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Newest 'linear-algebra' Questions - Quantum Computing Stack Exchange
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Learn more about TeamsQuestions tagged [linear-algebra]
For questions about vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc.
722 questions
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0
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0
answers
36
views
Monotonicity of a function of eigenvalues from tensor product of diagonal density matrices
Let
$$
\rho = \begin{pmatrix} \cos^2 \theta & 0 \\ 0 & \sin^2 \theta \end{pmatrix}
$$
be a single-qubit diagonal density matrix, and consider the $n$-qubit product state
$$
\rho^{\otimes n}.
$$...
6
votes
3
answers
195
views
When is $UAU^\dagger B\ge0$ for all $U$?
Let $A$ and $B$ be such that for all unitaries $U$, $UAU^\dagger B$ is positive semidefinite.
Does this imply that either $A$ or $B$ must be proportional to the identity?
(Motivated by the question ...
4
votes
1
answer
243
views
Complexity of finding a fixed-point of a unitary / eigenstate with eigenvalue 1?
Recently, while working on a problem I've been trying to solve I came across a formulation that requires me to find $|\phi\rangle$, such that
$$U|\phi\rangle = |\phi\rangle$$
Of course this ...
2
votes
0
answers
88
views
Simplifiying trace of Paulis
Let $\mathcal{P}$ denote all Pauli operators (Heisenberg-Weyl operators in general). I want to see that
$\frac{1}{d^4} \sum_{A_i, B_i \in \mathcal{P}} tr(A_1 U^\dagger B_1 U A_2 U^\dagger B_2 U \...
-1
votes
1
answer
94
views
What is ${\rm tr}((A \otimes B) \text{SWAP} (C \otimes D )\text{SWAP})$ equal to?
The SWAP operator in the computational basis is given by $\sum_{ij} |i \rangle \langle j | \otimes|j \rangle \langle i| $.
It satisfies for any operators $A,B, \; tr((A \otimes B) \text{SWAP}) = tr(...
0
votes
0
answers
45
views
Equivalence of projective measurement with general measurement
Nielsen and Chuang in page 91 say a very important thing: "projective measurements augmented by unitary operations turn out to be completely equivalent to general measurements, as shown in ...
1
vote
0
answers
19
views
How can I generate a sample of a unitary 1-design defined via Pauli twirling and a change of basis in Python?
I am working with a linear map $V: A \to B$, and I want to generate a sample of a unitary 1-design over the Hilbert space $A$ in Python. Specifically, the 1-design is defined as:
$$
\chi_0 = \left\{ Q ...
1
vote
0
answers
46
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Performing a Partial Trace on Arbitrary Subsystems with Python/QuTiP/Cirq
I have a composite quantum system with density matrix rho represented as a NumPy array of shape $(D,D)$, where $D = \prod_i d_i$ is the product of the dimensions of ...
0
votes
1
answer
43
views
Why are all singular values of an isometry equal to 1, and what does this mean for stabilizer code encodings?
In the context of quantum error-correcting codes, particularly stabilizer codes, the encoding map $V$ : $H_L \rightarrow H_P$ is often described as an isometry, meaning $V^{\dagger} V=I$.
I'm trying ...
1
vote
0
answers
46
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How many independent stabilizers are in a hypergraph product code?
I am trying to understand why (3) of this paper holds.
Given a classical code with parameters $[n_i, k_i, d_i]$ and parity check matrix $H_i$, one can also define the dual code by looking at the ...
1
vote
1
answer
87
views
Understanding Quantum Singular Value Transformation from Gilyen et al, 2018
I try to understand the first pages of Gilyen et al, 2018 article introducing the Quantum Singular Value Transformation.
Here is a snapshot of the paragraph I have difficulties with :
This paragraph ...
4
votes
2
answers
219
views
Looking for a proof of a trace identity and (possibly) relationship to Kraus operators
$\DeclareMathOperator\PSU{PSU}\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\O{O}\DeclareMathOperator\SO{SO}\DeclareMathOperator\GL{GL}\DeclareMathOperator\ASL{ASL}\...
2
votes
3
answers
254
views
Do the coefficients of the dual weight enumerator polynomial add up to $2^{n+k}$ for any $[\![n,k]\!]$ code?
Consider an $ [\![n,k]\!] $ quantum error correcting code encoding $ k $ logical qubits into $ n $ physical qubits. Define the dual weight enumerator polynomial $ B(x) $ of the code as
$$
B(x):=B_0+...
4
votes
2
answers
182
views
Maximum overlap between an entangled state and a product state
Say we have an entangled state $\left| \phi \right\rangle_{AB}$ and a product state $ \left| \psi \right\rangle_A \otimes \left| \psi \right\rangle_B$ , how can we show that their maximum overlap is ...
0
votes
1
answer
54
views
Eigenvalues of the maximally entangled state
How can one derive the eigenvalues of the maximally entangled state
$ \frac{1}{d}\sum_{ij} |ii \rangle \langle jj| $.
$\sum_{ij} |ii \rangle \langle jj| = \sum_{i} |ii \rangle \langle ii| + \sum_{i\...
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