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Questions tagged [schmidt-decomposition]

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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 $\...
researcher101's user avatar
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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 ...
Alessandro Romancino's user avatar
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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 ...
qityhd's user avatar
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How to write Schmidt decomposition for pure tripartite state?

Supposing a pure tripartite state, i wrote schmidt decomposition as follows $\sum_{}^{r}p_{ijk}|i\rangle|j\rangle|k\rangle$ Did i write it correctly? Like pure bipartite states If r>1 then the ...
reza's user avatar
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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 ...
JMark's user avatar
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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, ...
JMark's user avatar
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is the main purpose of using Schmidt decompositon to visualize entanglement?

Is it correct to say that an entanglement state is entangled in any basis, but this fact may not be evident (like in the state $ \frac{1}{ 2} | 00 \rangle + \frac{1}{ 2}| 01 \rangle + \frac{1}{ 2}| 10 ...
Hadamard's user avatar
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Can we use a Werner state for quantum teleportation? [duplicate]

Some background: The quantum teleportation protocol requires first that Alice and Bob share an entangled state, say a Bell state $|\psi^{+}\rangle_{AB}$. There is another state $|\psi\rangle_{A'}$ to ...
Physkid's user avatar
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schmidt coefficients are the square root of the eigenvalue of the two partial trace of a density matrix

Let $\psi\rangle_{AB} = \sum_{i=1}^{d}\lambda_{i}|i_{A}\rangle |i_{B}\rangle$ be a state vector of a pure bipartite syste. Now, $\rho_{AB} = |\psi\rangle\langle\psi| = \sum_{i=1}^{d} \lambda_{i}^{2}|...
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General Bell state expression: What condition for mixture of Bell states to be entangled?

Convention: $|qubit_{A}, qubit_{B}\rangle$ The general Bell state equation: $|\beta(a,b)\rangle = \frac{1}{\sqrt{2}}\sum_{k=0}^{1}(-1)^{ka}|k, k\oplus b\rangle = \frac{1}{\sqrt{2}}[|0,0 \oplus b\...
Physkid's user avatar
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Why does Schmidt decomposition (2 qubits) requires density matrix of each system?

This is in reference to Nielson & Chang (page 109). Schmidt decomposition: suppose $|\psi\rangle$ is a pure state of a composite syste, AB. Then there exists orthonormal state $|i_{A}\rangle$ for ...
Physkid's user avatar
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4 votes
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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\|< \...
QuantumHumanLearner's user avatar
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For tetrapartite state, and another way of decomposition, is the Schmidt basis separable?

Consider two tetrapartite quantum states $|\phi\rangle^{AA^\prime BB^\prime}$ and $|\psi_1\rangle^{AA^\prime}|\psi_2\rangle^{BB^\prime}$ in a finite dimentional Hilbert space $\mathcal{H}^A\otimes\...
Takimoto.R's user avatar
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2 answers
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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\...
Takimoto.R's user avatar
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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. ...
Steve J.'s user avatar
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3 votes
1 answer
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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 \...
SescoMath's user avatar
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1 answer
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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 ...
Ben's user avatar
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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 ...
Marco Fellous-Asiani's user avatar
6 votes
1 answer
407 views

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} = \...
nervxxx's user avatar
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