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Questions tagged [hilbert-space]

Understanding the geometric (tensor composition, vectors, holistic character) or algebraic (observables, commutative subspaces) properties of Hilbert spaces described in Quantum Information and Quantum Computation Science

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

Given some joint state $\rho_{AB}$, one can find either the marginal state $\rho_A$ or the marginal state $\rho_B$ through a CPTP map. The proof being that partial tracing is indeed CPTP. Is a CPTP ...
user1936752's user avatar
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6 votes
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Does Neumark's/Naimark's extension theorem only apply to rank-1 POVMs?

Starting with the definitions used. A PVM is a set $\mathcal{P} = \{P_i: P_i^2 = P_i, P_iP_j = \delta_{ij}P_j, \sum{P_i} = \mathbf{I}\}_{i,j=1}^n$, where $n\leq d$ on a Hilbert space $\mathcal{H}^d$ ...
junfan02's user avatar
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5 votes
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How to splice Hamiltonians corresponding to channels $\Phi_1$ and $\Phi_2$ so as to obtain a Hamiltonian corresponding to $\Phi_2\circ\Phi_1$?

Suppose I have two quantum channels $\Phi_1:B(\mathcal{H}_1)\rightarrow B(\mathcal{H}_2)$ and $\Phi_2:B(\mathcal{H}_2)\rightarrow B(\mathcal{H}_3)$, and let $\Phi=\Phi_2\circ \Phi_1$. Stinespring ...
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4 votes
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How is the surface of a Bloch sphere a Hilbert space?

In the linear algebra section of the Qiskit textbook appears the following claim regarding the Bloch sphere: The surface of this sphere, along with the inner product between qubit state vectors, is a ...
Ohad's user avatar
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3 votes
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What is the actual Hilbert space of a $N$-qubit system?

This question seems slightly naive. The Hilbert pace of any 2-level quantum system is given by the Bloch sphere and the algebra of observables arises from $SU(2)$, the Lie group generated by the three ...
Marion's user avatar
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With $\vert\Psi^+\rangle$ the Bell state, can $\sqrt{\rho}\vert\Psi^+\rangle\langle\Psi^+\vert\sqrt{\rho}$ be simplified?

Let $\vert\Psi^+\rangle_{AB} = \frac{1}{\sqrt n}\sum_{i=1}^n\vert i\rangle_A\vert i\rangle_B$ be the maximally entangled state in Hilbert space $\mathcal{H}(AB)$ and $\rho_A$ be some state in Hilbert ...
Jammy's user avatar
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2 votes
2 answers
219 views

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 $|\psi\...
Takimoto.R's user avatar
2 votes
1 answer
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$\mathbb{C}^2 \otimes \mathbb{C}^2$ vs $\mathbb{C}^4$

Is there a difference between the following two Hilbert spaces: $H_1 = \mathbb{C}^2 \otimes \mathbb{C}^2$ and $H_2 = \mathbb{C}^4$? Here's my confusion. For the following bases, $H_1 = H_2$ holds: $\...
Mohan's user avatar
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unitary that transforms one Hilbert space to another Hilbert space

Let $H = A \otimes B$. If there exists a unitary operator $U$ that transforms the Hilbert space $H$ into another Hilbert space $H' = A' \otimes B'$ (meaning that $U$ maps each basis of $H$ to each ...
Mohan's user avatar
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2 votes
1 answer
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How many dimensions does an n-qubit system have?

How many dimensions does an $n$-qubit system have? What is definition of dimension for a quantum state? Is it related to the number of rows and columns of a density matrix? My guess is that it has $2^...
reza's user avatar
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2 votes
1 answer
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How the circuit covers the Hilbert Space

I am refreshing my functional analysis knowledge to learn quantum machine learning and I am getting confused on Hilbert spaces. What does it mean for a "circuit to cover the Hilbert Space" I ...
epsilonolispe's user avatar
2 votes
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Minimum number of qubits to express given commutation relations (and linear dependences) of Pauli terms

I'm interested in the question written in the title. To explain what I mean, let's take the following set of 9 Pauli terms for 3 qubits: \begin{equation} X_1X_2, X_2X_3, X_3X_1,~ Y_1Y_2, Y_2Y_3, ...
Jun_Gitef17's user avatar
2 votes
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qudits vs bipartite system states [duplicate]

Suppose we have a bipartite system of two qubits. It will form a 4d hilbert space. Also, suppose I have just one quantum system and it is a 4-level system. It will also form a 4d Hilbert space. What ...
Chetan Waghela's user avatar
1 vote
2 answers
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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?

Let's suppose, that applying $\mathbf{H}$ (Hadamard operator) to the first register of the state $c \cdot \sum_{x}|x\rangle|f(x)\rangle$ ($f$ is a permutation, $c$ is a normalization factor), and ...
Georgy Firsov's user avatar
1 vote
1 answer
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If $\rho_{AB}$ is a separable then the partial transpose w.r.t to A is PSD

Def: The partial transpose of a linear operator $\rho_{AB}$ over a Hilbert space $H_A \otimes H_B$ w.r.t A is defined for a linear operator $\rho_{AB}=\rho_A \otimes\rho_B$ as $\rho^{T_A}_{AB}=\rho_A^...
some_math_guy's user avatar
1 vote
1 answer
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If $\text{tr}_B \rho \in A$, then $\rho \in A \otimes B$?

Let our Hilbert space be $H = (A \otimes B) \oplus (A \otimes B)^{\perp}$. If $\rho \in A \otimes B$, then we have $\text{tr}_B \rho \in A$. Is the converse true: if $\text{tr}_B \rho \in A$, then $\...
karavan's user avatar
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1 vote
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Neumark dilation for qubit tetrahedron SIC-POVM

I would like to know if an analytic solution is known for the Neumark dilation of the qubit tetrahedron SIC-POVM defined by $$ M_0= \frac{1}{4\sqrt{3}} \Big( \sqrt{3}I + X +Y +Z \Big), \qquad M_1= \...
quantum_theo's user avatar
1 vote
0 answers
35 views

Quadratic forms on finite linear combinations of pure symmetric (+) or antisymmetric (−) tensor products of basis vectors

I'm trying to solve this problem, I am not sure how to go about it. Some help would be highly appreciated. Let $\mathcal{H}$ be a (one-body) Hilbert space and let $\{u_\alpha\}^\infty_{\alpha=1}$ be ...
monkeyboy's user avatar
0 votes
1 answer
63 views

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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1 answer
50 views

Comparing Hilbert spaces of coupled and uncoupled qubits

Imagine two situations. In one, there are two qubits that are next to each other, that is, they have non-zero coupling terms in their Hamiltonian, and thus suffer from cross-talk and energy can leak ...
psitae's user avatar
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