The Theorem

Theorem 11.8 of Nielsen and Chuang collects the basic properties of the von Neumann entropy of a quantum state. On a system whose state space has dimension $d$ , how small and how large can the entropy be, and which states reach each bound? When two systems together are in a pure state, how do the entropies of the two systems on their own compare? And what is the entropy of a mixture of states with probabilities $p_i$ when the states live on mutually orthogonal subspaces, or when the $i$ th state is the product of a pure state $|i\rangle$ of one system with a state $\rho_i$ of another, the $|i\rangle$ orthonormal?

The library answers them in seven theorems on states of quantum systems, each bound and each case of equality a theorem of its own.

A Quantum System

A quantum system $A$ is a physical system whose state space is a finite dimensional complex Hilbert space.^✦ The composite system $A \otimes B$ of two systems $A$ and $B$ is the system whose state space is the tensor product of the state spaces of $A$ and $B$ .^✦

A state $\rho$ of $A$ is a positive operator of unit trace on the state space of $A$ .^✦ A pure state $|\psi\rangle$ of $A$ is a unit vector in the state space of $A$ .^✦ The pure state $|\psi\rangle$ determines the operator $|\psi\rangle\langle\psi|$ on that space, the rank one projector onto $|\psi\rangle$ .^✦ A state $\rho$ of $A$ is pure when $\rho = |\psi\rangle\langle\psi|$ for some pure state $|\psi\rangle$ of $A$ .^✦

The maximally mixed state of a system $A$ whose state space is nonzero, of dimension $d$ , is $I/d$ , the identity on that space divided by $d$ .^✦

Reduced States, Products and Mixtures

The reduced state $\rho_A$ of a state $\rho$ of $A \otimes B$ is the state of $A$ given by its partial trace $\operatorname{tr}_B \rho$ over $B$ , an operator on the state space of $A$ .^✦ The reduced state $\rho_B = \operatorname{tr}_A \rho$ of $\rho$ on $B$ is its partial trace over $A$ .^✦

The product state $\rho \otimes \sigma$ of a state $\rho$ of $A$ and a state $\sigma$ of $B$ is the tensor product of the two operators on the state space of $A \otimes B$ , acting as $\rho$ on the first factor and as $\sigma$ on the second.^✦

The mixture of states $\rho_i$ of $A$ , indexed by a finite set, with probabilities $p_i$ , nonnegative and summing to one, is the convex combination $\sum_i p_i \rho_i$ .^✦

The support of a state $\rho$ of $A$ is the range of $\rho$ , the subspace spanned by its eigenvectors with nonzero eigenvalues.^✦

The Entropy

The Shannon entropy of probabilities $p_i$ is $H(p) = -\sum_i p_i \log p_i$ , with $0 \log 0 = 0$ .^✦ The von Neumann entropy of a state $\rho$ of $A$ is the real number $S(\rho) = -\operatorname{tr}(\rho \log \rho),$ (1)^✦ the Shannon entropy of the eigenvalues of $\rho$ . Logarithms are natural throughout, and entropies are measured in nats.

The Claim

Theorem 1. Let $A$ and $B$ be quantum systems, $d$ the dimension of the state space of $A$ , and $\rho$ a state of $A$ . Then $S(\rho) \geq 0,$ (2)^✦ $S(\rho) = 0 \quad\text{if and only if $\rho$ is pure,}$ (3)^✦ $S(\rho) \leq \log d,$ (4)^✦ and, if the state space of $A$ is nonzero, $S(\rho) = \log d \quad\text{if and only if}\quad \rho = I/d.$ (5)^✦ If $|\psi\rangle$ is a pure state of $A \otimes B$ , its two reduced states have equal entropy: $S\big(\operatorname{tr}_B |\psi\rangle\langle\psi|\big) = S\big(\operatorname{tr}_A |\psi\rangle\langle\psi|\big).$ (6)^✦ Let $p_i$ be probabilities indexed by a finite set. If $\rho_i$ are states of $A$ with pairwise orthogonal supports, then $S\Big(\sum_i p_i \rho_i\Big) = H(p) + \sum_i p_i\, S(\rho_i).$ (7)^✦ If the pure states $|i\rangle$ of $A$ , indexed by the same finite set as the $p_i$ , form an orthonormal family and the $\rho_i$ are any states of $B$ , then $S\Big(\sum_i p_i\, |i\rangle\langle i| \otimes \rho_i\Big) = H(p) + \sum_i p_i\, S(\rho_i).$ (8)^✦

What Is Checked

The statements are made over the constructions defined here: the states of a system and of a composite system, the projector $|\psi\rangle\langle\psi|$ of a pure state and the condition of purity, the entropy (1), the maximally mixed state, the mixture $\sum_i p_i \rho_i$ , the support, the product state and the two reduced states.

The library proves that these constructions are states: the projector $|\psi\rangle\langle\psi|$ onto a unit vector, the mixture $\sum_i p_i \rho_i$ , the product $\rho \otimes \sigma$ of two states, the reduced states $\operatorname{tr}_B \rho$ and $\operatorname{tr}_A \rho$ of a state of $A \otimes B$ , and $I/d$ are each positive operators of unit trace.