Lean, informalized

Nielsen & Chuang · Chapter 9, Distance measures for quantum information · p. 407

Theorem 9.3 - The trace distance of two mixtures, bounded by their weights and their components

How far apart two mixtures of states can be.

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The Theorem

Two sources prepare the same quantum system SS. The first draws an index ii from a finite set with probability pip_i and prepares the state ρi\rho_i; the second draws ii from the same set with probability qiq_i and prepares σi\sigma_i. To someone who receives the system without learning the index, each source produces the mixture of its states with its weights. Theorem 9.3 bounds the trace distance, the measure of how far apart two states are, between the two mixtures by the distance between the two distributions and the distances between the states ρi\rho_i and σi\sigma_i of each pair. The bound is known as the strong convexity of the trace distance.

A Quantum System

A quantum system SS is described by its state space, a finite dimensional complex Hilbert space.✦ A state ρ\rho of SS is a positive operator of unit trace on the state space of SS; pure and mixed states are states alike.✦

Mixtures and Distances

The mixture of states ρi\rho_i of SS, indexed by ii in a finite set, with weights pip_i that are nonnegative and sum to one, is the state of SS whose density operator is the convex combination ∑ipiρi.\sum_i p_i \rho_i .(1)✦

The trace distance D(ρ,σ)D(\rho, \sigma) between two states ρ\rho and σ\sigma of SS is half the trace norm of their difference, D(ρ,σ)=12tr⁡|ρ−σ|=12∑k|λk|,D(\rho, \sigma) = \frac{1}{2} \operatorname{tr} |\rho - \sigma| = \frac{1}{2} \sum_k |\lambda_k| ,(2)✦ where the λk\lambda_k are the eigenvalues of ρ−σ\rho - \sigma.

The classical trace distance D(px,qx)D(p_x, q_x) between two real functions pp and qq on a finite set, such as two probability distributions, is half the sum of the absolute differences of their values, D(px,qx)=12∑x|px−qx|.D(p_x, q_x) = \frac{1}{2} \sum_x |p_x - q_x| .(3)✦

The Claim

Theorem 1. Let pip_i and qiq_i be probability distributions on the same finite index set, and let ρi\rho_i and σi\sigma_i be states of a quantum system SS indexed by that set. Then D(∑ipiρi,∑iqiσi)≤D(pi,qi)+∑ipiD(ρi,σi).D\Big( \sum_i p_i \rho_i , \sum_i q_i \sigma_i \Big) \le D(p_i, q_i) + \sum_i p_i \, D(\rho_i, \sigma_i) .(4)✦

What Is Checked

The claim (4) is stated over the mixtures (1) of states of SS and over the two trace distances, (2) between states and (3) between the weights.

The library proves of the mixture that a convex combination of density operators, with nonnegative weights summing to one, is again a positive operator of unit trace, so that each mixture is a state of SS.