[Paper Review] Pretty simple bounds on quantum state discrimination

Ashley Montanaro|arXiv (Cornell University)|Aug 22, 2019

Quantum Information and Cryptography15 references4 citations

TL;DR

This paper demonstrates that the pretty good measurement (PGM) provides efficient, explicit solutions to worst-case quantum state discrimination: for mixed states with pairwise fidelity bounded away from 1, O(log n / ε) copies suffice to achieve failure probability δ; for pure states, the success probability is bounded by the operator norm of the Gram matrix G, enabling Õ(∥G∥/ε) copies to achieve δ failure probability, with tighter bounds when ∥G∥ is small.

ABSTRACT

We show that the quantum measurement known as the pretty good measurement can be used to identify an unknown quantum state picked from any set of $n$ mixed states that have pairwise fidelities upper-bounded by a constant below 1, given $O(\log n)$ copies of the unknown state, with high success probability in the worst case. If the unknown state is promised to be pure, there is an explicit measurement strategy which solves this worst-case quantum state discrimination problem with $\widetilde{O}(\|G\|)$ copies, where $G$ is the Gram matrix of the states.

Motivation & Objective

• To provide explicit, constructive bounds for worst-case quantum state discrimination, avoiding nonconstructive minimax theorems.
• To show that the pretty good measurement (PGM) achieves near-optimal copy complexity for discriminating mixed states with bounded pairwise fidelity.
• To establish a dependence on the Gram matrix norm ∥G∥ for pure state discrimination, enabling improved scaling when ∥G∥ is small.
• To extend prior results on average-case discrimination to the worst-case setting, particularly for random pure states.
• To clarify the limitations of extending ∥G∥-based bounds to mixed states, showing they do not universally apply.

Proposed method

• Applies the pretty good measurement (PGM) defined as μ_i = Σ^{-1/2} ρ_i Σ^{-1/2}, where Σ = ∑_i ρ_i, to copies of the unknown state.
• Uses the Gram matrix G of the states (or their weighted eigenvectors) to bound the success probability of the PGM via operator norm ∥G∥.
• Employs a two-stage protocol for pure states: first apply PGM to k = O(∥G∥ log(1/δ)) copies to identify candidate states, then verify via projective measurements.
• Applies union bounds and exponential tail estimates to bound failure probability δ, using fidelity constraints (tr(ρ_i ρ_j) ≤ 1−ε) between distinct states.
• Derives bounds on the trace of μ_i ρ_i using spectral properties of G, showing ∥G∥^{-1} ≤ tr(μ_i ρ_i) ≤ ∥G^{-1}∥ tr(ρ_i^2).
• Analyzes the behavior of ∥G∥ for random pure states (e.g., Haar or ±1 amplitude states), showing ∥G∥ = O(1) with high probability when n = O(d).

Experimental results

Research questions

• RQ1 Can the pretty good measurement (PGM) be used to constructively solve worst-case quantum state discrimination with copy complexity matching known nonconstructive bounds?
• RQ2 How does the operator norm ∥G∥ of the Gram matrix of pure states influence the success probability and required number of copies in state discrimination?
• RQ3 Can the PGM achieve O(log n / ε) copy complexity for mixed states with pairwise fidelity bounded by 1−ε?
• RQ4 Is it possible to achieve sub-linear dependence on n in the copy complexity for pure state discrimination when ∥G∥ is small?
• RQ5 What are the limitations of extending ∥G∥-based bounds to mixed state discrimination, and when do they fail?

Key findings

• For any set of n mixed states with pairwise fidelity ≤ 1−ε, the PGM solves worst-case state discrimination with failure probability δ using O(log(n/δ)/ε) copies.
• For pure states, the PGM applied to one copy achieves a success probability of at least ∥G∥^{-1}, where G is the Gram matrix of the states.
• When all pure states have pairwise inner products ≤ 1−ε, the PGM can be combined with verification to solve the problem with Õ(∥G∥/ε) copies and failure probability δ.
• The bound ∥G∥ = O(1) holds with high probability for random pure states in d dimensions when n = O(d), implying constant copy complexity in such cases.
• The PGM fails to provide useful bounds for mixed states when ∥G∥ is small, as shown by the counterexample of n maximally mixed states where ∥G∥ = 1 but success probability is 1/n.
• For equiangular states with constant pairwise inner product c, the bound ∥G∥ = 1 + c(n−1) gives Õ(n) copies, but tighter analysis shows only O(1) copies are needed, indicating the bound is not always tight.