11
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
$$ \langle A,B\rangle=\mathrm{Tr}(A^\...
- 1,036
10
votes
Accepted
Inverting the depolarizing channel
The existence of the inverse of a linear map is independent of the way the map affects the trace. Moreover, if an invertible map preserves a property then its inverse necessarily also preserves the ...
- 18.2k
9
votes
Expected value of a Haar random quantum state multiplied by a unitary
I'm writing an alternate proof because it uses some interesting tools, computes the value of these expressions, and gives some insights into how we can interpret the quantities in consideration.
The ...
- 1,753
8
votes
Accepted
Closeness of purifications of states
No dimension-independent bound is possible.
Consider states $\rho_A$ and $\sigma_A$ that are close in $p$-norm (for $p>1$) but have relatively low fidelity. Specifically, assume
$$
\|\rho_A - \...
- 4,883
8
votes
Accepted
Why is the transpose of a density matrix positive and trace preserving?
Transposing a matrix is trace preserving since for $\rho = \sum_{a,b} \rho_{a,b} | a \rangle \langle b |$:
$$\text{Tr}(\rho)= \sum_c \langle c| \big( \sum_{a,b} \rho_{a,b} | a \rangle \langle b | \...
7
votes
Accepted
Does $\mathcal E^{\otimes n}$ admit a more efficient Stinespring dilation than the one used for $\mathcal E$?
No.
The minimal size of the environment is just the rank of the Choi matrix of $\mathcal E$, call it $J(\mathcal E)$. Since $J(\mathcal E^{\otimes n}) = \big(J(\mathcal E)\big)^{\otimes n}$ and $\text{...
- 2,196
7
votes
Accepted
How to show that Bell states are orthonormal
First, note that $|0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $|1\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ and therefore $\langle 0 |1\rangle = \begin{pmatrix} 1 & 0 \end{pmatrix} \...
- 13.1k
7
votes
Why is the transpose of a density matrix positive and trace preserving?
Trace preservation
The trace must be preserved in the transpose of a matrix, because the trace is the sum of the diagonal elements. When transposing a matrix, you do not change the diagonals at all! ...
- 12.7k
7
votes
In Stinespring dilation, can we always use a mixed state as the ancilla?
No, that doesn't work. It's fine to use an arbitrary pure state because the unitary $U$ can always be used to take it to any pure state you want. This argument doesn't work for a mixed state, as ...
- 2,196
7
votes
Accepted
Cliffordness of the qutrit Hadamard gate
The answer is no. Define
X=[[0,1,0],[0,0,1],[1,0,0]]
Z=[[1,0,0],[0,w,0],[0,0,w^2]], w^3=1
Then the Pauli group is generated by X and Z and is of order 27.
With H ...
- 1,485
7
votes
Accepted
How to express real matrices as linear combinations of unitaries?
You can select a basis of unitary matrices with respect to which you can decompose your matrix. For example, if your matrix $A$ is $2^n\times 2^n$, then you can select the Pauli basis
$$
\sigma_y,\...
- 51.1k
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 ...
- 18.2k
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 $I \otimes B$.
$$[A\otimes I,I\otimes B]=(A\otimes I) (I\...
- 2,224
7
votes
Accepted
Can we always simultaneously diagonalize $H_A \otimes \mathbb{1}$ and $\mathbb{1} \otimes H_B$?
TL;DR: You can always achieve simultaneous diagonalization of $H_A\otimes\mathbb{1}$ and $\mathbb{1}\otimes H_B$ even if $[H_A, H_B]\ne 0$. And yes, this does follow from the fact that $[H_A\otimes\...
- 18.2k
7
votes
Accepted
How to square a density matrix?
Squaring matrices works the same way as squaring numbers, i.e. you multiply the matrix by itself. Formally, $A^2=A\cdot A$. However, every density matrix $\rho$ is Hermitian, i.e. $\rho=\rho^\dagger$, ...
- 18.2k
6
votes
Accepted
How do I prove that $P_\pm=\frac12(1\pm U)$ if $U^2=I$?
