Author: Otto Schmidt
In quantum many-body physics, one often studies families of quantum states that are defined by algebraic constraints. These families naturally form algebraic varieties, i.e. solution sets of polynomial equations.
1. From Quantum States to Varieties
- A pure state of (N) particles in local Hilbert space (\mathbb{C}^d) is a vector
- In coordinates, (\psi) is specified by complex amplitudes (\psi_{i_1 i_2 \dots i_N}).
- Imposing algebraic relations among these amplitudes defines a variety inside projective space (\mathbb{P}^{d^N - 1}).
Example:
- Product states are tensors of rank one. The set of all product states is the Segre variety
2. Varieties from Entanglement Structure
- Matrix Product States (MPS): Defined by polynomial relations between amplitudes and tensor parameters. The closure of the set of MPS of fixed bond dimension forms an algebraic variety.
- Symmetry constraints: Imposing invariance under group actions leads to varieties cut out by polynomial invariants.
- Rank conditions: Reduced density matrices with bounded rank correspond to determinantal varieties.
3. Why Varieties Matter
- Classification: Algebraic geometry provides a language to classify families of quantum states.
- Geometry ↔ Physics: Singularities, dimension, and degree of a variety reflect physical features such as degeneracy, correlations, or phase transitions.
- Computational tools: Polynomial equations allow symbolic and numerical methods (Gröbner bases, tensor networks) to analyze many-body states.
4. Outlook
The study of algebraic varieties arising in quantum many-body physics bridges:
- Physics: entanglement, correlations, tensor network ansätze.
- Mathematics: projective geometry, invariant theory, commutative algebra.
This perspective opens pathways for exact classifications of quantum states and new algorithms for simulation and optimization.
