The Exercise

Two states of a system with four levels are given by their spectra. The first, $\rho_\psi$, has eigenvalues $0.4, 0.4, 0.1, 0.1$; the second, $\rho_\varphi$, has eigenvalues $0.5, 0.25, 0.25, 0$. Can local operations and classical communication (LOCC) turn $\rho_\psi$ into $\rho_\varphi$? Now borrow a qubit in the state $\rho_c$ with eigenvalues $0.6, 0.4$ and ask the same of the pair: can $\rho_\psi \otimes \rho_c$ be turned into $\rho_\varphi \otimes \rho_c$, the qubit handed back in the state it came in? The exercise is that the first answer is no and the second yes. The qubit is a catalyst: it is not consumed, and without it the conversion does not happen.

Systems and States

A quantum system $S$ is a physical object whose mathematical realisation is its state space, a finite dimensional complex Hilbert space.^✦ The system modelled on a finite dimensional complex Hilbert space $H$ is the system whose state space is $H$.^✦ The qudit with $d$ levels is the system modelled on $\mathbb{C}^d$, and its computational basis $\{|i\rangle\}$ is the standard basis of $\mathbb{C}^d$; the qubit is the qudit with two levels.

A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$.^✦ The composite system $S \otimes T$ of two systems $S$ and $T$ is the system whose state space is the tensor product of the state spaces of $S$ and $T$.^✦ The product state $\rho \otimes \sigma$ of a state $\rho$ of $S$ and a state $\sigma$ of $T$ is the state of $S \otimes T$ whose density operator is the tensor product of the density operators of $\rho$ and $\sigma$, acting as $\rho$ on the first factor and as $\sigma$ on the second.^✦

The Three States

The spectra of the three states are $$\lambda_\psi = (0.4,\, 0.4,\, 0.1,\, 0.1), \qquad \lambda_\varphi = (0.5,\, 0.25,\, 0.25,\, 0), \qquad \lambda_c = (0.6,\, 0.4),$$(1) each a list of nonnegative numbers that sum to one. The start state $\rho_\psi$ and the target state $\rho_\varphi$ are the states of the qudit with four levels diagonal in its computational basis with entries $\lambda_\psi$ and $\lambda_\varphi$, $$\rho_\psi = \sum_{i=0}^{3} \lambda_{\psi,i}\, |i\rangle\langle i| , \qquad \rho_\varphi = \sum_{i=0}^{3} \lambda_{\varphi,i}\, |i\rangle\langle i| ,$$(2) and the catalyst $\rho_c$ is the state of the qubit diagonal in its computational basis with entries $\lambda_c$, $$\rho_c = 0.6\, |0\rangle\langle 0| + 0.4\, |1\rangle\langle 1| .$$(3) With the catalyst, the pair lives on the composite of the qudit and the qubit, in the product states $\rho_\psi \otimes \rho_c$ and $\rho_\varphi \otimes \rho_c$.

Conversion by LOCC

A measurement on $S$ with a finite set of outcomes is a family of operators $M_j$ on the state space of $S$, one for each outcome $j$, satisfying the completeness relation $$\sum_j M_j^\dagger M_j = 1 .$$(4)^✦ A state $\rho$ of $S$ is convertible by LOCC to a state $\sigma$ of $S$, written $\rho \rightsquigarrow \sigma$, when a single measurement on $S$ turns $\rho$ into $\sigma$ whatever its outcome: there is a measurement $M_j$ on $S$ and a probability distribution $p_j \ge 0$, $\sum_j p_j = 1$, on its outcomes with $$M_j\, \rho\, M_j^\dagger = p_j\, \sigma \qquad \text{for every outcome } j .$$(5)^✦

The Claim

What Is Checked

The claim is stated over the start state $\rho_\psi$ and the target state $\rho_\varphi$ of the qudit with four levels, the catalyst $\rho_c$ of the qubit, the product states $\rho_\psi \otimes \rho_c$ and $\rho_\varphi \otimes \rho_c$ of their composite, and the relation $\rho \rightsquigarrow \sigma$ of (5), a single measurement with a probability distribution on its outcomes.

The operator diagonal in the computational basis with entries from a spectrum in (1) is positive with unit trace. The tensor product of two density operators is again positive with trace one, so the product of two states is a state of the composite system.