[Paper Review] Quantum Fidelities, Their Duals, And Convex Analysis

Keiji Matsumoto|arXiv (Cornell University)|Aug 15, 2014

Quantum Computing Algorithms and Architecture2 references3 citations

TL;DR

This paper introduces and analyzes three quantum fidelities—Uhlmann's fidelity, the negative f-divergence with f(x) = -√x, and a novel fidelity via reverse testing—using convex optimization. It defines their duals (polar forms), proves monotonicity under unital completely positive maps, and establishes concavity and duality via classical-to-quantum and quantum-to-classical maps, with explicit formulas for qubit cases showing inverse dependence on observable overlap.

ABSTRACT

We study tree kinds of quantum fidelity. Usual Uhlmann's fidelity, minus of f-divergence when $f(x)=-\sqrt{x}$, and the one introduced by the author via reverse test. All of them are quantum extensions of classical fidelity, where the first one is the largest and the third one is the smallest. We characterize them in terms of convex optimization, and introduce their 'dual' quantity, or the polar of the minus of the fidelity. They turned out to be monotone increasing by unital completely positive maps, concave, and linked to its classical version via optimization about classical-to-quantum maps and quantum-to-classical maps.

Motivation & Objective

• To characterize three quantum fidelities—Uhlmann’s, f-divergence-based, and reverse-test-based—using convex analysis.
• To define and analyze the duals (polar forms) of these fidelities.
• To establish monotonicity under unital completely positive maps and concavity.
• To connect quantum fidelities to classical fidelities through optimization over classical-to-quantum and quantum-to-classical maps.
• To derive explicit formulas for the maximum and minimum fidelities in the qubit case, showing inverse dependence on observable overlap.

Proposed method

• Uses convex optimization to characterize the three fidelities and their duals.
• Defines the dual of minus-fidelity as the polar of the fidelity, showing it is monotone increasing under unital CP maps.
• Applies unital completely positive maps to relate quantum fidelities to classical fidelities via measurement and preparation maps.
• Derives explicit expressions for the maximum and minimum fidelities in the qubit case using trace and spectral properties.
• Applies the pinching operation and generalized inverses to handle non-invertible operators.
• Uses the structure of POVMs and CPTP maps to express quantum-to-classical and classical-to-quantum transformations.

Experimental results

Research questions

• RQ1 How do the three quantum fidelities—Uhlmann’s, f-divergence-based, and reverse-test-based—relate to each other and to classical fidelity?
• RQ2 What is the dual (polar) of the minus-fidelity, and how does it behave under unital completely positive maps?
• RQ3 How are quantum fidelities linked to classical fidelities via optimization over quantum-to-classical and classical-to-quantum maps?
• RQ4 What are the explicit forms of the maximum and minimum fidelities for qubit states under specific trace and norm constraints?
• RQ5 Under what conditions is a CP unital map guaranteed to exist between two pairs of qubit operators?

Key findings

• The maximum fidelity $ F_{ ext{max}}(L_0, L_1) $ and minimum fidelity $ F_{ ext{min}}(L_0, L_1) $ are decreasing functions of $ \mathrm{tr}\,L_0L_1 $ when $ \mathrm{tr}\,L_0^2 = \mathrm{tr}\,L_1^2 = \mathrm{tr}\,(L_0')^2 = \mathrm{tr}\,(L_1')^2 $ and $ \|L_0 - L_1\| $ is fixed.
• For qubit states satisfying $ L_0 = I_2 + a\sigma_x + b\sigma_z $, $ L_1 = I_2 + a\sigma_x - b\sigma_z $, the minimum fidelity is $ \hat{F}_{\min}(L_0, L_1) = 2\sqrt{(1 - r^2)(1 - a^2/r^2)} $ when $ r^2 \geq |a| $, and $ 2(1 - |a|) $ otherwise.
• The maximum fidelity $ \hat{F}_{\max}(L_0, L_1) $ is given by a closed-form expression involving the trace and determinant of $ L_0 $ and $ L_1 $, specifically $ \frac{2(1 - r^2)^2}{\mathrm{tr}\,L_0L_1 + \sqrt{(\mathrm{tr}\,L_0L_1)^2 - 4(1 - r^2)^2}} $.
• The dual fidelities $ \hat{F}_{\max} $ and $ \hat{F}_{\min} $ are monotone increasing under unital completely positive maps, while the original fidelities are monotone decreasing under such maps.
• When $ \mathrm{rank}(L_0) = 1 $ or $ \mathrm{rank}(L_1) = 1 $, a CP unital map exists between two pairs of operators only if the target pair also has at least one rank-1 operator.
• For qubit systems with equal trace norms, a CP unital map exists between $ (L_0, L_1) $ and $ (L_0', L_1') $ if and only if $ \mathrm{tr}\,L_0L_1 = \mathrm{tr}\,L_0'L_1' $, implying unitary equivalence.