# CSE 599Q: Homework #5

## Quantum probability and information

Due: Sun, Dec 11 @ 11:59pm

## 1. Positive semidefinite matrices [12 points]

A Hermitian matrix $M \in \C^{d \times d}$ is said to be positive semidefinite, denoted $M \succeq 0$ if it holds that $\bra{u}M\ket{u} \geq 0$ for all $\ket{u} \in \C^d$.

1. [3 pts] Prove that $M \succeq 0$ if and only if $\bra{u}M\ket{u} \geq 0$ holds for all unit vectors $\ket{u} \in \C^d$.

2. [3 pts] Let $M \in \C^{d \times d}$ be a diagonal matrix. Verify that $M$ is Hermitian if and only if all its diagonal entries are real and $M \succeq 0$ if and only if all diagonal entries is nonnegative.

3. [3 pts] Consider any matrix $A \in \C^{d \times d'}$. Show that $A^* A$ is Hermitian and $A^*A \succeq 0$.

4. [3 pts] Recall the Frobenius inner product on matrices $A,B \in \C^{d\times d}$:

$\langle A,B\rangle \seteq \tr(A^*B).$

Prove that if $A \succeq 0$ and $B \succeq 0$, then $\langle A,B\rangle \geq 0$.

You may use the fact that every Hermitian matrix $M \in \C^{d \times d}$ can be written as $M = \sum_{i=1}^d \lambda_i \ket{u_i}\bra{u_i}$ where $\lambda_1,\ldots,\lambda_d$ are the real eigenvalues of $M$ and $\ket{u_1},\ldots,\ket{u_d}$ is an orthornormal basis of eigenvectors for $M$.

## 2. Purification of quantum states [10 points]

Suppose that $\rho^A$ is a $d \times d$ density matrix. Write $\rho^A$ in its eigenbasis:

$\rho^A = \sum_{j=1}^n \lambda_j \ket{v_j}\bra{v_j}\,.$

Let $\mathbb{C}^B$ denote a $d$-dimensional Hilbert space with basis $\ket{1},\ket{2}\,\ldots,\ket{d}$ and define

\begin{align*} \ket{u^{AB}} & \seteq \sum_{j=1}^n \sqrt{\lambda_j} \ket{v_j} \ket{j} \\ \rho^{AB} &\seteq \ket{u^{AB}} \bra{u^{AB}} \end{align*}

Show that $\rho^A = \tr_B(\rho^{AB})$. In other words, the mixed state $\rho^A$ can be seen as arising from taking a joint system in the pure state $\ket{u^{AB}}$ and then discarding the $B$-part of the space.

## 3. Quantum uncertainty splits evenly [10 points]

Recall that if $X$ is a classical random variable such that $p_i \seteq \Pr[X=i]$ for $i=1,2,\ldots,d$, then the Shannon entropy of $X$ is defined by

$H(X) \seteq \sum_{i=1}^d p_i \log \frac{1}{p_i}.$

This is a measure of the uncertainty of $X$ measured in bits (or measured in “nats” if we use the natural logarithm).

Define the von Neumann entropy of a $d \times d$ density matrix $\rho$ by

$\mathcal{S}(\rho) \seteq \sum_{j=1}^d \lambda_j \log \frac{1}{\lambda_j}\,.$

Suppose that $\rho = \ket{u^{AB}} \bra{u^{AB}}$ is a pure state and $\rho^A = \tr_B(\rho), \rho^B = \tr_A(\rho)$. Show that

$\mathcal{S}(\rho^A) = \mathcal{S}(\rho^B)\,.$

Note that the two states don’t necessarily have the same dimension, so they could each have a different number of eigenvalues.

[ Hint: Show first that if $U$ is a $d \times d$ matrix, then $UU^*$ and $U^*U$ have the same non-zero eigenvalues. ]

## 4. Negative conditional entropy [10 points]

1. [5 pts] In classical probability theory, if $A$ and $B$ are two random variables, one defines the entropy of $A$ conditioned on $B$ by the formula

$H(A \mid B) \seteq \sum_{x} \Pr[B=x] \cdot H(A \mid \{B=x\})\,,$

where $A \mid \{B=x\}$ is the random variable $A$ condition on $X$. This quantity is nonnegative because the entropy $H(A \mid \{B=x\})$ is always nonnegative.

Prove that

$H(A \mid B) = H(A,B) - H(B)\,.$

In particular, right-hand side is always nonnegative, and therefore

$H(B) \leq H(A,B)\,.$

This asserts the relatively obvious fact that the pair of random variables $\{A,B\}$ has more uncertainty than the single random variable $B$. In other words, it is easier to predict $B$ than to simultaneously predict both $A$ and $B$.

2. [5 pts] You will show that this fails dramatically in the quantum setting where the conditional entropy can be negative! Using Problem 2, Show that there is a state $\rho^{AB}$ with

$\mathcal{S}(\rho^B) > \mathcal{S}(\rho^{AB})\,,$

where $\rho^B \seteq \tr_A(\rho^{AB})$. In other words, the entropy of the subsystem is actually bigger than the entropy of the full system.