For questions about geometrical features of the set of quantum states.
Questions tagged [state-space-geometry]
25 questions
31
votes
5 answers
Can the Bloch sphere be generalized to two qubits?
The Bloch sphere is a nice visualization of single qubit states. Mathematically, it can be generalized to any number of qubits by means of a high-dimensional hypersphere. But such things are not easy to visualize.
What attempts have been made to…
James Wootton
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14
votes
2 answers
What is the difference between the "Fubini-Study distances" $\arccos|\langle\psi|\phi\rangle|$ and $\sqrt{1-|\langle\psi|\phi\rangle|}$?
I sometimes see the "Fubini-Study distance" between two (pure) states $|\psi\rangle,|\phi\rangle$ written as
$$
d(\psi,\phi)_1=\arccos(|\langle\psi|\phi\rangle|),
$$
for example in the Wikipedia page.
Other sources (e.g. this paper in pag. 16), use…
glS
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14
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1 answer
Does the trace distance have a geometric interpretation?
Consider the trace distance between two quantum states $\rho,\sigma$, defined via
$$D(\rho,\sigma)=\frac12\operatorname{Tr}|\rho-\sigma|,$$
where $|A|\equiv\sqrt{A^\dagger A}$.
When $\rho$ and $\sigma$ are one-qubit states, the trace distance can be…
glS
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11
votes
2 answers
What are useful resources about the geometric of qutrits and its relation with Gell-Mann matrices?
I need some useful sources about the geometry of qutrit. Specifically related to the Gell-Mann matrix representation.
Azadeh Zohrabi
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- 4
9
votes
1 answer
Do pure qudit states lie on a hypersphere in the Bloch representation?
It is known that every state $\rho$ of a $d$-level system (or if you prefer, qudits living in a $d$-dimensional Hilbert space) can be mapped into elements of $\mathbb R^{d^2-1}$ through the mapping provided by the Bloch representation, by writing it…
glS
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9
votes
2 answers
Homeomorphism or stereographic projection corresponding to the set of mixed states within the Bloch sphere
The Bloch sphere is homeomorphic to the Riemann sphere, and there exists a stereographic projection $\Bbb S^2\to \Bbb C_\infty$. But this only holds for pure states. To quote Wikipedia:
Quantum mechanics is mathematically formulated in Hilbert…
Sanchayan Dutta
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7
votes
2 answers
Is the set of all states with negative conditional Von Neumann entropy convex?
I have read somewhere / heard that the set of all states that have non-negative conditional Von Neumann entropy forms a convex set. Is this true? Is there a proof for it?
Can anything be said about the reverse - set of all states that have negative…
Mahathi Vempati
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6
votes
2 answers
Why is the boundary of the set of states in the generalised Bloch representation comprised of singular matrices?
Consider an arbitrary qudit state $\rho$ over $d$ modes.
Any such state can be represented as a point in $\mathbb R^{d^2-1}$ via the standard Bloch representation:
$$\rho=\frac{1}{d}\left(\mathbb I +\sum_k c_k\sigma_k\right)$$
with $\sigma_k$…
glS
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6
votes
1 answer
What is the connection between Bures metric and (finite) Bures distance?
The Wikipedia page discussing the Bures metric introduces it as the Hermitian 1-form operator $G$ defined implicitly by $\rho G+G\rho = \mathrm d\rho$, and which induces the corresponding Bures distance, which reads
$$[d(\rho,\rho+\mathrm d\rho)]^2…
glS
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5
votes
3 answers
How are orthogonal sets of pure states arranged in state space?
It is well known that the state of a (pure) qubit can be described as a point on a two-dimensional sphere, the so-called Bloch sphere.
The mapping $\lvert\psi\rangle\mapsto \boldsymbol r_\psi$ that sends each state into its representative in the…
glS
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5
votes
3 answers
Purity of mixed states as a function of radial distance from origin of Bloch ball
@AHusain mentions here that the purity of a qubit state can be expressed as a function of the radius from the center of a Bloch sphere. The state corresponding to the origin is maximally mixed whereas the states corresponding to the boundary points…
Sanchayan Dutta
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4
votes
2 answers
Do states with the same purity always have the same rank?
The purity of a state $\rho$ is $\newcommand{\tr}{\operatorname{Tr}}\tr(\rho^2)$, which is known to equal $1$ iff $\rho$ is pure.
Another quantity that can be used to quantify how close a state $\rho$ is to be pure is its rank. This equals the…
glS
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4
votes
2 answers
Is any multi-qubit unitary operation a rotation about a specific unit vector?
I understand that any single qubit unitary operation can be expressed as a rotation around a three dimensional unit vector. Is it possible to do the same for multi-qubit unitaries? Can I express an $n$-qubit unitary operation as a rotation about…
smi
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4
votes
1 answer
How to go from finite to infinitesimal form of the Fubini-Study metric?
As mentioned e.g. in the Wikipedia page, given a pair of pure states $\psi,\phi\in\mathbb{CP}^{N-1}$, the geodesic distance between them is
$$\gamma(\psi,\phi) =…
glS
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4
votes
0 answers
Relation between geometric and discrete circuit complexity
Geometric complexity of a unitary, as introduced for example here https://arxiv.org/abs/quant-ph/0502070, measures the length of a geodesic connecting the identity matrix and a given unitary in the metric where directions corresponding to…
Nikita Nemkov
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