The Theorem

A source prepares a quantum system in a state $\rho$ , again and again, each time independently of the others. After $n$ rounds the $n$ systems share a state space whose dimension grows exponentially in $n$ . Theorem 12.5 of Nielsen and Chuang, the typical subspace theorem, is about how much of that space their joint state occupies. One picks a subspace and measures whether the $n$ systems are found in it: how small can the subspace be while the answer is yes with probability close to one, once $n$ is large?

Systems, States and Copies

A quantum system $Q$ is described by its state space $\mathcal{H}_Q$ , a finite dimensional complex Hilbert space, of dimension $d$ .^✦ A state $\rho$ of $Q$ is a positive operator of unit trace on $\mathcal{H}_Q$ .^✦

The composite system $\bigotimes_i Q_i$ of finitely many systems $Q_i$ is the system whose state space is the tensor product of the spaces $\mathcal{H}_{Q_i}$ .^✦ For states $\rho_i$ of the $Q_i$ , the product state $\bigotimes_i \rho_i$ is the state of $\bigotimes_i Q_i$ whose density operator is the tensor product of the $\rho_i$ .^✦ The system $Q^{\otimes n}$ of $n$ copies of $Q$ is the composite of $n$ systems each equal to $Q$ ^✦, and the state $\rho^{\otimes n}$ of $Q^{\otimes n}$ is the product state with $\rho$ in every position.

Spectrum and Entropy

The spectrum $\lambda$ of $\rho$ is the list $\lambda_1, \dots, \lambda_d$ of its eigenvalues, repeated according to multiplicity, with an orthonormal basis $|e_1\rangle, \dots, |e_d\rangle$ of $\mathcal{H}_Q$ such that $\rho|e_i\rangle = \lambda_i|e_i\rangle$ .^✦ The $\lambda_i$ are nonnegative and sum to one, a probability distribution on $\{1, \dots, d\}$ . The von Neumann entropy $S(\rho)$ of the state $\rho$ is the Shannon entropy of the distribution $\lambda$ , in nats: $S(\rho) = -\operatorname{tr}(\rho \ln \rho) = -\sum_{i=1}^{d} \lambda_i \ln \lambda_i .$ (1)^✦

The Typical Subspace

The eigenbasis of $\rho^{\otimes n}$ is the orthonormal basis of the state space of $Q^{\otimes n}$ formed by the product vectors $|e_x\rangle = |e_{x_1}\rangle \otimes \cdots \otimes |e_{x_n}\rangle, \qquad x = (x_1, \dots, x_n) \in \{1, \dots, d\}^n .$ (2)^✦ The vector $|e_x\rangle$ has eigenvalue $\lambda_{x_1} \cdots \lambda_{x_n}$ , the probability of the sequence $x$ when its $n$ indices are drawn independently from $\lambda$ .

A sequence $x$ of length $n$ is typical for $\lambda$ with tolerance $\epsilon$ , in Shannon’s sense, when its probability $\lambda_{x_1} \cdots \lambda_{x_n}$ lies within the window of width $\epsilon$ in the exponent around $e^{-nS(\rho)}$ . Let $T(n,\epsilon)$ be the set of these sequences and $|T(n,\epsilon)|$ their number.^✦ The typical subspace of $\rho^{\otimes n}$ is the span of the $|e_x\rangle$ with $x \in T(n,\epsilon)$ . Its dimension is $|T(n,\epsilon)|$ , and $P(n,\epsilon)$ denotes the orthogonal projector onto it.

Measurements

A measurement on $Q$ with a finite set of outcomes is a family of measurement operators $M_i$ on $\mathcal{H}_Q$ , one for each outcome $i$ , that satisfies the completeness relation $\sum_i M_i^\dagger M_i = 1 .$ (3)^✦ The positive operator valued measure (POVM) of the measurement is the family of effects $E_i = M_i^\dagger M_i$ . Performed on a state $\rho$ , the measurement gives the outcome $i$ with the Born probability $p(i) = \operatorname{tr}(M_i \rho M_i^\dagger) = \operatorname{tr}(E_i \rho) .$ (4)^✦ A measurement $\{M_i\}$ is projective when every $M_i$ is an orthogonal projector, a Hermitian operator equal to its own square; then $E_i = M_i$ and $p(i) = \operatorname{tr}(M_i \rho)$ .^✦

The block measurement of an orthonormal basis $\{|b_\sigma\rangle\}$ of $\mathcal{H}_Q$ and a labelling that assigns an outcome to every index $\sigma$ is the measurement on $Q$ whose operator $P_m$ for the outcome $m$ is the orthogonal projector onto the span of the $|b_\sigma\rangle$ labelled $m$ .^✦ The typical subspace measurement on $Q^{\otimes n}$ is the block measurement of the eigenbasis (2) that labels $x$ yes when $x \in T(n,\epsilon)$ and no otherwise.^✦ Its operator for yes is $P(n,\epsilon)$ .

The yes subspace of a measurement on $Q^{\otimes n}$ with outcomes yes and no is the range of its operator $M_{\mathrm{yes}}$ .

The Claim

Theorem 1. Let $\rho$ be a state of a quantum system $Q$ , and let $\delta > 0$ and $\epsilon > 0$ .

(i) For all sufficiently large $n$ , the typical subspace measurement gives the outcome yes on $\rho^{\otimes n}$ with probability $\operatorname{tr}\bigl(P(n,\epsilon)\, \rho^{\otimes n}\bigr) \ge 1 - \delta .$ (5)^✦

(ii) For all sufficiently large $n$ , the number of typical sequences satisfies $(1 - \delta)\, e^{n(S(\rho) - \epsilon)} \le |T(n,\epsilon)| \le e^{n(S(\rho) + \epsilon)} .$ (6)^✦

(iii) Let $R < S(\rho)$ . For all sufficiently large $n$ , every projective measurement on $Q^{\otimes n}$ with outcomes yes and no whose yes subspace has dimension at most $e^{nR}$ gives the outcome yes on $\rho^{\otimes n}$ with probability $\operatorname{tr}\bigl(M_{\mathrm{yes}}\, \rho^{\otimes n}\bigr) \le \delta .$ (7)^✦

What Is Checked

The claim is stated over the state $\rho^{\otimes n}$ of the $n$ copies $Q^{\otimes n}$ , the entropy (1), the typical sequences of the spectrum of $\rho$ and their number $|T(n,\epsilon)|$ , the typical subspace measurement, and the projective measurements on $Q^{\otimes n}$ with two outcomes and the dimension of their yes subspace, every probability being the Born probability (4).

Of these constructions the library proves that $\rho^{\otimes n}$ is a state, its density operator positive and of unit trace because each factor is, and that the projectors of a block measurement satisfy the completeness relation (3), so that the typical subspace measurement is a measurement.