Conservative Quantum Logic Gate

Question

In "Quantum Computation and Quantum Information," by Nielsen & Chuang, pg 156, a conservative logic gate is defined as one which preserves the number of zeros and ones. This definition makes sense for bits with definite 1's and 0's, but how does this definition extend to qubits?

For instance, consider the unitary operator $$ U = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 2^{-1/2} & 2^{-1/2} & 0 \\ 0 & 2^{-1/2} & 2^{-1/2} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$ This gate takes a state $|01\rangle$ to a balanced superposition of $|01\rangle$ and $|10\rangle$, has it "conserved the number of 0's and 1's? (I think so.)

What about a gate that takes $|0011\rangle$ to a balanced superposition of $|0111\rangle$ and $|0001\rangle$?

EDIT: The question is really in the last line. Is a balanced superposition of $|0111\rangle$ and $|0001\rangle$ really "conserving" the number of 1's from $|0011\rangle$. While it commutes with the counting operator, measurement will definitely give a different number of 1's! So is that still called conservation?

Answer 1

"The number of zeros" and "the number of ones" are, respectively, the matrices $$ N_0=\begin{pmatrix}2&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&0 \end{pmatrix} \quad\text{and}\quad N_1=\begin{pmatrix}0&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&2 \end{pmatrix}, $$ in the $\{|00\rangle, |01\rangle, |10\rangle, |11\rangle \}$ basis, and they add to a multiple of the identity, $N_0+N_1=2=2\,\mathbb I$. Saying that a unitary $U$ "conserves the number of zeros" (or equivalently the number of ones) is, by definition, saying that it commutes with $N_0$ (and equivalently with $N_1$), and therefore that it takes any eigenvector of $N_0$ and $N_1$ (itself necessarily of the form $|00\rangle$, $a|01\rangle+b|10\rangle$, or $|11\rangle$) to another eigenvector of the same eigenvalue. The same is true, with a transparent generalization, to the tensor product of any number of qubits.

As such, if a unitary takes $|0011\rangle$ to a superposition of $|0111\rangle$ and $|0001\rangle$ then, regardless of how even-weighted the superposition, is not considered to conserve $N_1$, because it takes vectors from one eigenspace and puts them outside that eigenspace, and therefore it cannot commute with $N_1$.