### Exercise 3.

- The Bell states \(| \Phi^+_{AB} \rangle\), \(| \Phi^-_{AB} \rangle\), \(| \Psi^+_{AB} \rangle\), \(| \Psi^-_{AB} \rangle\), can be characterized by two classical bits, namely the parity bit and the phase bit. Show that the latter are eigenvalues of two commuting observables.
- The Bell states form an orthonormal basis of the two-qubit Hilbert space. It is referred to as the Bell basis. Let us denote it by \(B_1\). A sequence of two operations can be used to convert states of the computational basis \(B_2 := \{| ij \rangle: i,j\in \{0,1\}\}\) to the Bell states. State what these operations are. Can they also be used to convert states of \(B_1\) to \(B_2\)? Justify your answer.
- Prove that the Schmidt rank of a pure state
*cannot*be increased by local operations and classical communication (LOCC), clearly stating any theorem that you use. - It is known that a matrix \(A\) is doubly stochastic if and only if \(\vec x \prec \vec y\) for all vectors \(\vec y\), where \(x = A \vec y\).

Let \(\rho\in \mathcal{D}(\mathcal{H})\) be a state, where \(\dim \mathcal{H}=d\), and let \(\Lambda:\mathcal{D}(\mathcal{H})\to \mathcal{D}(\mathcal{H})\) be a*unital*channel. Let \(\vec r = (r_1,r_2,\dotsc,r_d)\) and \(\vec s = (s_1,s_2,\dotsc,s_d)\) respectively denote the vectors of eigenvalues of \(\rho\) and \(\sigma = \Lambda(\rho)\), arranged in non-increasing order. Using the above result, prove that \(\vec s\prec \vec r\).*Note:*the question originally said \(\vec r\prec \vec s\), which was an error.

The parity bit is \(0\) if it is in a \(| \Phi_{AB}^\pm \rangle\) state, and \(1\) if its in a \(| \Psi^\pm_{AB} \rangle\) state. The phase bit is \(0\) if its in a \(| \alpha^+ \rangle\) state (for \(\alpha \in \{\Phi,\Psi\}\)), and \(1\) if it’s in a \(| \alpha^- \rangle\) state. Let \(X_{AB} = \sigma_x^{(A)}\otimes \sigma_x^{(B)}\) where \(\sigma_x\) is the Pauli \(x\) matrix, and likewise \(Z_{AB} = \sigma_z^{(A)} \otimes \sigma_z^{(B)}\). Then one can compute \([X_{AB},Z_{AB}]= 0\)

