# 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 ...
279 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 ...
552 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 $\mathcal{H}^d$ ...
121 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 ...
123 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 ...
346 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 ...
47 views

### 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 ...
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 ...
1 vote
40 views

### 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= \...
1 vote
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 ...