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8 votes
Accepted

Is there a CPTP map that takes $\rho_{AB}$ to $\rho_A\otimes\rho_B$?

No, because such a map would be non-linear. You want to perform the operation $$ \rho_{AB} \mapsto \operatorname{tr}_B(\rho_{AB}) \otimes \operatorname{tr}_A(\rho_{AB}). $$ Take for example $\rho_{AB} ...
Mateus Araújo's user avatar
6 votes
Accepted

How is the surface of a Bloch sphere a Hilbert space?

The surface of a Bloch sphere is not a Hilbert space. Maybe they meant to write that it's a valid projective Hilbert space (in particular it's isomorphic to $\mathbb{CP}^1$)? It's not a vector space, ...
glS's user avatar
  • 25.4k
5 votes

What is the actual Hilbert space of a $N$-qubit system?

First off, the Bloch sphere is the complex projective line $\mathbb{C}P^1$, which is homeomorphic to $S^2$, while $SU(2)$ is homeomorphic to $S^3$. $SU(2^N)$ is the group of operators on pure states, ...
Cody Wang's user avatar
  • 1,223
5 votes
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What is the actual Hilbert space of a $N$-qubit system?

Just a small remark for part of the question: Letting two Hilbert spaces $\mathcal{H}_1$ and $\mathcal{H}_2$ (this can be generalized to any linear space) the tensor product $\mathcal{H}_1 \otimes \...
R.W's user avatar
  • 2,347
5 votes

Does Neumark's/Naimark's extension theorem only apply to rank-1 POVMs?

You can assume without loss of generality that the POVM's are rank one because $\sum_i A_i=I$, so it's not necessary just more convenient. The enlargement of the space in the Naimark dilation theorem ...
Condo's user avatar
  • 2,048
4 votes
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Does proximity of two bipartite states in a norm force high overlap between the elements of the Schmidt bases?

No. Here is an example without small Schmidt coefficients. To this end, consider $$ \lvert\phi\rangle = a\lvert0\rangle\lvert0\rangle + b \lvert1\rangle\lvert1\rangle\ , $$ and $$ \lvert\psi\rangle = ...
Norbert Schuch's user avatar
3 votes
Accepted

How to splice Hamiltonians corresponding to channels $\Phi_1$ and $\Phi_2$ so as to obtain a Hamiltonian corresponding to $\Phi_2\circ\Phi_1$?

TL;DR: In general, this cannot be done exactly, because of the relationship between the eigenvectors of $A$ and $e^A$. That said, the Feynman's clock construction gets pretty close to realizing the ...
Adam Zalcman's user avatar
  • 22.9k
3 votes
Accepted

$\mathbb{C}^2 \otimes \mathbb{C}^2$ vs $\mathbb{C}^4$

You don't have to directly express the basis of one space in another to know they are the "same". We say two Hilbert spaces $A$ and $B$ are isomorphic if there exists a linear isomorphism ...
Rammus's user avatar
  • 5,863
3 votes
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If $\rho_{AB}$ is a separable then the partial transpose w.r.t to A is PSD

The main result that you need to complete all the steps that you mention is that if $\rho$ is a density matrix, then $\rho^T$ is also a density matrix. So, what are the key properties of a density ...
DaftWullie's user avatar
  • 58.8k
2 votes
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What happens to $|y\rangle \sum_{x}|x\rangle|f(x) + g(y)\rangle$ when we throw away the first register?

After throwing the first register away the state of the second and third registers becomes $$|\psi_{23}\rangle=\sum_x|x\rangle|f(x)+s\rangle\tag1$$ where $s$ is a fixed value. More precisely, $s:=g(y)$...
Adam Zalcman's user avatar
  • 22.9k
2 votes

What happens to $|y\rangle \sum_{x}|x\rangle|f(x) + g(y)\rangle$ when we throw away the first register?

If $y$ is a bitstring, meaning $|y\rangle$ is a basis state, yes, you can just throw it away since it is not entangled with the rest of the state. However, if you had something like $\sum_y|y\rangle\...
Tristan Nemoz's user avatar
  • 6,462
2 votes
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unitary that transforms one Hilbert space to another Hilbert space

No. Suppose $A$ and $B'$ are qutrits, $A'$ and $B$ are qubits and $U$ is the SWAP unitary that sends $|ab\rangle$ to $|ba\rangle$ for every $a\in\{0,1,2\}$ and $b\in\{0,1\}$. Clearly, SWAP transforms $...
Adam Zalcman's user avatar
  • 22.9k
2 votes

For tetrapartite state, and another way of decomposition, is the Schmidt basis separable?

This is immediately disproved by the structure of the counterexamples to your previous question Does proximity of two bipartite states in a norm force high overlap between the elements of the Schmidt ...
Norbert Schuch's user avatar
2 votes

With $\vert\Psi^+\rangle$ the Bell state, can $\sqrt{\rho}\vert\Psi^+\rangle\langle\Psi^+\vert\sqrt{\rho}$ be simplified?

TL;DR: Yes, $\Omega_{AB}(\rho)$ can be simplified. It turns out that it's one $n$th of the projector onto the ray spanned by the vectorization of $\rho^{1/2}$, see $(3)$ below. First, observe that $\...
Adam Zalcman's user avatar
  • 22.9k
2 votes

Does Neumark's/Naimark's extension theorem only apply to rank-1 POVMs?

No, any POVM, including POVMs whose elements do not have unit rank, can be via interpreted Naimark's theorem as a projective measurement in a higher-dimensional space. Derivation of the dilated ...
glS's user avatar
  • 25.4k
2 votes

Does proximity of two bipartite states in a norm force high overlap between the elements of the Schmidt bases?

TL;DR: No such relation exists, because the upper bound on the norm fails to impose any constraints whatsoever on the basis elements corresponding to very small Schmidt coefficients $\sqrt{p_i}$ and $\...
Adam Zalcman's user avatar
  • 22.9k
2 votes
Accepted

How many dimensions does an n-qubit system have?

Yes, $n$ qubits are represented by a vector in $2^n$ dimensional Hilbert space (which is, in finite dimensions, just the same as a vector space). So for $n = 1$, the system is described by a two ...
banercat's user avatar
  • 763
1 vote
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If $\text{tr}_B \rho \in A$, then $\rho \in A \otimes B$?

It's true in an appropriate formulation. First of all, don't confuse pure states $|\psi\rangle$ from $H$ and density matrices $\rho$ on $H$ (which are $|\psi\rangle\langle\psi|$ for pure states). Let $...
Danylo Y's user avatar
  • 7,342
1 vote

How the circuit covers the Hilbert Space

From the context and wording its not entirely clear what they are talking about. I suspect it is something similar to this: Say $U(\theta)$ is a $d$-dimensional unitary describing a parameterized ...
forky40's user avatar
  • 7,123
1 vote

Comparing Hilbert spaces of coupled and uncoupled qubits

According to the Composite system postulate: "The Hilbert space of a composite system is the Hilbert space tensor product of the state spaces associated with the component systems. For a non-...
Condo's user avatar
  • 2,048

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