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No-Signalling Fixes the Hilbert-Space Inner Product

Arun Kumar Pati

TL;DR

The paper investigates whether the Hilbert-space inner product can be generalized via a positive operator $A>0$ without violating no-signalling. It shows that any nontrivial $A\neq \mathbb{I}$ induces a state-dependent norm change under linear, unitary evolution, leading to nonlinear, ensemble-dependent dynamics and potential superluminal signalling. Consequently, no-signalling enforces the standard inner product $A=\mathbb{I}$ (up to scale), establishing a unique compatibility between quantum kinematics and relativistic causality. A concrete two-qubit Bell-state example demonstrates the signalling mechanism, while the appendix formalizes the trace normalization and nonlinear state-update map, linking the result to broader discussions on PT-symmetric quantum mechanics and the foundational rigidity of quantum theory.

Abstract

We investigate whether the inner product structure of quantum mechanics can be modified without violating fundamental physical principles. We consider a generalized inner product defined by a positive operator and assume local unitary dynamics, existence of entangled states and the no-signalling principle. We show that any nontrivial choice of inner product different from standard one inevitably leads to superluminal signalling, in contradiction with relativistic causality. Therefore, the standard Hilbert-space inner product is uniquely enforced by no-signalling.

No-Signalling Fixes the Hilbert-Space Inner Product

TL;DR

The paper investigates whether the Hilbert-space inner product can be generalized via a positive operator without violating no-signalling. It shows that any nontrivial induces a state-dependent norm change under linear, unitary evolution, leading to nonlinear, ensemble-dependent dynamics and potential superluminal signalling. Consequently, no-signalling enforces the standard inner product (up to scale), establishing a unique compatibility between quantum kinematics and relativistic causality. A concrete two-qubit Bell-state example demonstrates the signalling mechanism, while the appendix formalizes the trace normalization and nonlinear state-update map, linking the result to broader discussions on PT-symmetric quantum mechanics and the foundational rigidity of quantum theory.

Abstract

We investigate whether the inner product structure of quantum mechanics can be modified without violating fundamental physical principles. We consider a generalized inner product defined by a positive operator and assume local unitary dynamics, existence of entangled states and the no-signalling principle. We show that any nontrivial choice of inner product different from standard one inevitably leads to superluminal signalling, in contradiction with relativistic causality. Therefore, the standard Hilbert-space inner product is uniquely enforced by no-signalling.
Paper Structure (8 sections, 1 theorem, 27 equations)

This paper contains 8 sections, 1 theorem, 27 equations.

Table of Contents

  1. Introduction
  2. Example
  3. Case 1: Diagonal basis $\{|0\rangle,|1\rangle\}$
  4. Case 2: Diagonal basis $\{|+\rangle,|-\rangle\}$
  5. Discussion and Conclusions
  6. Appendix
  7. Trace Condition
  8. The Nonlinear State-Update Map

Key Result

Theorem 1

Let the inner product on a Hilbert space ${\cal H}$ be modified to where $A>0$ is a fixed positive operator. If $A \neq I$, while allowing standard local unitaries and entanglement, then the no-signalling principle is violated.

Theorems & Definitions (2)

  • Theorem 1: Non-standard inner products imply signalling
  • proof