# 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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### 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 $\mathcal{H}^d$ ...
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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= \...
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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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### 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 ...
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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 ...
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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 ...
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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 ...
### 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 ...