10 votes
Accepted

How are arbitrary $2\times 2$ matrices decomposed in the Pauli basis?

The Pauli matrices form an orthogonal basis of $\mathcal{M}_2$, this vector space can be endowed with a scalar product called the Hilbert-Schmidt inner product $$ \langle A,B\rangle=\mathrm{Tr}(A^\...
  • 1,006
9 votes
Accepted

Inverting the depolarizing channel

The existence of the inverse of a linear map is independent of the way the map affects the trace. Moreover, if an invertible map preserves a property then its inverse necessarily also preserves the ...
  • 15.6k
8 votes
Accepted

Closeness of purifications of states

No dimension-independent bound is possible. Consider states $\rho_A$ and $\sigma_A$ that are close in $p$-norm (for $p>1$) but have relatively low fidelity. Specifically, assume $$ \|\rho_A - \...
  • 4,713
8 votes
Accepted

Why is the transpose of a density matrix positive and trace preserving?

Transposing a matrix is trace preserving since for $\rho = \sum_{a,b} \rho_{a,b} | a \rangle \langle b |$: $$\text{Tr}(\rho)= \sum_c \langle c| \big( \sum_{a,b} \rho_{a,b} | a \rangle \langle b | \...
7 votes
Accepted

Does $\mathcal E^{\otimes n}$ admit a more efficient Stinespring dilation than the one used for $\mathcal E$?

No. The minimal size of the environment is just the rank of the Choi matrix of $\mathcal E$, call it $J(\mathcal E)$. Since $J(\mathcal E^{\otimes n}) = \big(J(\mathcal E)\big)^{\otimes n}$ and $\text{...
7 votes

Expected value of a Haar random quantum state multiplied by a unitary

I'm writing an alternate proof because it uses some interesting tools, computes the value of these expressions, and gives some insights into how we can interpret the quantities in consideration. The ...
7 votes

Why is the transpose of a density matrix positive and trace preserving?

Trace preservation The trace must be preserved in the transpose of a matrix, because the trace is the sum of the diagonal elements. When transposing a matrix, you do not change the diagonals at all! ...
  • 12.2k
7 votes

In Stinespring dilation, can we always use a mixed state as the ancilla?

No, that doesn't work. It's fine to use an arbitrary pure state because the unitary $U$ can always be used to take it to any pure state you want. This argument doesn't work for a mixed state, as ...
7 votes
Accepted

How to express real matrices as linear combinations of unitaries?

You can select a basis of unitary matrices with respect to which you can decompose your matrix. For example, if your matrix $A$ is $2^n\times 2^n$, then you can select the Pauli basis $$ \sigma_y,\...
  • 49.4k
7 votes
Accepted

Does the controlled Pauli Z gate cause entanglement?

Yes, CZ gate can entangle its inputs. For example $$ CZ|+\rangle|+\rangle = \frac{|0\rangle|+\rangle+|1\rangle|-\rangle}{\sqrt2} $$ which is entangled because the reduced density matrix of either ...
  • 15.6k
7 votes
Accepted

Do the linear operators $M\otimes I$ and $I\otimes N$ commute?

Let $A,B$ be any two bounded operators acting over a Hilbert space and $I$ the identity operator. Then, $A\otimes I$ always commutes with $I \otimes B$. $$[A\otimes I,I\otimes B]=(A\otimes I) (I\...
  • 2,112
6 votes
Accepted

How to show that Bell states are orthonormal

First, note that $|0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $|1\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ and therefore $\langle 0 |1\rangle = \begin{pmatrix} 1 & 0 \end{pmatrix} \...
  • 12.9k
6 votes
Accepted

How do I prove that $P_\pm=\frac12(1\pm U)$ if $U^2=I$?

First, we can start with $U = P_+ - P_-$, since the Hermitian is the sum of the projection operators of the eigenspaces scaled by their eigenvalues. If $U^2 = I$, that means $I = (P_+ - P_-)(P_+ - P_-)...
6 votes
Accepted

What is the matrix for a SWAP operation on two qubits?

In the general case I think it's easier to consider the matrix in the form $$ M = \sum_{i_1,\dots,i_n, j_1, \dots j_n} c_{i_1,\dots,j_n} |i_1 \dots i_n\rangle \langle j_1 \dots j_n|, $$ where the $i_1,...
  • 4,271
6 votes
Accepted

Why can any density operator be written this way? (quantum tomography)

From linear algebra, if $v_1, \dots, v_n$ is a basis of the vector space $V$ then every vector $v\in V$ can be written as a linear combination $$ v = a_1 v_1 + \dots + a_n v_n\tag1 $$ where the ...
  • 15.6k
6 votes
Accepted

Does a basis of maximally entangled states exist for two-qubit or two-qutrit system so that the density matrices of the basis states don't commute?

