Trace distance between two probability distributions - prove

[Emil_M]

Homework Statement

Let $\{p_x\}$ and $\{q_x\}$ be two probability distributions over the same index set $\{x\}={1,2,...,N}$. Then, the trace distance between them is given by $D(p_x,q_x):=\frac{1}{2} \sum_x |p_x-q_x|$.

Prove that $D(p_x,q,_x)=max_S |p(S)-q(S)|=max_S | \sum_{x \in S} p_x - \sum_{x \in S} q_x|$, where the maximization is over all subsets $S$ of the index set $\{x\}$.

Homework Equations

See above

The Attempt at a Solution

$\begin{align*} \frac{1}{2} \sum_x |p_x-q_x| &\geq | \sum_{x \in S} p_x - \sum_{x \in S} q_x|\\ &= |\sum_{x \in S} p_x-q_x| \end{align*}$

Then, I have tried playing around with the triangle inequality, but that didn't go anywhere...

Thanks for you help!

 

[RUber]

The key piece of information here is that all the probabilities will sum to 1.
If p(1) - q(1) = z, then the sum of q(2) to q(N) minus the sum of p(2) to p(N) will also be z.

 

[Ray Vickson]

Homework Statement

Let $\{p_x\}$ and $\{q_x\}$ be two probability distributions over the same index set $\{x\}={1,2,...,N}$. Then, the trace distance between them is given by $D(p_x,q_x):=\frac{1}{2} \sum_x |p_x-q_x|$.

Prove that $D(p_x,q,_x)=max_S |p(S)-q(S)|=max_S | \sum_{x \in S} p_x - \sum_{x \in S} q_x|$, where the maximization is over all subsets $S$ of the index set $\{x\}$.

Homework Equations

See above

The Attempt at a Solution

$\begin{align*} \frac{1}{2} \sum_x |p_x-q_x| &\geq | \sum_{x \in S} p_x - \sum_{x \in S} q_x|\\ &= |\sum_{x \in S} p_x-q_x| \end{align*}$

Then, I have tried playing around with the triangle inequality, but that didn't go anywhere...

Thanks for you help!

If you let $p_x - q_x = r_x$, the $r_x$ sum to zero. We can re-write $D(p_x,q_x)$ as $(1/2) \sum_x |r_x|$ and $p(S) - q(S) = \sum_{x \in S} r_x$.

We can write
$$\sum_{x \in S} r_x = \underbrace{\sum_{x \in S, r_x > 0} r_x }_ {\geq 0} + \underbrace{\sum_{x \in S, r_x < 0} r_x }_ {\leq 0}$$
Think about what properties $S$ must have in order that the absolute value of the above sum be maximal, say at the subset $S = S_0$.

 

[Emil_M]

Thanks for your help! I get it now