# 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

20 questions
Filter by
Sorted by
Tagged with
27 views

### 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: X_1X_2, X_2X_3, X_3X_1,~ Y_1Y_2, Y_2Y_3, ...
• 183
1 vote
108 views

1 vote
104 views

### 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 ...
1 vote
85 views

### 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 ...
• 179
141 views

### What is the difference between $\mathbb{C}^2 \otimes \mathbb{C}^2$ and $\mathbb{C}^4$?

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: \...
• 179
64 views

495 views

• 305