Alternatively, one can consider the projections \[ P_{\text{parity}} = | \Psi^+_{AB} \rangle\langle \Psi^+_{AB} |+| \Psi^-_{AB} \rangle\langle \Psi^-_{AB} | \] and \[ P_\text{phase} = | \Phi_{AB}^- \rangle\langle \Phi_{AB}^- | + | \Psi_{AB}^- \rangle\langle \Psi_{AB}^- |. \] Then the eigenvalues of these operators are exactly the phase and parity bit, in the sense that \[ P_{\text{parity}} | \Psi^\pm_{AB} \rangle = | \Psi^\pm_{AB} \rangle, \qquad P_{\text{parity}} | \Phi_{AB}^\pm \rangle = 0, \] and for \(\alpha\in\{\Phi,\Psi\}\), \[ P_{\text{phase}} | \alpha^- \rangle = | \alpha^- \rangle, \qquad P_{\text{phase}} | \alpha^+ \rangle = 0. \] Moreover these projections commute.^{1}. Moreover, \[\begin{aligned} X_{AB} | \alpha^+_{AB} \rangle &= | \alpha^+_{AB} \rangle \\ X_{AB} | \alpha^-_{AB} \rangle &= -| \alpha^-_{AB} \rangle \end{aligned}\] and thus the eigenvalues of \(X_{AB}\) are the phase bit. Likewise, \[\begin{aligned} Z_{AB} | \Phi^{\pm}_{AB} \rangle &= | \Phi^{\pm}_{AB} \rangle\\ Z_{AB} | \Psi^{\pm}_{AB} \rangle &= - | \Psi^{\pm}_{AB} \rangle \end{aligned}\] so the eigenvalues of \(Z_{AB}\) are the parity bit.- Hadamard then CNOT. It is reversible.
- Nielsen’s Majorization Theorem states that a bipartite state \(| \psi_{AB} \rangle\) can be converted to \(| \phi_{AB} \rangle\) if and only if \[ \lambda_\psi \prec \lambda_\phi \] where \(\lambda_\psi\) and \(\lambda_\phi\) are the vectors of eigenvalues of \(\psi_A := \operatorname{tr}_B | \psi_{AB} \rangle\langle \psi_{AB} |\) and \(\phi_A = \operatorname{tr}_B | \phi_{AB} \rangle\langle \phi_{AB} |\) and \(\prec\) is the majorization pre-order. Let \(n(\psi)\) and \(n(\phi)\) denote the Schmidt ranks of two pure states \(| \psi_{AB} \rangle\) and \(| \phi \rangle_{AB}\), such that \(| \psi \rangle_{AB} \xrightarrow{\text{LOCC}} | \phi_{AB} \rangle\). Assume \[ n(\psi) < n(\phi). \](1) Let \(\lambda_\psi = (\nu_1,\dotsc,\nu_d)\) and \(\lambda_\phi = (\mu_1,\dotsc,\mu_d)\), where \(d = \dim \mathcal{H}_A\) is the dimension of the Hilbert space for system \(A\). The assumption (1) implies that there exists some integer \(m\leq d\) such that \(\mu_m \neq 0\) but \(\nu_m = 0\). Hence \[ \sum_{i=1}^{m-1}\nu_i = 1, \qquad\text{but}\qquad \sum_{i=1}^{m-1} \mu_i < 1. \] This contradicts that \(\lambda_\psi \prec \lambda_\phi\). Therefore, (1) cannot hold, and therefore the Schmidt rank cannot increase by LOCC operations.
- Let us write the eigendecompositions \[\begin{aligned} \rho &= \sum_{i=1}^d r_i | e_i \rangle\langle e_i |\\ \sigma &= \sum_{i=1}^d s_j | f_j \rangle\langle f_j | \end{aligned}\] where the eigenvalues are arranged in non-increasing order. We have that \[\begin{aligned} s_j &= \operatorname{tr}[ \sigma | f_j \rangle\langle f_j |]\\ &= \operatorname{tr}[\Lambda(\rho) | f_j \rangle\langle f_j |] &= \sum_i r_i \operatorname{tr}[\Lambda(| e_i \rangle\langle e_i |)| f_j \rangle\langle f_j |]. \end{aligned}\] Now define a matrix \(D\) with entries \(D_{ji} = \operatorname{tr}[\Lambda(| e_i \rangle\langle e_i |)| f_j \rangle\langle f_j |]\). Then we’ve shown \[ s_j = \sum_i D_{ji} r_i. \] Thus, it remains to show that \(D\) is doubly-stochastic, by the result quoted in the question. We have \[\begin{aligned} \sum_j D_{ji} &= \sum_j \operatorname{tr}[\Lambda(| e_i \rangle\langle e_i |)| f_j \rangle\langle f_j |] \\ &= \operatorname{tr}[\Lambda(| e_i \rangle\langle e_i |) I] = \operatorname{tr}[\Lambda(| e_i \rangle\langle e_i |)] = 1 \end{aligned}\] since \(\Lambda\) is trace-preserving. Next, \[\begin{aligned} \sum_i D_{ji} &= \sum_i \operatorname{tr}[\Lambda(| e_i \rangle\langle e_i |)| f_j \rangle\langle f_j |] \\ &= \operatorname{tr}[\Lambda( I)| f_j \rangle\langle f_j |] = \operatorname{tr}[| f_j \rangle\langle f_j |] = 1 \end{aligned}\] since \(\Lambda\) is unital. Therefore, \(D\) is doubly stochastic. Since \(\vec s= D \vec r\), we have \(\vec s \prec \vec r\).

Takes a little time and expansion; this question was from an earlier iteration of the class where this was treated.↩