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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4 votes
1 answer
119 views

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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2 votes
0 answers
39 views

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 ...
3 votes
1 answer
118 views

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 ...
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7 votes
1 answer
60 views

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 ...
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4 votes
1 answer
255 views

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 ...
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2 votes
2 answers
252 views

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 ...
  • 545
1 vote
0 answers
24 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 ...
4 votes
2 answers
456 views

Confusion regarding Neumark's/Naimark's extension of POVM

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 $\mathcal{H}^d$ ...
0 votes
1 answer
41 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 ...
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