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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Kraus decomposition in the infinite-dimensional case
The action of a quantum channel $\Lambda$ on a generic state $\rho_S$ coupled to a pure environment state $|0\rangle_E$ can be written in the Stinespring representation as
$\Lambda(\rho_S) = \text{tr}...
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$\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: $\...
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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\...
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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\...
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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^...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\mathcal{H}^d$ ...
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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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