[Paper Review] On the Spin Dependence of Detection Times and the Nonmeasurability of Arrival Times

Sheldon Goldstein, Roderich Tumulka+1 more|arXiv (Cornell University)|Sep 21, 2023

Quantum Mechanics and ApplicationsPhysics and Astronomy3 references3 citations

TL;DR

This paper demonstrates that the spin-dependent arrival time distributions proposed by Das and Dürr in Bohmian mechanics cannot be measured due to a fundamental incompatibility with Positive Operator-Valued Measures (POVMs). The key result is that these distributions violate a necessary condition for measurability—specifically, the sum of probabilities for opposite spin directions depends on the direction, ruling out any physical measurement apparatus.

ABSTRACT

According to a well-known principle of quantum physics, the statistics of the outcomes of any quantum experiment are governed by a Positive Operator-Valued Measure (POVM). In particular, for experiments designed to measure a specific physical quantity, like the time of a particle's first arrival at a surface, this principle establishes that if the probability distribution of that quantity does not arise from a POVM, no such experiment exists. Such is the case with the arrival time distributions proposed by Das and Dürr [arXiv:1802.07141], due to the nature of their spin dependence.

Motivation & Objective

• To assess the measurability of arrival time distributions for spin-1/2 particles in a symmetric quantum system as computed by Das and Dürr using Bohmian mechanics.
• To determine whether the spin-dependent arrival time statistics derived in [1] can arise from a valid quantum measurement process governed by a POVM.
• To investigate the implications of axial, inversion, and chiral symmetries on the structure of measurable arrival time distributions.
• To quantify the minimal error in any approximate measurement of such non-POVM-compatible arrival time distributions.
• To challenge the assumption in [1] that detectors could measure these arrival times with only mild disturbance.

Proposed method

• Derives a necessary condition for spin-dependent outcome statistics to arise from a POVM: the sum of probabilities for opposite spin directions must be independent of the direction.
• Applies this condition to the arrival time distributions from [1], showing that for axial symmetry and $\mathscr{P}_{\uparrow} = \mathscr{P}_{\downarrow}$, the condition $\mathscr{P}_{\uparrow} = \mathscr{P}_{\rightarrow}$ must hold for measurability.
• Uses the total variation norm to define a lower bound on the error of any approximating POVM, given by $\Delta/4$, where $\Delta = \sup_{{\bf n},{\bf m}}\| (\mathscr{P}_{\bf n}+\mathscr{P}_{\bf -n}) - (\mathscr{P}_{\bf m}+\mathscr{P}_{\bf -m}) \|$.
• Analyzes the impact of additional symmetries—specifically inversion symmetry ($\mathscr{P}_{\bf n} = \mathscr{P}_{\bf -n}$) and chiral symmetry ($P_{\uparrow} = P_{\downarrow}$)—on the structure of measurable statistics.
• Shows that under axial and chiral symmetry, any detector respecting these symmetries must yield identical statistics regardless of spin direction, thus failing to detect the purported spin dependence.
• Establishes that the family of distributions $\mathscr{P}_{\bf n}$ in [1] cannot be approximated by any POVM with error less than $\|\mathscr{P}_{\rightarrow} - \mathscr{P}_{\uparrow}\|/2$.

Experimental results

Research questions

• RQ1 Can the spin-dependent arrival time distributions computed by Das and Dürr in Bohmian mechanics be realized as the outcome of a physical quantum measurement?
• RQ2 Does the requirement that measurement statistics arise from a POVM impose a fundamental constraint on the form of spin-dependent arrival time distributions?
• RQ3 What is the minimal error in approximating the arrival time statistics from [1] using any valid POVM?
• RQ4 How do axial, inversion, and chiral symmetries affect the possibility of detecting spin-dependent arrival times?
• RQ5 Why does the assumption of detector measurability in [1] fail despite the apparent physical plausibility of the setup?

Key findings

• The arrival time distributions from [1] violate the necessary POVM condition that $\mathscr{P}_{\bf n} + \mathscr{P}_{\bf -n}$ be independent of $\bf n$, proving they cannot arise from any quantum measurement.
• Under axial symmetry and $\mathscr{P}_{\uparrow} = \mathscr{P}_{\downarrow}$, the condition $\mathscr{P}_{\uparrow} = \mathscr{P}_{\rightarrow}$ must hold for measurability; since this fails in [1], the distributions are not measurable.
• Any approximating POVM must incur an error of at least $\Delta/4$, where $\Delta = \sup_{{\bf n},{\bf m}}\| (\mathscr{P}_{\bf n}+\mathscr{P}_{\bf -n}) - (\mathscr{P}_{\bf m}+\mathscr{P}_{\bf -m}) \|$, leading to a lower bound of $\|\mathscr{P}_{\rightarrow} - \mathscr{P}_{\uparrow}\|/2$ for the model in [1].
• Due to axial and chiral symmetry, any detector respecting these symmetries will observe identical statistics for all spin directions, rendering it incapable of detecting the purported spin dependence.
• Inversion symmetry ($\mathscr{P}_{\bf n} = \mathscr{P}_{\bf -n}$) combined with axial symmetry implies that measurable arrival time statistics must be completely spin-independent, contradicting the model’s predictions.
• The conclusion invalidates the assumption in [1] that detectors could measure these arrival times with only mild disturbance, as the nonmeasurability is fundamental and not a matter of experimental precision.