nLab matrix product state

Idea

What are called matrix product states (MPS) in quantum physics (specifically in solid state physics and in AdS/CFT) are those tensor network states of the form of a ring of tensors all of rank 3.

For example, given a metric Lie algebra $\mathfrak{g}$ (with string diagram-notation as discussed there) its Lie bracket-tensor $f \coloneqq [-,-] \in \mathfrak{g}^\ast \otimes \mathfrak{g}^\ast \otimes \mathfrak{g}$ gives rise to matrix product states of the following form:

graphics from Sati-Schreiber 19c

Related concepts

quantum probability theory – observables and states

• states

  • classical state

  • quantum state

    • space of states (in geometric quantization)

    • state on a star-algebra, quasi-state

    • qbit, Bell state

      dimer, tensor network state

    • quantum state preparation

    • probability amplitude, quantum fluctuation

    • pure state

      • wave function

      • bra-ket

      • Bell state

    • quantum superposition, quantum interference

    • quantum entanglement

    • quantum measurement

      wave function collapse

      Born rule

      deferred measurement principle

      quantum reader monad

      measurement problem

    • superselection sector

    • mixed state, density matrix

      entanglement entropy

      holographic entanglement entropy

    • coherent quantum state

  • ground state, excited state

  • quasi-free state

  • Fock space, second quantization

  • vacuum, vacuum state

    • Hadamard state

    • vacuum diagram

    • vacuum expectation value, vacuum amplitude, vacuum fluctuation

    • vacuum energy

    • vacuum polarization

    • interacting vacuum

    • thermal vacuum, KMS state

    • vacuum stability

      • false vacuum, tachyon, Coleman-De Luccia instanton

    • theta vacuum

    • perturbative string theory vacuum

      • non-geometric string theory vacuum

      • landscape of string theory vacua

  • entangled state

  • tensor network state

    • matrix product state

    • tree tensor network state

• observables

  • quantum observable, beable

  • algebra of observables, star-algebra

  • Bohr topos

  • quantum operator (in geometric quantization)

  • quantum operation, quantum effect, effect algebra

  • in quantum field theory

    • local observable

    • polynomial observable

      • linear observable

        • field observable

    • regular observable

    • microcausal observable

    • normal-ordered product, time-ordered products, retarded product

    • Wick algebra

    • scattering amplitude

    • interacting field algebra of observables, Bogoliubov's formula

• GNS construction

• theorems

  • order-theoretic structure in quantum mechanics

    • Gleason's theorem

    • Alfsen-Shultz theorem

    • Harding-Döring-Hamhalter theorem

    • Kochen-Specker theorem

    • Nuiten's lemma

  • Wigner's theorem

  • no-cloning theorem

  • Bell's theorem

References

For more see the references at tensor network state.

General

Review and exposition:

• Roman Orus, Section 5.1 of: A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States, Annals of Physics 349 (2014) 117-158 (arXiv:1306.2164)

• Jens Eisert, Section 2 of: Entanglement and tensor network states, Modeling and Simulation 3, 520 (2013) (arXiv:1308.3318)

• Jacob Biamonte, Ville Bergholm, Section 5 of: Tensor Networks in a Nutshell, Contemporary Physics (arxiv:1708.00006)

• Bei Zeng, Xie Chen, Duan-Lu Zhou, Xiao-Gang Wen:

  Sec. 8 of: Quantum Information Meets Quantum Matter – From Quantum Entanglement to Topological Phases of Many-Body Systems, Quantum Science and Technology (QST), Springer (2019) $[$arXiv:1508.02595, doi:10.1007/978-1-4939-9084-9$]$

• Jacob Biamonte, Chapter II of: Lectures on Quantum Tensor Networks (arXiv:1912.10049)

• Shi-Ju Ran, Emanuele Tirrito, Cheng Peng, Xi Chen, Luca Tagliacozzo, Gang Su, Maciej Lewenstein, Section 2.3.2 of: Tensor Network Contractions, Lecture Notes in Physics, Springer (2020) (arXiv:1708.09213, doi:10.1007/978-3-030-34489-4)

• Roman Orus, Chapter II of: Tensor networks for complex quantum systems, Nature Reviews Physics 1, 538-550 (2019) (arXiv:1812.04011, doi:10.1038/s42254-019-0086-7)

See also:

• Wikipedia, Matrix product state

• Frank Verstraete, J. Ignacio Cirac, V. Murg: Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems, Adv. Phys. 57 (2008) 143 [arXiv:0907.2796, doi:10.1080/14789940801912366]

In quantum computation

In the context of quantum computation:

• Yiqing Zhou, E. Miles Stoudenmire, Xavier Waintal, What limits the simulation of quantum computers? (arXiv:2002.07730)