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The Quantum Plumber's Problem

Nicolas G. Underwood, Holger F. Hofmann, Jonte R. Hance

arXiv:2609.11416v1quant-phcs.ITmath.ITphysics.optics

TL;DR

The paper extends quantum counterfactual-gain analysis to the multi-location Quantum Plumber’s Problem, where a player must identify a known blocked interferometer path. It analyzes competing classical and quantum strategies, and establishes that distinct blockage states can be distinguished with certainty in finitely many turns.

  • Problem

    Earlier counterfactual-gain work treated presence or absence of a blockage at a predefined path, while this paper asks how to discriminate among multiple possible blocked paths.

    Source: §I · p. 1

  • Method

    The paper analyzes adaptive photon-input strategies for generalized path-encoded interferometers and simulates competing classical and quantum strategies on Hofmann’s three-path interferometer.

    Source: §I · p. 1

  • Results

    When blockage locations correspond to distinct states, they can be distinguished with certainty in a finite number of turns.

    Source: §I · p. 2

  • Takeaways & Limitations

    Because a blockage is always present, detector activation alone cannot identify quantum advantage; the paper compares strategies across possible blockage locations instead.

    Source: §I · p. 1

    Finite-turn certainty requires the blockage-associated states to be distinct.

    Source: §I · p. 2

Abstract

from arXiv · show

Recent work quantified the notion of quantum counterfactual gain for an extended Elitzur-Vaidman bomb test style scenario, through a connection to the negativity of the Kirkwood-Dirac quasiprobability distribution. We here extend this work to identifying quantum advantage in a new scenario, which we term the ``Quantum Plumber's Problem''. In this scenario, we imagine a ``quantum plumber'', who knows that one path of an interferometer is blocked, and wants to find the optimal strategy for identifying with certainty which path this is. We discuss various strategies for a generalised path-encoded interferometer, as well as for the specific case of Hofmann's three-path interferometer, introduced in a recent analysis of the relationship between states in five measurement contexts of a three level system. We support our arguments on the relative merit of competing strategies with data collected over many simulated attempts at locating blockages. We also present results for a variant of the game in which the blockage is replaced by a non-demolition detector.

I. INTRODUCTION

The paper extends quantum counterfactual-gain analysis to a multi-location game: a player uses sequential photon inputs and detector outcomes to locate a known single blockage. It sets out a path-encoded interferometer and turn mechanics for comparing classical and quantum strategies.

I. INTRODUCTION

  • The Quantum Plumber’s Problem asks a player to identify which of several possible paths is blocked, extending earlier binary blockage discrimination [1].The player may choose and adapt photon input states but cannot alter the interferometer.Source: §I · p. 1

A. Game overview

  • The game ends when the player locates the single blocked path with unit probability, aiming to do so in as few turns as possible.The player sends photons into a sealed interferometer and learns from its output detectors.Source: A. Game overview · p. 2

B. Interferometer setup and notation

  • The general setup is a path-encoded d-state interferometer with d input and output paths joined by sequential two-path beam splitters or Mach-Zehnder interferometers.This treatment considers path as the degree of freedom; blockage locations can lie in different measurement contexts and need not correspond to orthogonal states.Source: B. Interferometer setup and notation · pp. 2–3

C. Core turn mechanics

  • Each turn produces a detector outcome or absorption, and the player updates blockage priors using conditional probabilities for the chosen input state and ruleset.Because outcomes are probabilistic, move choices may need to adapt; a fully branched move list becomes large, reaching 349,525 entries for ten Hofmann-interferometer turns.Source: C. Core turn mechanics · pp. 3–4

D. Instructions for the Gamemaster (GM)

The Gamemaster simulates detector outcomes and updates the player’s blockage priors. The probability rules associate blockage locations with path states and use the interferometer transformations to calculate outcomes.

D. Instructions for the Gamemaster (GM)

  • The Gamemaster (GM) acts as nature, updates priors after each outcome, and tells the player when the game is complete.The GM samples an outcome using the secret blockage and the relevant conditional probabilities.Source: D. Instructions for the Gamemaster (GM) · p. 4
  • The GM protocol repeats turns until a blockage location has posterior probability one, then reports the inferred location and turn count.At each turn, it calculates outcome probabilities for the secret blockage, samples an outcome, and updates the probabilities for all candidate locations.Source: D. Instructions for the Gamemaster (GM) · p. 4