First, we can start with $U = P_+ - P_-$, since the Hermitian is the sum of the projection operators of the eigenspaces scaled by their eigenvalues. If $U^2 = I$, that means $I = (P_+ - P_-)(P_+ - P_-)...
- 841
6
votes
Accepted
What is the matrix for a SWAP operation on two qubits?
In the general case I think it's easier to consider the matrix in the form
$$
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|,
$$
where the $i_1,...
- 4,516
6
votes
Accepted
Why can any density operator be written this way? (quantum tomography)
From linear algebra, if $v_1, \dots, v_n$ is a basis of the vector space $V$ then every vector $v\in V$ can be written as a linear combination
$$
v = a_1 v_1 + \dots + a_n v_n\tag1
$$
where the ...
- 18.2k
6
votes
Accepted
Does a basis of maximally entangled states exist for two-qubit or two-qutrit system so that the density matrices of the basis states don't commute?
No such (orthonormal) basis can exist. An orthonormal basis $\{|\psi_i\rangle\}$ requires $\langle \psi_i | \psi_j \rangle = 0$ for $i\neq j$, and so clearly
\begin{align}
[\rho_i, \rho_j] &= |\...
- 5,630
6
votes
Accepted
What does $ A - \langle A \rangle $ mean?
For any vector $v$ and scalar $\alpha$, we have $\alpha v = \alpha I v$, so multiplication by a scalar $\alpha$ behaves the same way as the linear operator $\alpha I$. Therefore, we interpret $A - \...
- 18.2k
5
votes
How to show that Bell states are orthonormal
The most basic but laborious way of checking that Bell states are orthonormal is to carry out the calculations for all sixteen inner products such as $\langle\Phi^+|\Psi^-\rangle$.
One way to do this ...
- 18.2k
5
votes
Accepted
Expected value of a Haar random quantum state multiplied by a unitary
With the chosen structure of $ U $, i think it's even possible to prove the stronger statement:
$$ \langle z| \rho|z \rangle = \langle z| \sigma_\rho|z \rangle, \hspace{0.2em} \text{where} \hspace{0....
- 1,366
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
$$
A=B=\begin{pmatrix}
1 &...
- 18.2k
5
votes
Accepted
Can we express $\mathrm{tr}_A((A\otimes B)\rho_{AB})$ in terms of $A$, $B$, $\rho_A$ and $\rho_B$?
In general, the knowledge of the marginals $\rho_A$ and $\rho_B$ and the operators $A$ and $B$ is insufficient to compute $\mathrm{tr}_A((A\otimes B)\rho_{AB})$. Indeed, we can find two different ...
- 18.2k
5
votes
Accepted
Prove that $|(\langle \psi|_{A} \otimes \langle \phi|_{B})|\theta\rangle_{AB}|^{2}<1$ for entangled $|\theta\rangle_{AB}$
TL;DR: There is no need to use Schmidt decomposition. The non-strict variant of the inequality follows directly from Cauchy-Schwarz inequality and equality is ruled out because $|\theta\rangle_{AB}$ ...
- 18.2k
5
votes
Accepted
What is the best way to write a tridiagonal matrix as a linear combination of Pauli matrices?
Summary: There is a solution for expressing a tridiagonal matrix of the form you've provided for arbitrary $n$ in terms of Pauli operators using recursion. This procedure is given at the bottom of ...
- 5,630
5
votes
Proof that the projector onto the symmetric subspace of the Swap $F$, with $n=2$, equals $\frac{1}{2}(I+F)$
I'm assuming $F$ is the Swap operator here, acting on some finite-dimensional space $H\otimes H$. In bra-ket notation, this reads $F\equiv \sum_{ij} |ij\rangle\!\langle ji|$.
Observe that if $\dim H=...
glS♦
- 21.3k
4
votes
Accepted
Depolarization of density operator with zeros in diagonal
Quantum channels are foremost, linear operators. So given a basis for the Hilbert-Schmidt operator space (for example the states $\{|0\rangle\langle 0|,|0\rangle\langle 1|,|1\rangle\langle 0|,|1\...
- 1,753
4
votes
Accepted
Diamond norm distance bound on Stinespring dilations of channels
Yes, in fact there exists Stinespring dilations such that
$$\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 ...
- 1,761
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