A universal complementarity identity for polarized double-slit interferometry

Jos\'e J. Gil

arxiv: 2604.18760 · v2 · pith:KWEQIC5Bnew · submitted 2026-04-20 · 🪐 quant-ph · physics.optics

A universal complementarity identity for polarized double-slit interferometry

Jos\'e J. Gil This is my paper

Pith reviewed 2026-05-10 04:16 UTC · model grok-4.3

classification 🪐 quant-ph physics.optics

keywords complementarity identitydouble-slit interferometryfringe visibilitypath predictabilityquantum mixednesspolarizationdensity matrixmaximum entropy

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The pith

Four measurable quantities in polarized double-slit experiments satisfy the exact identity V_A squared plus V_N squared plus P squared plus I squared equals one.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the in-phase and quadrature parts of fringe visibility, together with path predictability and the mixedness of the reduced state, always sum their squares to exactly one. This relation follows directly as an algebraic consequence of the positivity and normalization of the path-polarization density matrix for any such experiment. A sympathetic reader would care because the identity unifies earlier complementarity bounds, splits visibility into two operationally distinct components that can be measured separately with phase shifts, and supplies a maximum-entropy interpretation of the path information. It therefore gives a single, universal accounting of the trade-offs among coherence, predictability, and decoherence in the setup.

Core claim

We establish an exact identity among four dimensionless invariants accessible by standard polarimetric and interferometric measurements in a polarized double-slit experiment: the in-phase and quadrature components V_A and V_N of fringe visibility, the path predictability P, and the mixedness I of the path-reduced state satisfy V_A^2 + V_N^2 + P^2 + I^2 = 1. The identity is a universal algebraic consequence of the positivity of the reduced state and holds for every normalized path-polarization density matrix. It unifies the Englert-Greenberger-Yasin and Jakob-Bergou relations, separates the two operationally distinct components of visibility measurable by phase-shifted interferometry, and adm

What carries the argument

The positivity and normalization of the reduced path-polarization density matrix, which algebraically enforces the identity among the four invariants V_A, V_N, P, and I.

Load-bearing premise

The reduced path-polarization density matrix must be positive semidefinite and normalized; if this fails the identity does not hold.

What would settle it

An experimental run in which the measured values of in-phase visibility, quadrature visibility, path predictability, and mixedness satisfy V_A squared plus V_N squared plus P squared plus I squared not equal to one.

Figures

Figures reproduced from arXiv: 2604.18760 by Jos\'e J. Gil.

**Figure 1.** Figure 1: Geometrical representation of identity (3). (a) Bloch sphere V 2 A + V 2 N + P 2 ≤ 1 of the path-reduced state, with the equatorial trajectory of preparatory-phase rotation (solid), the radial path of phase decoherence (dashed), and the meridian arc of predictability–coherence interpolation (dotted). The residual mixedness I equals the distance from the state point to the spherical surface. (b) Detected in… view at source ↗

read the original abstract

An exact identity is established among four experimentally accessible quantities in polarized double-slit interferometry: the phase-reference-dependent in-phase and quadrature components $V_A$ and $V_N$ of fringe visibility, the path predictability $\mathcal{P}$, and the mixedness $\mathcal{I}$ of the reduced path state satisfy $V_A^2+V_N^2+\mathcal{P}^2+\mathcal{I}^2=1$. The identity is an algebraic consequence of positivity and holds for every normalized path--polarization density matrix. It contains the Greenberger--Yasin predictability bound and, for globally pure path--polarization states, the Jakob--Bergou complete-complementarity equality; it is also connected with Englert's distinguishability relation when polarization carries which-path information. The separation $V^2=V_A^2+V_N^2$ resolves visibility into two components measurable by phase-shifted interferometry. Within a fixed real basis and a fixed phase convention, the quadrature-sensitive component is read from the antisymmetric sector of the Hermitian decomposition $\rho=A+iN$. A maximum-entropy reconstruction is included as an interpretation of how measurements sensitive to the two sectors constrain an inferred state, but the identity itself does not depend on that reconstruction.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper's four-term identity is just the Bloch-vector normalization for any qubit state, repackaged with interferometry language and a Jaynes reading.

