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]
703 questions
53
votes
4 answers
How do I show that a two-qubit state is an entangled state?
The Bell state $|\Phi^{+}\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle )$ is an entangled state. But why is that the case? How do I mathematically prove that?
user72
36
votes
5 answers
How to derive the CNOT matrix for a 3-qubit system where the control & target qubits are not adjacent?
In a three-qubit system, it's easy to derive the CNOT operator when the control & target qubits are adjacent in significance - you just tensor the 2-bit CNOT operator with the identity matrix in the untouched qubit's position of…
ahelwer
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22
votes
1 answer
Is the Pauli group for $n$-qubits a basis for $\mathbb{C}^{2^n\times 2^n}$?
The $n$-fold Pauli operator set is defined as $G_n=\{I,X,Y,Z \}^{\otimes n}$, that is as the set containing all the possible tensor products between $n$ Pauli matrices. It is clear that the Pauli matrices form a basis for the $2\times 2$ complex…
Josu Etxezarreta Martinez
- 4,136
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18
votes
2 answers
Can arbitrary matrices be decomposed using the Pauli basis?
Is it possible to decompose a Hermitian and unitrary matrix $A$ into the sum of the Pauli matrix Kronecker products?
For example, I have a matrix 16x16 and want it to be decomposed into something like $$A…
C-Roux
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13
votes
1 answer
How does the number of copies affect the diamond distance?
Suppose we are given two maps $\Phi$ and $\Psi$ such that
$$\|\Phi-\Psi\|_{\diamond}\leqslant\varepsilon.$$
What can we say about $\left\|\Phi^{\otimes t}-\Psi^{\otimes t}\right\|_{\diamond}$? Is it upper-bounded by $t\varepsilon$? Is the result…
Tristan Nemoz
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12
votes
2 answers
What does it mean for a density matrix to "act on a Hilbert space $\mathcal{H}"$?
For a Hilbert space $\mathcal{H}_A$, I have seen the phrase
density matrices acting on $\mathcal{H}_A$
multiple times, e.g. here.
It is clear to me that if $\mathcal{H}_A$ has finite Hilbert dimension $n$, then this makes sense mathematically,…
Peter
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11
votes
3 answers
Is the tensor product of two states commutative?
I'm reading "Quantum Computing Expained" by David McMahon, and encountered a confusing concept.
At the beginning of Chapter 4, the author described the tensor product as below:
To construct a basis for the larger Hilbert space, we simply form…
akawarren
- 111
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10
votes
2 answers
Is there an algorithm for determining if a given vector is separable or entangled?
I'm trying to understand if there is some sort of formula or procedural way to determine if a vector is separable or entangled, i.e., whether or not a vector of size $mn$ could be represented by the tensor product of two vectors of size $m,n$,…
A Poor
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10
votes
1 answer
How can we be sure that for every $A$, $A^\dagger A$ has a positive square root?
In the Polar Decomposition section in Nielsen and Chuang (page 78 in the 2002 edition),
there is a claim that any matrix $A$ will have a decomposition $UJ$ where $J$ is positive and is equal to $\sqrt{A^\dagger A}$.
Firstly, how can we be sure that…
Mahathi Vempati
- 1,731
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10
votes
1 answer
How to keep track of entanglements when emulating quantum computation?
I am trying to build a quantum computation library as my university project. I am still learning all the aspects of the Quantum Computing field. I know there are efficient libraries already for quantum emulation. I just want to make my own, which…
Midhun XDA
- 305
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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
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9
votes
4 answers
How do I prove that the Hadamard satisfies $H\equiv e^{i\pi H/2}$?
I am trying to solve this exercise from Qiskit's textbook, problem set 2 "Basic Synthesis of Single-Qubit Gates":
Show that the Hadamard gate can be written in the following two forms
\begin{equation}
H = \frac{X + Z}{\sqrt{2}} \equiv \exp\left(i…
walid
- 335
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9
votes
1 answer
How are arbitrary $2\times 2$ matrices decomposed in the Pauli basis?
I read in this article (arXiv) Appendix III p.8, that for $A\in \mathcal{M}_2$, since the normalized Pauli matrices $\{I,X,Y,Z\}/\sqrt{2}$ form an orthogonal matrix basis.
$$A=\frac{Tr(AI)I+Tr(AX)X+Tr(AY)Y+Tr(AZ)Z}{2} $$
I don't understand, where…
lufydad
- 491
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9
votes
3 answers
Why is the state of multiple qubits given by their tensor product?
How did we derive that the state we get by $n$ qubits is their tensor product? You can use $n=2$ in the explanation for simplicity.
PiMan
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9
votes
0 answers
Is the Pauli norm of an observable minimized in its eigenbasis?
Consider an observable $O = \sum_i \lambda_i P_i$ decomposed into Paulistrings $P_i$ and a unitary $U$ each acting on $n$ qubits. The Pauli-norm of $O$ is defined as the 1-norm of the Pauli vector, i.e.
$$\|O\|_P = \sum_i |\lambda_i|.$$
I can write…
Refik Mansuroglu
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