[Paper Review] A three state invariant

M. Fannes D. Vanpeteghem|ArXiv.org|Feb 5, 2004

Random Matrices and Applications6 references3 citations

TL;DR

This paper introduces a quantum three-state invariant called 'phase' to uniquely reconstruct sequences of pure quantum states from pairwise fidelities and phases, overcoming the ambiguity of fidelity-only reconstruction. It proves that fidelities alone cannot distinguish between classical probability measures, pure, or mixed quantum states, but adding phases enables full reconstruction of pure state sequences.

ABSTRACT

For triples of probability measures, pure quantum states and mixed quantum states we obtain the exact constraints on the fidelities of pairs in the sequence. We show that it is impossible to decide between a quantum model, either pure or mixed, and a classical model on the basis of the fidelities alone. Next, we introduce a quantum three state invariant called phase and show that any sequence of pure quantum states is uniquely reconstructible given the fidelities and phases.

Motivation & Objective

• To determine the constraints on pairwise fidelities for triples of classical probability measures, pure quantum states, and mixed quantum states.
• To demonstrate that fidelity values alone are insufficient to distinguish between classical and quantum models (pure or mixed).
• To introduce a new quantum invariant—'phase'—for pure states to enable unique reconstruction of state sequences.
• To extend the concept of phase to mixed states via purification and unitary freedom in the GNS construction.
• To establish that the minimal set of phases (Φ₁ₖⱼ for 1 < k < j) provides enough information to reconstruct pure state sequences uniquely.

Proposed method

• Uses fidelity as a measure of similarity between states, defined via overlaps: F(μ,λ) = (∑√(λⱼμⱼ))² for classical measures and |⟨φ|ψ⟩|² for pure quantum states.
• Applies Uhlmann's fidelity definition for mixed quantum states: F(ρ,σ) = (Tr√(ρ¹ᐟ²σρ¹ᐟ²))².
• Analyzes triplets of states to derive necessary and sufficient constraints on pairwise fidelities, showing they are identical across classical, pure, and mixed settings.
• Introduces the phase invariant Φ₁₂ⱼ via the complex phase of the triple product ⟨φ₁|φ₂⟩⟨φ₂|φⱼ⟩⟨φⱼ|φ₁⟩, normalized by the geometric mean of fidelities.
• Uses the Schmidt decomposition of purified states to define a generalized phase for mixed states via unitary optimization over purifications.
• Demonstrates that specifying fidelities and phases Φ₁ₖⱼ (1 < k < j) allows unique reconstruction of the state sequence through recursive amplitude determination.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient constraints on the pairwise fidelities of triples of classical probability measures, pure quantum states, and mixed quantum states?
• RQ2 Can the fidelity values alone distinguish between classical and quantum models (pure or mixed) for a triple of states?
• RQ3 Is it possible to uniquely reconstruct a sequence of pure quantum states from pairwise fidelities and additional invariants?
• RQ4 How can the concept of phase be generalized from pure states to mixed quantum states using purification and unitary freedom?
• RQ5 What is the minimal set of phase invariants required to reconstruct a sequence of pure quantum states from fidelities?

Key findings

• The constraints on pairwise fidelities for triples are identical across classical probability measures, pure quantum states, and mixed quantum states, making fidelity alone insufficient to distinguish the model type.
• For any sequence of pure quantum states, the full sequence can be uniquely reconstructed from the pairwise fidelities and the phases Φ₁ₖⱼ (1 < k < j), which are defined via the phase of the triple product of inner products.
• The number of required phase invariants is (n-1)(n-2)/2 for a sequence of length n, which matches the deficit in degrees of freedom compared to the n(n-1)/2 fidelities.
• The phase invariant Φ₁₂ⱼ is defined as the phase of the normalized triple product: e⁻ⁱΦ₁₂ⱼ⟨φ₁|φ₂⟩⟨φ₂|φⱼ⟩⟨φⱼ|φ₁⟩ = F₁₂¹ᐟ² F₁ⱼ¹ᐟ² F₂ⱼ¹ᐟ².
• The reconstruction process recursively determines the complex amplitudes cⱼₖ using the fidelities and phases, with the final amplitude cⱼⱼ determined by normalization.
• A generalization of the phase invariant to mixed states is proposed via the infimum over unitary transformations of the purified state triple, using the GNS construction and Schmidt decomposition.