Communications in Mathematical Physics

Translation invariant states in quantum mechanics

R. Beaume, J. Manuceau, A. Pellet, M. Sirugue

Published Mar 1, 1974• 23 citations

DOI: 10.1007/bf01651547

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Communications in Mathematical Physics

Translation invariant states in quantum mechanics

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Published Mar 1, 1974• 23 citations

Publisher

DOI: 10.1007/bf01651547

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Interdisciplinary Information SciencesJan 1, 20046 citations

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On Translation-Invariant Matrix Product States and advances in MPS representations of the $W$ -stateOn Translation-Invariant Matrix Product States and advances in MPS representations of the $W$-state

Klimov, Petr, Richik Sengupta, Jacob Biamonte

This work is devoted to the study Translation-Invariant (TI) Matrix Product State (MPS) representations of quantum states with periodic boundary conditions (PBC). We pursue two directions: we introduce new methods for constructing TI MPS representations of a certain class of TI states and study their optimality in terms of their bond dimension. We pay particular attention to the $n$ -party $W$ -state and construct a TI MPS representation of bond dimension $\left \lfloor \dfrac{n}{2} \right \rfloor +1$ for it. We further study properties of this class and show that we can can always achieve a bond dimension of $n$ for TI MPS representation of states in this class. In the framework of studying optimality of TI MPS representations with PBC, we study the optimal bond dimension $d(ψ)$ for a given state $ψ$ . In particular we introduce a deterministic algorithm for the search of $d(ψ)$ for an arbitary state. Using numerical methods, we verify the optimality of our previous construction for the $n$ -party $W$ -state for small $n$ .This work is devoted to the study Translation-Invariant (TI) Matrix Product State (MPS) representations of quantum states with periodic boundary conditions (PBC). We pursue two directions: we introduce new methods for constructing TI MPS representations of a certain class of TI states and study their optimality in terms of their bond dimension. We pay particular attention to the $n$-party $W$-state and construct a TI MPS representation of bond dimension $\left \lfloor \dfrac{n}{2} \right \rfloor +1$ for it. We further study properties of this class and show that we can can always achieve a bond dimension of $n$ for TI MPS representation of states in this class. In the framework of studying optimality of TI MPS representations with PBC, we study the optimal bond dimension $d(ψ)$ for a given state $ψ$. In particular we introduce a deterministic algorithm for the search of $d(ψ)$ for an arbitary state. Using numerical methods, we verify the optimality of our previous construction for the $n$-party $W$-state for small $n$.

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On Translation-Invariant Matrix Product States and advances in MPS representations of the

W

-state

n

-party

W

-state and construct a TI MPS representation of bond dimension

\left \lfloor \dfrac{n}{2} \right \rfloor +1

for it. We further study properties of this class and show that we can can always achieve a bond dimension of

n

for TI MPS representation of states in this class. In the framework of studying optimality of TI MPS representations with PBC, we study the optimal bond dimension

d(ψ)

for a given state

ψ

. In particular we introduce a deterministic algorithm for the search of

d(ψ)

for an arbitary state. Using numerical methods, we verify the optimality of our previous construction for the

n

-party

W

-state for small

n

p

-periodic with

p>1

. At the corresponding classical-bound points, the ground-state sectors are highly degenerate and support many commensurate periods. Along the interpolation paths analyzed in this work, entering the contextual regime is accompanied by the lifting of this classical period degeneracy in favor of a quantum-selected period. We also identify finite periodic-boundary-condition benchmarks at the selected periods: for each model studied, the finite-ring witness reproduces the same classical bound and quantum value as the corresponding infinite-chain witness, and in several cases the resulting finite inequalities are tight. These reductions turn the infinite-chain contextuality certification into compact energy-estimation benchmarks requiring only local correlator measurements. We establish the mechanism analytically in representative two- and three-body witness models and corroborate it more broadly using translation-invariant semidefinite-program relaxations together with variational matrix-product-state calculations.