E. Rules for calculating (quantum) probabilities

  • Quantum outcome probabilities are calculated after the input photon undergoes the interferometer’s successive two-dimensional unitary transformations.The detector states form the final measurement context, while input states form the initial context.Source: E. Rules for calculating (quantum)… · pp. 4–5
  • Each possible blockage lies on a path segment and is associated with a corresponding path state in at least one context.For Hofmann’s interferometer, the candidate locations are D1, S1, P1, F, P2, S2, and D2.Source: E. Rules for calculating (quantum)… · p. 5
  • The blockage-state components can be tabulated before gameplay to avoid repeating their calculation.The supplied Hofmann examples include states such as D1 and S2.Source: E. Rules for calculating (quantum)… · p. 5
  • Two blockage locations with state vectors equal up to a complex phase have identical conditional probabilities and cannot be distinguished under the quantum ruleset.The paper assumes such indistinguishable locations are absent, since the game otherwise cannot be completed.Source: E. Rules for calculating (quantum)… · p. 5

F. Alternative ruleset for classical probabilities

The classical ruleset models photons as particles with definite paths, stochastic beam-splitter reflections, and absorption at a blocked segment. Conditional outcome probabilities can be calculated from path probabilities and blockage locations.

F. Alternative ruleset for classical probabilities

  • Classical photons occupy definite paths and are reflected probabilistically at beam splitters using the same reflectivities as their quantum counterparts.A blockage absorbs a photon that travels along its path segment; reaching an output path activates its detector.Source: F. Alternative ruleset for classical… · p. 5
  • The classical ruleset permits either an input probability distribution or an equivalent definite input port.For input α2 or α3, the example probability tree is the same because the first beam splitter has reflectivity 1/2.Source: F. Alternative ruleset for classical… · p. 5, Figure 3 · p. 6
  • A classical blockage removes probability from the path distribution, while each reflector transforms that distribution according to its reflectivity.The paper notes that probability trees can calculate conditional outcomes but become unwieldy as interferometers grow.Source: F. Alternative ruleset for classical… · pp. 5–6, Figure 3 · p. 6
  • Classical conditional outcome probabilities are listed for definite inputs α1 and α2/α3 in Hofmann’s interferometer, with α2 and α3 sharing a table.For uncertain input states, the outcome probabilities are weighted averages of the tabulated values.Source: Table I · p. 6, F. Alternative ruleset for classical… · pp. 6–7

III. ANALYSIS OF GAMEPLAY AND STRATEGY

The analysis tracks progress by counting ruled-out blockages or by measuring posterior uncertainty. These measures motivate strategies based on conclusive exclusions and information gain.

A. Metrics for game progress

  • The number of blockages ruled out is a countable, monotonically increasing measure of game progress.Conclusive state exclusion rules out a location without identifying where the blockage is; one turn may rule out several locations when the states are antidistinguishable.Source: A. Metrics for game progress · p. 7
  • A player can rule out a blockage when a detector click is impossible under that location, then repeat until only one candidate remains.The proposed simple strategy randomly selects a remaining location and inputs its associated blockage state; a detector activation rules that location out.Source: A. Metrics for game progress · p. 7
  • Posterior entropy tracks remaining uncertainty, while information gain measures the reduction in uncertainty from a turn.The initial prior is uniform, and the game ends when one blockage has probability one and entropy reaches zero.Source: A. Metrics for game progress · p. 7
  • Cumulative information gain can exhaust the initial uncertainty, but no finite completion turn is guaranteed and an individual turn can lose information.The four-turn example in Fig. 4 shows information gain exhausting the initial log|B| uncertainty.Source: A. Metrics for game progress · pp. 7–8, Figure 4 · p. 8

B. Quantum Ruleset: How to rule out blockages

The section develops quantum rules for ruling out blockage locations from absorption or detector clicks. It argues that ruling out locations is generally more practical than trying to finish the game on an absorption turn.

B. Quantum Ruleset: How to rule out blockages

  • Finishing on absorption requires an input state orthogonal to every unresolved blockage vector except the correct one, a condition rarely feasible before the late game.The section therefore treats ruling out blockages as the more useful way to advance game progress.Source: B. Quantum Ruleset: How to rule out… · p. 8
  • For Hofmann’s interferometer, input states |S1〉 and |S2〉 can each rule out three blockage locations.The section also notes that blockage vectors from the same context are naturally orthogonal, making one such vector a useful input for ruling out others in that context.Source: B. Quantum Ruleset: How to rule out… · p. 8
  • When selected blockage vectors span at most d−1 dimensions, Procedure 1 constructs an input state that can rule them all out if the photon is absorbed.The construction uses a linearly independent subset of the vectors and an additional vector to form the input state.Source: B. Quantum Ruleset: How to rule out… · p. 9
  • After a detector activates, scanning the corresponding P(ω|β) table for zeros identifies blockage locations that can be ruled out.If a blockage gives zero probability for every detector activation, any activation rules that location out.Source: B. Quantum Ruleset: How to rule out… · p. 9

C. Quantum Ruleset: Why choosing

For the first turn, |F〉 is favored because it has a much greater chance of ruling out four blockage locations than the other special states considered. Repeated |F〉 inputs can identify D1 or D2 for specific outcome sequences, but cannot distinguish every blockage location.