read the letter

The core result is that V_A² + V_N² + P² + I² = 1 for the path-reduced state in a polarized double-slit setup. This is an immediate rewriting of the fact that any valid 2×2 density matrix satisfies |r|² + (1 − |r|²) = 1 once the off-diagonal coherence is split into real and imaginary parts, the diagonal imbalance is called predictability, and the purity deficit is called I. The identity therefore holds for every normalized positive semidefinite qubit state with no extra assumptions needed. The paper does a clean job of separating the two measurable visibility components (in-phase versus quadrature) and showing how the Englert-Greenberger-Yasin and Jakob-Bergou relations sit inside the same parametrization. The Jaynes maximum-entropy framing is also a reasonable way to view the three path invariants as the minimal exponential family. Those interpretive moves are useful for people who actually run polarized interferometers. The main limitation is that the algebraic step itself is standard quantum mechanics rather than a new derivation; once the parameters are defined, the sum of squares equals one by construction. No new predictions or experimental tests appear in the abstract, and the unification is achieved by re-labeling rather than by solving a dynamical problem. Readers already comfortable with qubit geometry will see the identity as tautological, while experimental groups working on visibility and which-path information may still find the operational split convenient. The work is coherent on its own terms and engages the cited literature directly. I would send it to peer review so that referees can check whether the full derivation adds any non-obvious steps and whether the experimental discussion is substantive enough to justify publication in a quantum-optics venue.

Referee Report

0 major / 0 minor

Summary. The manuscript establishes an exact identity V_A² + V_N² + P² + I² = 1 among four dimensionless invariants in polarized double-slit interferometry: the in-phase (V_A) and quadrature (V_N) components of fringe visibility, the path predictability P, and the mixedness I of the path-reduced state. This is derived as a universal algebraic consequence of the positivity and normalization of the reduced path-polarization density matrix, holding for every valid 2×2 state. The work unifies the Englert-Greenberger-Yasin and Jakob-Bergou relations, operationally separates the two visibility components via phase-shifted interferometry, and interprets the invariants within the Jaynes maximum-entropy framework, with I² as the residual mixedness saturating the positivity bound.

Significance. If the result holds, the identity supplies a geometrically transparent reparametrization of complementarity bounds in terms of the Bloch vector of the reduced qubit state, which may aid experimental characterization of decoherence in quantum optics by isolating phase-sensitive coherence. The unification of prior relations and the Jaynes interpretation add conceptual clarity, while the separation of V_A and V_N offers an operational handle on environmental coupling; however, the core algebraic content is a direct rewriting of the standard |r|² ≤ 1 constraint for qubit states.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful summary of our results, the positive assessment of their potential utility for decoherence characterization, and the recommendation for minor revision. We address the principal observation raised in the report below.

read point-by-point responses

Referee: however, the core algebraic content is a direct rewriting of the standard |r|² ≤ 1 constraint for qubit states.

Authors: We agree that the identity V_A² + V_N² + P² + I² = 1 is algebraically equivalent to the Bloch-vector norm bound |r| ≤ 1 on the reduced path qubit. Our derivation begins from this positivity constraint and simply expresses it in the basis of interferometric observables. The manuscript's contribution, however, consists in (i) decomposing the visibility into its operationally distinct in-phase and quadrature components that are separately measurable by phase-shifted interferometry, (ii) unifying the Englert–Greenberger–Yasin and Jakob–Bergou relations as special cases of the same four-term identity, and (iii) supplying a Jaynes maximum-entropy interpretation in which the three path invariants parametrize the minimal exponential family while I² quantifies the residual mixedness required by positivity. These elements provide an experimentally actionable reparametrization that is not contained in the abstract qubit bound alone. revision: no

Circularity Check

0 steps flagged

No significant circularity; identity is direct algebraic consequence of qubit Bloch vector normalization

full rationale

The paper derives the identity V_A² + V_N² + P² + I² = 1 strictly as an algebraic rewriting of the condition |r|² + (1 - |r|²) = 1 on the Bloch vector r of any normalized 2×2 density matrix. All four invariants are defined component-wise from the same matrix elements (P from diagonal imbalance, V_A/V_N from real/imaginary off-diagonal coherence, I from the normalized purity deficit complementing |r|²), so the sum-of-squares relation holds identically once the state is required to be positive semidefinite. No parameters are fitted, no quantity is defined in terms of another inside the claimed derivation, and the unification of EGY/JB relations follows immediately from the reparametrization without invoking self-citations or external uniqueness theorems. The result is therefore self-contained against standard quantum mechanics and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests solely on the positivity of the reduced density matrix, a standard domain assumption in quantum mechanics; no free parameters or new entities are introduced.

axioms (1)

domain assumption The path-polarization density matrix is positive semidefinite and normalized.
The identity is stated to be an algebraic consequence of this property.

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