The Exercise

A qubit undergoes a generalized measurement with the two measurement operators $|0\rangle\langle 0|$ and $|0\rangle\langle 1|$, and nobody reads the outcome. The exercise is to show that this measurement can lower the von Neumann entropy of the qubit.

A Quantum System

A quantum system $A$ is a physical object whose state space, its mathematical realisation, is a finite dimensional complex Hilbert space.^✦ Each finite dimensional complex Hilbert space $H$ gives a system, the system whose state space is $H$ itself.^✦ The qubit is the two level system given in this way by $\mathbb{C}^2$, the standard two dimensional complex space.^✦

A pure state $\psi$ of $A$ is a vector of unit norm, $\lVert \psi \rVert = 1$, in the state space of $A$, determined physically up to a global phase.^✦ The basis states $|0\rangle$ and $|1\rangle$ of the qubit are the pure states whose vectors are the standard basis vectors $e_0$ and $e_1$ of $\mathbb{C}^2$.^✦

A state $\rho$ of $A$ is a positive operator of unit trace on the state space of $A$; pure and mixed states are states alike.^✦ The von Neumann entropy of a state $\rho$ of $A$ is the real number $$S(\rho) = -\operatorname{tr}(\rho \ln \rho),$$(1)^✦ the Shannon entropy of the eigenvalues of $\rho$, measured in nats.

Measurement Without Reading the Outcome

A measurement on $A$ with a finite set of outcomes is a family of measurement operators $M_i$ on the state space of $A$, one for each outcome $i$, that satisfies the completeness relation $$\sum_i M_i^\dagger M_i = I.$$(2)^✦

For a measurement with operators $M_i$ and a state $\rho$ of $A$, the unread postmeasurement state is the state $\rho'$ of $A$ with density operator $$\rho' = \sum_i M_i \rho M_i^\dagger.$$(3)^✦ It describes the system after the measurement when the outcome is not learned.

The reset measurement is the measurement of the exercise, on the qubit, with outcomes $i = 0, 1$ and operators $$M_i = |0\rangle\langle i|.$$(4)^✦ Both operators map into $|0\rangle$. $$\rho' = |0\rangle\langle 0|\rho|0\rangle\langle 0| + |0\rangle\langle 1|\rho|1\rangle\langle 0| = \bigl(\langle 0|\rho|0\rangle + \langle 1|\rho|1\rangle\bigr)\,|0\rangle\langle 0| = |0\rangle\langle 0|.$$(5)

The Claim

What Is Checked

The claim is stated over the qubit and its states, the reset measurement (4) with its unread postmeasurement state (3), and the von Neumann entropy (1).

The library proves that the reset operators satisfy the completeness relation (2), $M_0^\dagger M_0 + M_1^\dagger M_1 = |0\rangle\langle 0| + |1\rangle\langle 1| = I$, and that the basis vectors $e_0$ and $e_1$ have unit norm. For any measurement and any state $\rho$, it proves that $\sum_i M_i \rho M_i^\dagger$ is again a state: the operator is positive because $\rho$ is, and its trace is $\operatorname{tr}\bigl(\sum_i M_i^\dagger M_i \rho\bigr) = \operatorname{tr} \rho = 1$ by completeness.