Quantum Information Axioms

Information-theoretic axioms describe physical possibilities and limitations in terms of distinguishability, composition, communication, and reversible control. Their purpose is to explain the structure of a theory through experimentally meaningful assumptions. Calling an axiom “informational” does not make it independent of physical assumptions about systems and measurements.

Helpful background. GPTs supplies states, effects, and composite systems.

Distinguishability and capacity

A set of states $\{\omega_i\}_{i=1}^N$ is perfectly distinguishable if one allowed measurement satisfies

$$e_j(\omega_i)=\delta_{ij}.$$

The maximum $N$ is an operational capacity. It differs from the real dimension $K$ of the unnormalized state space. A classical $N$-level system has $K=N$; a complex quantum $N$-level system has $K=N^2$. A qubit therefore has two perfectly distinguishable states despite requiring three real parameters after normalization.

This distinction prevents an ambiguity in “information content.” Capacity counts messages that can be identified without error in one use. Tomographic dimension counts independent real data required to characterize an unknown preparation. Neither is the number of bits in an arbitrary numerical description at unlimited precision.

Principles with different logical jobs

Causality in operational reconstructions often means a unique deterministic effect: preparation probabilities cannot depend on the choice of a later measurement. It is not, by itself, relativistic local causality.

Local tomography requires product measurement statistics to determine a composite state. It restricts composition and gives $K_{AB}=K_AK_B$ under the usual finite-dimensional assumptions.

Continuous reversibility demands continuous reversible paths relating pure states. A classical simplex has reversible permutations of finitely many vertices, so its reversible transformations cannot provide such a nontrivial continuous path.

Purification requires every mixed state to arise as a marginal of a pure state, with an appropriate uniqueness clause. Its force comes from constraints on composites, not from redescribing a mixture with an arbitrary mathematical label.

These principles are not an interchangeable menu. In the reconstruction by Chiribella, D’Ariano, and Perinotti, purification operates alongside causality, distinguishability, compression, local distinguishability, and pure conditioning. The theorem concerns the stated package.

Reading a reconstruction critically

Separate the ambient framework from the named axioms. Finite dimensionality, convex mixing, allowed classical controls, and assumptions about composition can carry substantial content before the advertised postulates appear.

Then ask what the conclusion reconstructs: states only, probabilities, reversible transformations, or all allowed transformations. Finally distinguish a theorem from its physical interpretation. A derivation of density matrices from operational principles does not establish that the wavefunction is ontic, epistemic, or observer relative.

Exercises

Does $K_{AB}=K_AK_B$ alone prove that a theory is quantum?

Solution

No. Classical probability also satisfies the dimension product rule, with $K=N$. Local tomography constrains composites but must be combined with assumptions about local state spaces and transformations. Compare the full packages in reconstructions.

References

• Chiribella, Giulio, Giacomo Mauro D’Ariano, and Paolo Perinotti, “Informational derivation of quantum theory,” Physical Review A 84, 012311 (2011). Paper.
• Hardy, Lucien, “Quantum Theory From Five Reasonable Axioms,” arXiv:quant-ph/0101012 (2001). Paper.