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.
Questions tagged [linear-algebra]
515 questions
10
votes
3 answers
Quantum tensor product closer to Kronecker product?
Coming more from a computer science background, I never really studied tensor products, covariant/contravariant tensors etc. So until now, I was seeing the "tensor product" operation mostly as (what appears to be) a Kronecker product between the…
Léo Colisson
- 656
- 4
- 12
4
votes
1 answer
How should I interpret $|2\rangle|3\rangle$?
I am a beginner at QC. I was going through the basics of multi-qubits I encountered a state $|2\rangle|3\rangle$.
I want clarification on the following points:
Can I write $|2\rangle$ as $|10\rangle = |1\rangle|0\rangle$ always?
If $|2\rangle =…
Adam Levine
- 451
- 2
- 10
4
votes
4 answers
What role does the non-commutativity of the tensor product play in experimental quantum computation?
We know that $H_A\otimes H_B\neq H_B\otimes H_A$ (in general). Theoretically, we know the formalism and what observables to construct from the two compositions possible, but we never talk about both the possibilities. I wish to know that how…
Siddhant Singh
- 1,745
- 8
- 21
4
votes
1 answer
Relating eigenvalues after summing over multiple unitary transformations
I am interested in the relation of the eigenvalues of two Hermitian operators $A$ and $B$ that are related via
$$A = \sum_j c_j U_j B U^\dagger_j.$$
Is there anything useful I can say about the spectrum? It looks suspiciously similar to the case of…
Korbinian
- 145
- 5
4
votes
1 answer
What do normalization term and partial measurement represent when tracing out ancillary qubits?
I am reading a paper and I am having trouble following some equations.
The system in this paper has $N$ qubits, with $N_A$ ancillary and the rest ($N - N_A$) as data qubits. For the purpose of this question, any subscript $t$ can be ignored.…
James Ellis
- 103
- 3
3
votes
1 answer
Solving linear systems over finite fields using Quantum Computation
I have a generalised approach to formulation of cryptanalysis of diverse problems as a problem of solving a linear system of equations Ax=b over a finite (most often the binary) field. The size of the linear system is of polynomial order (matrix…
Viren Sule
- 31
- 2
2
votes
1 answer
Derive one equation from the other
Equation 1.31 in Quantum Computation and Quantum Information a textbook by Isaac Chuang and Michael Nielsen is as follows,
$\left|\psi_2 \right> = \frac{1}{2}[\alpha(\left|0
\right>+\left|1
\right>)(\left|00\right>+\left|11\right>) +…
rranjik
- 123
- 3
2
votes
2 answers
Tensor product between operators
If the state of one qubit can be described by a ray in $\mathbb{C}^2$, then the combined state of an $n$-qubit system can be described by a ray in $(\mathbb{C}^2)^{\otimes n}=\mathbb{C}^{2 n}$.
However, if $G_1$ is the Pauli group of one qubit,…
Carucel
- 125
- 4
2
votes
0 answers
Decomposing an operator as a minimal sum of tensor products
Suppose we want to express an N-qubit operator as
$$U=\sum_i \lambda_i \bigotimes_{k=1}^N W_{i,k}$$
where $W_{i,k}$ are each a two-by-two matrix. How can one find a minimal decomposition, that is one that has as few terms in the sum as possible?
I…
user34722
- 421
- 1
- 5
2
votes
0 answers
Find supremum in an expression containing trace
I am working on a problem in which I need to find the supremum of an expression. Namely, the expression below:
$$
\sup_{w > 0}\Big\{\operatorname{tr}[ \rho \log w] - \log\operatorname{tr}[\sigma w ]\Big\}
$$
where $\rho$ is a density matrix…
Pegi
- 165
- 3
1
vote
1 answer
How to represent general isometries and unitaries:
I have the following exercise:
Let $V : H_A → H_A ⊗ H_E$ denote an isometry and $|ψ_E⟩ ∈ H_E$ a normalized
vector. Show that there exists a unitary $U : H_A ⊗ H_E → H_A ⊗ H_E$ such
that
$$U(1_{H_A} ⊗ |ψ_E⟩) = V.$$
My question is how do i choose to…
Pink Elephants
- 131
- 4
1
vote
1 answer
show that $\mathrm{tr}_A \left[\rho_A \lvert \phi^+ \rangle_{AB} \langle \phi^+ \rvert\right]$ equals to $\rho_B^T$
I have difficulties calculating with partial traces in terms of quantum operations.
For me it is not clear how to derive the equality stated in the question title for a quantum mechanical system whose state space is the tensor product $H_A \otimes…
gehbiszumeis
- 119
- 4
1
vote
1 answer
How doesn't combining two eigenvectors that have the same eigenvalue for a specific matrix represent every vector left in the plane?
If we have a 2D plane and the hermitian matrix $L$ where:
$$L|\lambda_1\rangle=\lambda|\lambda_1\rangle$$
$$L|\lambda_2\rangle=\lambda|\lambda_2\rangle$$
Given that $|\lambda_1\rangle$ and $|\lambda_2\rangle$ are linearly independent, we can make…
zizaaooo
- 83
- 5
1
vote
1 answer
Proof that tensor product of unit vectors is a unit vector
I am trying to find a simple proof that $\|v \otimes u\| = 1 $ if $\|v\|=1$ and $\|u\|=1$.
I have a proof by induction, where I can fix the length of $u$ and show by induction on the length of $v$ that the previous statement is true. The base case…
Lucie Nashwild
- 25
- 3
1
vote
2 answers
What's $(\langle 0|\otimes I)(|00\rangle + |11\rangle)$ simplified?
It's a rather simple question. I think I am confused by the fact that using $\langle0|$ on a qubit doesn't result in another qubit. So I'm not sure if I should interpret $\langle 0|$ as the $1\times2$ matrix $(1 \space 0)$.
kerf
- 35
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