointerRefutation.claim (✓ std3).

Citation. Alastair A. Abbott; Ralph Silva; Julian Wechs; Nicolas Brunner; Cyril Branciard (2019). Anomalous Weak Values Without Post-Selection. DOI: 10.22331/q-2019-10-14-194. URL: https://arxiv.org/abs/1805.09364v3.

Commentary.

The conjecture: for every dimension d, every number n + 1 of observables, every family of orthogonal projections A_i on C^d (A_i^2 = A_i and A_i^* = A_i, the projection observables with eigenvalues 0 and 1) and every unit vector psi, the mean product of the pointer positions is at least -1/8. The paper proves this bound for two observables.

Theorem 1.4 (A sequence of three projections below minus one eighth).

$$\neg claim$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Measurement/SequentialWeakPointerRefutation.result (✓ std3). ∎

Resolves. Problems/abbott-2019-sequential-weak-pointer-bound (refuted) by D5/S3/Quantum/Measurement/SequentialWeakPointerRefutation.result.

Source. Repository-derived.

Acknowledgement. Alastair A. Abbott; Ralph Silva; Julian Wechs; Nicolas Brunner; Cyril Branciard (2019). Anomalous Weak Values Without Post-Selection. DOI: 10.22331/q-2019-10-14-194. URL: https://arxiv.org/abs/1805.09364v3.

Commentary.

Take d = 3, the state e_3, and the rational projections P_1 = (1/3)[[1, 1, -1], [1, 1, -1], [-1, -1, 1]] onto (1, 1, -1), P_2 = [[1, 0, 0], [0, 1/2, 1/2], [0, 1/2, 1/2]] = I - v v^* with v = (0, 1, -1)/sqrt(2), and P_3 = [[1/2, 0, 1/2], [0, 1, 0], [1/2, 0, 1/2]] = I - w w^* with w = (1, 0, -1)/sqrt(2). Each satisfies P^2 = P = P^*, checked entrywise, and e_3 is a unit vector. Evaluating the nested anticommutator gives (1/4) <e_3, {P_1, {P_2, P_3}} e_3> = -1/6 < -1/8, so the conjectured bound fails for three observables.

References

• Truth anchor: D5/S3/Quantum/Measurement/SequentialWeakPointerRefutation.claim
• Truth anchor: D5/S3/Quantum/Measurement/SequentialWeakPointerRefutation.nestedAnti
• Truth anchor: D5/S3/Quantum/Measurement/SequentialWeakPointerRefutation.pointerMean
• Truth anchor: D5/S3/Quantum/Measurement/SequentialWeakPointerRefutation.result