Quantum State Assignment Flows
Abstract
:1. Introduction
1.1. Overview and Motivation
1.2. Contribution and Organization
1.3. Basic Notation
Canonical basis vectors of | |
Euclidean inner vector product | |
Euclidean norm | |
Unit matrix of | |
Component-wise vector multiplication | |
Component-wise division | |
Space of Hermitian matrices (cf. (22)) | |
Trace of matrix A | |
Matrix inner product , | |
Commutator | |
The diagonal matrix with vector v as entries | |
The vector of the diagonal entries of a square matrix V | |
The matrix exponential | |
The matrix logarithm | |
The set of discrete probability vectors of dimension c (cf. (6)) | |
The relative interior of , i.e., the set of strictly positive probability | |
vectors (cf. ) | |
The product manifold (cf. ) | |
The set of symmetric positive definite matrices (cf. (17)) | |
The subset of matrices in whose trace is equal to 1 (cf. (18)) | |
The product manifold (cf. (96)) | |
Barycenter of the manifold | |
Barycenter of the manifold | |
Matrix | |
The Riemannian metrics on (cf. (8), (54), (25)) | |
The tangent spaces to (cf. (10), (54), (21)) | |
Orthogonal projections onto (cf. (11), (24)) | |
Replicator operators associated with the assignment flows | |
on (cf. (12), (58), (64), (105)) | |
∂ | Euclidean gradient operator: |
grad | Riemannian gradient operator with respect to the Fisher–Rao metric |
, etc. | Square brackets indicate a linear operator that acts in a non-standard |
way, e.g., row-wise to a matrix argument. |
2. Information Geometry
- The relative interior of probability simplices, each of which represents the categorical (discrete) distributions of the corresponding dimension; and
- The set of positive definite symmetric matrices with trace one.
2.1. Categorical Distributions
2.2. Density Matrices
2.3. Alternative Metrics and Geometries
2.3.1. Affine-Invariant Metrics
2.3.2. Log-Euclidean Metric
2.3.3. Comparison to Bogoliubov-Kubo-Mori Metric
3. Assignment Flows
3.1. Single-Vertex Assignment Flow
3.2. Assignment Flows
3.3. Reparameterized Assignment Flows
4. Quantum State Assignment Flows
- Determination of the form of the Riemannian gradient of functions with respect to the BKM metric (25), the corresponding replicator operator and exponential mappings Exp and exp, together with their differentials (Section 4.1);
- Definition of the single-vertex quantum state assignment flow (Section 4.2);
- Determination of the general quantum state assignment flow equation for an arbitrary graph (Section 4.3) and its alternative parameterization (Section 4.4), which generalizes Formulation (62) of the assignment flow accordingly.
4.1. Riemannian Gradient, Replicator Operator and Further Mappings
4.2. Single-Vertex Density Matrix Assignment Flow
4.3. Quantum State Assignment Flow
4.4. Reparameterization and Riemannian Gradient Flow
4.5. Recovering the Assignment Flow for Categorical Distributions
- (a)
- The submanifold with the induced BKM metric is isometric to ;
- (b)
- If , then the tangent subspace is contained in the subspace defined by (32);
- (c)
- Let denote an orthonormal basis of such that for every , there are that form a basis of . Then, there is an inclusion of commutative subsets that corresponds to an inclusion .
- (i)
- If , then .
- (ii)
5. Experiments and Discussion
- Structure-preserving feature patch smoothing without accessing data at individual pixels (Section 5.3);
5.1. Geometric Integration
Algorithm 1:Geometric Integration Scheme |
- A reasonable convergence criterion that measures how close the states are to a rank-one matrix is ;
- A reasonable range for the step size parameter is ;
- In order to remove spurious non-Hermitian numerical rounding errors, we replace each matrix with ;
- The constraint of (18) can be replaced by with any constant . This ensures that for larger matrix dimensions c, the entries of vary in a reasonable numerical range and that the stability of the iterative updates.
5.2. Labeling 3D Data on Bloch Spheres
5.3. Basic Image Patch Smoothing
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
Appendix A. Proofs
Appendix A.1. Proofs of Section 2
Appendix A.2. Proofs of Section 3
Appendix A.3. Proofs of Section 4
- (b)
- Let and . Suppose that vector X is represented by a curve such that and . In view of the definition (123) of , we have
- (a)
- The bijection is explicitly given by
- (c)
- Part (c) is about the commutativity of the diagram.
- (i)
- Due to the commutativity of the components of , we can simplify the expression for the vector field of the QSAF as follows.
- (ii)
- We write for all with and express in terms of as
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Assignment Flow (AF) | Quantum State AF (QSAF) |
---|---|
Single-vertex AF (Section 3.1) | Single-vertex QSAF (Section 4.2) |
AF approach (Section 3.2) | QSAF approach (Section 4.3) |
Riemannian gradient AF (Section 3.3) | Riemannian gradient QSAF (Section 4.4) |
Recovery of the AF from the QSAF by restriction (Section 4.5) |
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Schwarz, J.; Cassel, J.; Boll, B.; Gärttner, M.; Albers, P.; Schnörr, C. Quantum State Assignment Flows. Entropy 2023, 25, 1253. https://doi.org/10.3390/e25091253
Schwarz J, Cassel J, Boll B, Gärttner M, Albers P, Schnörr C. Quantum State Assignment Flows. Entropy. 2023; 25(9):1253. https://doi.org/10.3390/e25091253
Chicago/Turabian StyleSchwarz, Jonathan, Jonas Cassel, Bastian Boll, Martin Gärttner, Peter Albers, and Christoph Schnörr. 2023. "Quantum State Assignment Flows" Entropy 25, no. 9: 1253. https://doi.org/10.3390/e25091253