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
Questions tagged [hilbert-space]
31 questions
22
votes
6 answers
Quantum states are unit vectors... with respect to which norm?
The most general definition of a quantum state I found is (rephrasing the definition from Wikipedia)
Quantum states are represented by a ray in a finite- or infinite-dimensional Hilbert space over the complex numbers.
Moreover, we know that in…
Adrien Suau
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votes
2 answers
What does it mean for a density matrix to "act on a Hilbert space $\mathcal{H}"$?
For a Hilbert space $\mathcal{H}_A$, I have seen the phrase
density matrices acting on $\mathcal{H}_A$
multiple times, e.g. here.
It is clear to me that if $\mathcal{H}_A$ has finite Hilbert dimension $n$, then this makes sense mathematically,…
Peter
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8
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2 answers
When would I consider using an outer product of quantum states, to describe aspects of a quantum algorithm?
I know the inner product has a relationship to the angle between two vectors and I know it can be used to quantify the distance between two vectors. Similarly, what's an use case for the outer product? You can exemplify with the simplest case. It…
R. Chopin
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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 map that outputs $\rho_A\otimes\rho_B$ possible?
user1936752
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What kind of mathematics is common in quantum computing?
From what I have seen so far, there is a lot of linear algebra. Curious what other kinds of maths are used in QC & the specific fields in which they are most predominately invoked.
user820789
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votes
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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 $\mathcal{H}^d$ of dimension $d$
A POVM is a set $\mathcal{M} =…
junfan02
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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 Dilation says there are two auxiliary systems…
Sam Jaques
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5
votes
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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 valid Hilbert space.
It is pretty clear that by…
Ohad
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5
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What types of quantum systems use infinite values?
Background
I am curious to learn more about any work that has been done regarding quantum systems that deal with infinite values. I am primarily interested in photonic quantum computing; however I am open to learning about other systems where…
user820789
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4
votes
0 answers
Mutually unbiased bases in infinite dimensions
The problem of determining the maximal number of mutually unbiased bases in $d$ dimensional Hilbert space is open for any natural number $d$ which is not of the form $p^n$ where $p$ is prime. I'm interested in this problem in the infinite…
truebaran
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4
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Hilbert space to accurately represent 3x3 Rubik's Cube
What Hilbert space of dimension greater than 4.3e19 would be most convenient for working with the Rubik's Cube verse one qudit?
The cardinality of the Rubik's Cube group is given…
user820789
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4
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3 answers
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 Pauli matrices…
Marion
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What is the Kronecker product of two 1D vectors?
When working with $n$-qubit systems, we can use the Kronecker product $\otimes$ to build corresponding bigger Hilbert spaces. For example:
$$\begin{bmatrix} \alpha \\ \beta \end{bmatrix} \otimes \begin{bmatrix} \gamma \\ \delta \end{bmatrix} =…
Danyel
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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 space $\mathcal{H}(A)$.
I encountered the…
Jammy
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votes
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Set of `reachable` states from an initial density matrix with polynomial elements
I've been reading about the Bernoulli-factory problem and I'm particularly interested in deriving the results using the density matrix formalism, i.e., given required numbers of copies of the initial state
\begin{pmatrix}
p & \sqrt{p(1-p)}…
Yash Solanki
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