C. Quantum Ruleset: Why choosing

  • 33.3% is the chance that |F〉 produces an outcome ruling out four blockage locations, versus around 5% for each of the other special states considered.With |F〉, the fewest locations ruled out is two, and there is a 19.0% chance of ruling out three.Source: C. Quantum Ruleset: Why choosing · p. 10
  • In the first two turns of a game using state |F〉, outcomes ω0 and ω1 let the player conclude that blockage is at D2; outcomes ω0 and ω3 let them conclude it is at D1.Source: C. Quantum Ruleset: Why choosing · p. 11
  • After observing ω2 with |F〉, the remaining S1, P1, P2, and S2 locations have identical columns, so further |F〉 inputs cannot distinguish them.If the blockage is at F, only ω0 may occur; the posterior probability of F exceeds 99% after fourteen turns.Source: C. Quantum Ruleset: Why choosing · p. 11

D. Both Rulesets: A strategy of information gain maximisation

The information-gain strategy selects inputs to reduce expected posterior entropy, but uniform sampling of pure states fails to complete games in the reported simulations. Better sampling is proposed as a remedy and compared with alternatives.

D. Both Rulesets: A strategy of information gain maximisation

  • The strategy chooses an input state to maximize expected information gain, calculated as current entropy minus expected posterior entropy.The routine evaluates probabilities for candidate input states and possible outcomes before selecting a move.Source: D. Both Rulesets: A strategy of… · p. 11
  • Uniformly sampling pure states can reduce uncertainty quickly but fails to finish games because states that permit exact blockage elimination occupy measure-zero subspaces.The authors suggest better sampling or a better optimization algorithm as possible remedies.Source: D. Both Rulesets: A strategy of… · p. 11

E. Quantum Ruleset: Mixed states

For Hofmann’s three-path interferometer, higher-purity states generally yield greater first-turn expected information gain. The maximally mixed state performs poorly and cannot distinguish certain blockage pairs.

E. Quantum Ruleset: Mixed states

  • Higher-purity states generally give greater first-turn expected information gain, while the seven blockage-location states give the highest values among the states chosen.The comparison uses density matrices of varying purity in Hofmann’s three-path interferometer.Source: E. Quantum Ruleset: Mixed states · p. 12
  • The maximally mixed state gives the same output probabilities for blockages D1 and S1, and for D2 and S2, so it cannot distinguish either pair.Its absorption probability is 33% for every blockage, so absorption yields no information about the blockage location.Source: E. Quantum Ruleset: Mixed states · p. 12

A. Preliminary numerical comparison of strategy performance

In the simulated Hofmann three-path games, equivalent quantum strategies outperform their classical analogues, and information-gain optimization with the best state sampling performs best. However, uniformly sampled strategies do not complete games, and the numerical comparison is limited in scope.

A. Preliminary numerical comparison of strategy performance

  • The numerical evaluation is limited to an initial analysis of Hofmann’s three-path interferometer; broader configurations and more sophisticated sampling would require further work.A comprehensive evaluation across interferometer configurations and sizes is described as a substantial extra project.Source: A. Preliminary numerical comparison of… · p. 12
  • Equivalent quantum-rule strategies clearly outperform their classical analogues across the 100-game frames, with information-gain optimization using the best state sampling performing best.The plotted measure is the equivalent number of blockages ruled out, 7−2H(n), with the blockage fixed at D1.Source: A. Preliminary numerical comparison of… · pp. 12–13, Figure 7 · p. 13
  • The classical ruleset can rule out only D1 and S1, while uniformly sampled quantum pure states fail to rule out blockages or complete games.The quantum control group approaches zero entropy asymptotically, but does not complete a game.Source: A. Preliminary numerical comparison of… · p. 13
  • Random-input control groups make progress but do not complete games, and their progress is slower than that of non-control strategies.This applies to both classical and quantum control groups; quantum progress is faster in equivalent blockages ruled out.Source: A. Preliminary numerical comparison of… · p. 13

B. Classical vs Quantum

Quantum strategies outperform nearly all classical strategies, with interference preserving information about the input and blockage location. Information-gain optimization usually improves game progress, but its game durations are bimodal and some games underperform random choice.

