Questions tagged [schmidt-decomposition]
22 questions
7
votes
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Is it possible to derive a Schmidt decomposition for a mixed state?
It is relatively simple to derive the Schmidt decomposition of a pure state $|{\psi}\rangle \in H_A \otimes H_B$ with the SVD decomposition theorem. There are plenty of examples (lecture notes, books, videos, etc.) on the subject.
The question I…
JMark
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7
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1 answer
How to check if a $n$-qubit unitary is the tensor product of single-qubit unitaries
Let's assume I give you the expression of a unitary matrix acting on two qubits that is:
$$U=\sum_{i} A_i \otimes B_i$$
for some operators $A_i$ and $B_i$.
Is there a simple criterion allowing you to find out if it is actually simply…
Marco Fellous-Asiani
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6
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Problem 2.2 in Nielsen & Chuang - Properties of the Schmidt number
This question has been asked here: "Problem 2.2 in Nielsen & Chuang - Properties of the Schmidt number",
but no answer has been provided there yet, thus I move it here.
The problem is stated below. This is problem 2.2 (not exercise 2.2) in Nielsen &…
fagd
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6
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Schmidt decomposition for tripartite system $ABC$ with vanishing mutual information between $A$ and $C$
Suppose I have a tripartite system $ABC$ in a pure state $|\psi_{ABC}\rangle$ with mutual information $I(A:C)=0$. This implies that the reduced density matrix $\rho_{AC}$ factorizes as $\rho_{AC} = \rho_A \otimes \rho_C$.
How do I show that this…
nervxxx
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4
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Quantum teleportation and a maximally entangled state
I came across this question as part of my quantum mathematics exam preparation:
Consider the entangled state shared between Alice and Bob:
$|\psi^{AB}\rangle = \sqrt{p_1} |1\rangle^A |1\rangle^B + \cdots + \sqrt{p_n} |n\rangle^A |n\rangle^B$
where…
Amit Gabay
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4
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Is there a more efficient (pure-system bipartite) separability check than Qrack's?
Firstly, I am lead author and developer of unitaryfund/qrack, the open-source quantum computer simulator. We've had a method in the library, QInterface::TryDecompose(), since about 2018, whereby Qrack can check if a general pure quantum state…
Dan Strano
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4
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1 answer
Open neighborhood of an entangled state with non-decreasing Schmidt rank
Let $\psi\in H_A \otimes H_B$ be an entangled state, which means that it has Schmidt rank $r \geq 2$. Does there exist some $\epsilon>0$ for which all states $\varphi$ with $\|\psi - \varphi\|< \epsilon$ have Schmidt rank at least $r$? I think this…
QuantumHumanLearner
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3
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What's the Schmidt decomposition of $|\psi\rangle = 1/ \sqrt{3}( |0\rangle| 0\rangle + |0\rangle |1\rangle + |1\rangle |1\rangle)$?
$|\psi\rangle = 1/ \sqrt{3}( |0\rangle| 0\rangle + |0\rangle |1\rangle + |1\rangle |1\rangle) $
I absolutely cannot figure out the Schmidt decomposition of this state. I have looked at a ton of examples and have done the calculation multiple times,…
qityhd
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3
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A separable pure bipartite quantum state must be a product state
I'm looking for the simple argument to prove that a separable pure bipartite quantum state is in fact a product state. This question comes from a statement in Wikipedia on separable states: In the special case of pure states the definition…
JMark
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3
votes
1 answer
SWAPing Schmidt vectors
Can anything be said about the inner product of a bipartite entangled state with itself but with the Schmidt vectors swapped? That is, if the Schmidt decomposition of a state is given by
$$\vert \psi \rangle = \sum_{i=1}^d\sqrt{\lambda_i} \vert i_A…
Sergio Escobar
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3
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Truncating the bond dimension of an MPS -- how good is the approximation?
$\newcommand{\complex}{\mathbb{C}}\newcommand{\ket}[1]{|#1\rangle}$ Let $\ket{\psi}\in(\complex^d)^{\otimes n}$ be a pure quantum state. It is well-known that $\ket{\psi}$ is a matrix product state with bond dimension $r$ if and only if $\ket{\psi}$…
Ben
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2
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2 answers
How to calculate the Schmidt decomposition of a state without SVD
I have this state of two qubits here:
$$
|\psi_{AB}\rangle = \frac{1}{2}(|0\rangle_A |0\rangle_B + |1\rangle_A |1\rangle_B + |1\rangle_A |0\rangle_B - |0\rangle_A |1\rangle_B)
$$
Which means that the density matrix (with order $|00\rangle$,…
Alessandro Romancino
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2
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Does proximity of two bipartite states in a norm force high overlap between the elements of the Schmidt bases?
I want to know that there is a relation between the distance of two vectors and the corresponding elements of the Schmidt bases.
We assume that two bipartite vectors $|\phi\rangle^{AB}$ and $|\psi\rangle^{AB}$ can be decomposed such…
R. J. Ernest
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2
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1 answer
Prove that there are infinitely many two-qubit entanglement classes under LU
Dur, 2000 states that
(...)But even in the simplest systems, $|\psi\rangle$ and $|\phi\rangle$ are typically not related by LU, and continuous parameters are needed to label all equivalence classes.
I've found some similar explanation in…
Steve J.
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1
vote
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Why does it matter that Schmidt number is invariant under unitary transformations?
I am reading Nielsen & Chuang and they say this:
"The bases $|i_A\rangle$ and $|i_B\rangle$ are called the Schmidt bases for A and B, respectively, and the number of non-zero values $\lambda_i$ is called the Schmidt number for the state…
researcher101
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