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Against probability: A quantum state is more than a list of probability distributions

Ladina Hausmann, Renato Renner

TL;DR

The paper argues that representing quantum states solely by probability distributions across measurements is inherently non-robust topologically. By formalizing topological robustness and linking it to data-compression structure via an asymptotic entropy framework, it shows that any probability representation with nontrivial structure cannot be robust, with concrete results for product and efficient measurement sets. The authors extend the analysis to generalized probabilistic theories, constructing a keys-and-locks GPT where local tomography holds but the probability representation metrics induce different topologies from the trace-distance-inspired metric, underscoring a fundamental obstruction to robust probabilistic representations in broad theories. Together, these results motivate the search for an alternative, topologically robust framework for state representation that preserves subsystem structure and generalizes beyond quantum theory.

Abstract

The state $ρ$ of a quantum system can be represented by a vector $\mathbf{P}_{\mathcal{M}}(ρ)$ of outcome probabilities for a set of measurements $\mathcal{M}$. Such representations appear throughout physics, for example, in quantum field theory via correlation functions and in quantum foundations within generalized probabilistic frameworks. In this work, we identify an unavoidable tension: to enable operationally meaningful statements, the map ${ρ\mapsto \mathbf{P}_{\mathcal{M}}(ρ)}$ must be topologically robust $\unicode{x2013}$ preserving the notion of closeness between states. Yet, a probability representation that is topologically robust cannot simultaneously retain other essential structure, such as the subsystem structure.

Against probability: A quantum state is more than a list of probability distributions

TL;DR

The paper argues that representing quantum states solely by probability distributions across measurements is inherently non-robust topologically. By formalizing topological robustness and linking it to data-compression structure via an asymptotic entropy framework, it shows that any probability representation with nontrivial structure cannot be robust, with concrete results for product and efficient measurement sets. The authors extend the analysis to generalized probabilistic theories, constructing a keys-and-locks GPT where local tomography holds but the probability representation metrics induce different topologies from the trace-distance-inspired metric, underscoring a fundamental obstruction to robust probabilistic representations in broad theories. Together, these results motivate the search for an alternative, topologically robust framework for state representation that preserves subsystem structure and generalizes beyond quantum theory.

Abstract

The state of a quantum system can be represented by a vector of outcome probabilities for a set of measurements . Such representations appear throughout physics, for example, in quantum field theory via correlation functions and in quantum foundations within generalized probabilistic frameworks. In this work, we identify an unavoidable tension: to enable operationally meaningful statements, the map must be topologically robust preserving the notion of closeness between states. Yet, a probability representation that is topologically robust cannot simultaneously retain other essential structure, such as the subsystem structure.
Paper Structure (21 sections, 22 theorems, 116 equations, 3 figures, 1 table)

This paper contains 21 sections, 22 theorems, 116 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction and a first example
  2. A topological characterization of robustness
  3. Structure is incompatible with robustness
  4. Beyond Quantum
  5. Conclusion
  6. Topological properties
  7. Asymptotic entropy
  8. Proof of
  9. Proof of
  10. Proof of
  11. Related results
  12. Proof of
  13. GPTs
  14. States and measurements.
  15. System composition.
  16. ...and 6 more sections

Key Result

Proposition 1

The topologies induced by $d_{\mathcal{M}}$ and $\delta$ are identical on the space of density matrices $\mathcal{D}(\mathcal{H})$, except when $\mathcal{M}$ is not stable.

Figures (3)

  • Figure 1: Non-robustness of $\mathbf{P}_{\mathcal{M}_{\otimes}}$. A region of the quantum state space $\mathcal{D}(\mathcal{H})$ is shown, containing the sequence of states $\rho^{(n)}$ from the example (black) and the corresponding closest points in $\Sigma = \Sigma^{\mathrm{rand}}$ (blue). Panel (a) uses the trace distance $\delta$, for which the distance to $\Sigma$ stays constant. Panel (b) uses the metric $d_{\mathcal{M}_{\otimes}}$ induced by the product-measurement representation $\mathbf{P}_{\mathcal{M}_{\otimes}}$, for which the distance to $\mathbf{P}_{\mathcal{M}_{\otimes}}(\Sigma)$ shrinks with increasing $n$.
  • Figure 2: State-space versus representation-space approximations. Physical properties are often expressed by the proximity of a state $\rho$ to a set $\Sigma$. The diagram illustrates the requirement that approximate statements established at the level of a probability representation remain valid when pulled back to the level of density operators. According to \ref{['def:preservation']}, this is the case whenever the representation is topologically robust.
  • Figure 3: Sketch of the topological problem. The state space $\mathcal{D}(\mathcal{H})$ (in blue) is embedded into $\mathrm{span}(\mathcal{D}(\mathcal{H}))$. While for stable $\mathcal{M}$ the topologies induced by $\|\cdot\|_{\mathcal{M}}$ and $\|\cdot\|_{1}$ agree on $\mathcal{D}(\mathcal{H})$, they generally disagree on an open region (in green) around it.

Theorems & Definitions (69)

  • Example
  • Example : continued
  • Definition 1
  • Proposition 1
  • Proposition 1
  • Definition 2: Informal
  • Definition 3
  • Remark 3
  • Theorem 4
  • Corollary 4
  • ...and 59 more