No such (orthonormal) basis can exist. An orthonormal basis $\{|\psi_i\rangle\}$ requires $\langle \psi_i | \psi_j \rangle = 0$ for $i\neq j$, and so clearly \begin{align} [\rho_i, \rho_j] &= |\...
  • 5,175
6 votes
Accepted

Cliffordness of the qutrit Hadamard gate

The answer is no. Define X=[[0,1,0],[0,0,1],[1,0,0]] Z=[[1,0,0],[0,w,0],[0,0,w^2]], w^3=1 Then the Pauli group is generated by X and Z and is of order 27. With H ...
  • 1,400
6 votes
Accepted

What does $ A - \langle A \rangle $ mean?

For any vector $v$ and scalar $\alpha$, we have $\alpha v = \alpha I v$, so multiplication by a scalar $\alpha$ behaves the same way as the linear operator $\alpha I$. Therefore, we interpret $A - \...
  • 15.6k
5 votes

How to show that Bell states are orthonormal

The most basic but laborious way of checking that Bell states are orthonormal is to carry out the calculations for all sixteen inner products such as $\langle\Phi^+|\Psi^-\rangle$. One way to do this ...
  • 15.6k
5 votes
Accepted

Is there an identity for the partial transpose of a product of operators?

Your suspicion is correct, even when $A=B$. Consider the Hilbert space of two qubits and let $^{T_A}$ denote the partial transpose with respect to one of them. Suppose that $$ A=B=\begin{pmatrix} 1 &...
  • 15.6k
5 votes
Accepted

Can we express $\mathrm{tr}_A((A\otimes B)\rho_{AB})$ in terms of $A$, $B$, $\rho_A$ and $\rho_B$?

In general, the knowledge of the marginals $\rho_A$ and $\rho_B$ and the operators $A$ and $B$ is insufficient to compute $\mathrm{tr}_A((A\otimes B)\rho_{AB})$. Indeed, we can find two different ...
  • 15.6k
5 votes
Accepted

Prove that $|(\langle \psi|_{A} \otimes \langle \phi|_{B})|\theta\rangle_{AB}|^{2}<1$ for entangled $|\theta\rangle_{AB}$

TL;DR: There is no need to use Schmidt decomposition. The non-strict variant of the inequality follows directly from Cauchy-Schwarz inequality and equality is ruled out because $|\theta\rangle_{AB}$ ...
  • 15.6k
5 votes
Accepted

What is the best way to write a tridiagonal matrix as a linear combination of Pauli matrices?

Summary: There is a solution for expressing a tridiagonal matrix of the form you've provided for arbitrary $n$ in terms of Pauli operators using recursion. This procedure is given at the bottom of ...
  • 5,175
5 votes

Proof that the projector onto the symmetric subspace of the Swap $F$, with $n=2$, equals $\frac{1}{2}(I+F)$

I'm assuming $F$ is the Swap operator here, acting on some finite-dimensional space $H\otimes H$. In bra-ket notation, this reads $F\equiv \sum_{ij} |ij\rangle\!\langle ji|$. Observe that if $\dim H=...
  • 20.5k
4 votes
Accepted

Expected value of a Haar random quantum state multiplied by a unitary

With the chosen structure of $ U $, i think it's even possible to prove the stronger statement: $$ \langle z| \rho|z \rangle = \langle z| \sigma_\rho|z \rangle, \hspace{0.2em} \text{where} \hspace{0....
  • 1,336
4 votes
Accepted

Depolarization of density operator with zeros in diagonal

Quantum channels are foremost, linear operators. So given a basis for the Hilbert-Schmidt operator space (for example the states $\{|0\rangle\langle 0|,|0\rangle\langle 1|,|1\rangle\langle 0|,|1\...
4 votes
Accepted

Diamond norm distance bound on Stinespring dilations of channels

Yes, in fact there exists Stinespring dilations such that $$\frac{\|N_1-N_2\|_{cb}}{\sqrt{\|N_1\|_{cb}}+\sqrt{\|N_2\|_{cb}}}\leq \|V_1-V_2\|\leq \sqrt{\|N_1-N_2\|_{cb}}$$ where the distance between ...
  • 1,617
4 votes

How to show that Bell states are orthonormal

To avoid long computation of all pair-wise inner products in Bell basis, you can think as follow. Switching from computational basis to the Bell basis is done by transformation $$ CNOT(H \otimes I) $$ ...
4 votes
Accepted

Conjugation of $R_x(\theta)$ with $CNOT$

This is an application of the following identity $$ Be^AB^{-1} = e^{BAB^{-1}}\tag1 $$ where $A$ is any $n\times n$ real or complex matrix and $B$ is any invertible $n\times n$ real or complex matrix. ...
  • 15.6k

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