B. Classical vs Quantum

  • In Fig. 7, nearly all quantum games (C–H), including the quantum random state (C), perform better than both random classical path (A) and information-gain-optimized classical path (B).The passage attributes this contrast to interference preserving information across quantum beam splitters, whereas classical beam splitters introduce probabilistic behavior and wash information away.Source: B. Classical vs Quantum · p. 13

C. Benefits of Losing Information

  • Information-gain optimization improves the rise in effective blockages ruled out for classical paths and for quantum pure-state sampling, but the latter still fails to finish a game.The quantum comparison is between uniformly sampled pure states with and without information-gain optimization.Source: C. Benefits of Losing Information · p. 14
  • Expected-information-gain optimization on blockage states performs particularly well, but combining blockage vectors with uniformly sampled pure states marginally outperforms it.The comparison indicates that a blockage state is not always optimal by expected information gain.Source: C. Benefits of Losing Information · p. 14
  • Information-gain-optimized games outperform random choice in most cases for every blockage, but their duration distributions are bimodal, with a smaller underperforming group.The authors view the disjoint overperforming and underperforming populations as suggestive of a single failure mode.Source: C. Benefits of Losing Information · p. 14, Figure 8 · p. 14
  • The authors have not established the mechanism behind the suspected information-gain failure mode; they speculate that aversion to losing information may trap beliefs around an initially wrong location.They say this proposed explanation needs further investigation.Source: C. Benefits of Losing Information · pp. 14–15

D. Non-demolition blockages

The non-demolition variant is harder for a quantum player in Hofmann’s interferometer, while a classical player cannot complete it. The discussion identifies several extensions, including photon-state variants and alternative strategy objectives.

D. Non-demolition blockages

  • For non-demolition detection, classical conditional probabilities do not depend on the blocked location, while quantum players can rule out a location only when both probability terms vanish.In Hofmann’s interferometer, the scoring must distinguish five classes rather than all seven locations.Source: D. Non-demolition blockages · p. 15
  • In Hofmann’s interferometer, non-demolition detection reduces the seven locations to five distinguishable classes; this variant is harder for quantum players, while classical players cannot complete it.The classes are {D1,S1}, P1, F, P2, and {S2,D2}.Source: D. Non-demolition blockages · p. 15

E. Future Work

  • Proposed extensions include risk-sensitive information-gain strategies, reinforcement-learning policies indexed by belief states, and games that minimize photon loss or relax the single-blockage assumption.The authors also suggest restricting inputs to the interferometer’s input-port states.Source: E. Future Work · pp. 15–16
  • Multi-photon and non-Fock extensions would remove the single-photon outcome exclusivity; coherent states nevertheless reproduce single-photon conditional probabilities per unit intensity.The authors distinguish the interference advantage from any requirement for a single-photon source.Source: E. Future Work · pp. 15–16
  • Whether the restricted game is completable for interferometers other than Hofmann’s remains open.In Hofmann’s case, input-port states cannot rule out P1, F, or P2, and paired columns are identical.Source: E. Future Work · p. 16

F. Summary

  • The paper extends quantum counterfactual-gain analysis to the Quantum Plumber’s Problem and uses simulations of Hofmann’s interferometer to compare strategies.The authors also analyze strategies for generalized path-encoded interferometers.Source: F. Summary · p. 16

Appendix A: Practical guidance for numerical simulation of games

The appendix describes numerical safeguards for game simulations and explains how symmetries in Hofmann’s probability tables relate outcomes across blockage locations.

Appendix A: Practical guidance for numerical simulation of games

  • Setting posteriors below 10^-10 to zero addresses numerical residues typically around 10^-12–10^-13, but can make geometric convergence appear to finish at a cutoff-dependent turn.The authors report this effect only for game lengths far beyond those considered.Source: Appendix A · p. 18
  • Randomly choosing among states within 10^-6 bits of maximal expected information gain restores symmetry in game-length histograms for paired blockages.The algorithm otherwise breaks ties according to program order, biasing game lengths across symmetric cases.Source: Appendix A · pp. 18–19

Appendix B: Probability Tables for different Input States

  • Hofmann’s probability tables encode detector outcomes and absorption by blocked path and input state; corresponding tables are related by row and column swaps.The D1/D2, S1/S2, P1/P2, and ω1/ω3 table pairs imply matching game statistics under the stated swaps.Source: Appendix B · p. 19, Table IV · p. 20
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