bibkey: klimov2023wstate authors: Petr Klimov; Richik Sengupta; Jacob Biamonte year: 2023 title: “On Translation-Invariant Matrix Product States and advances in MPS representations of the W-state” doi: 10.48550/arXiv.2306.16456 url: https://arxiv.org/abs/2306.16456v2 claim: “A TI MPS representation with PBC of bond dimension d of the W-state of order n is a pair of complex d x d matrices A_0, A_1 with Tr(A_{i_1} … A_{i_n}) = 1/sqrt(n) when exactly one i_j is 1 and 0 otherwise. Theorem 1 gives such a representation of bond dimension floor(n/2) + 1 for every n. The paper conjectures that no TI MPS representation with PBC of the W-state of order n has bond dimension smaller than floor(n/2) + 1, and asks whether d(n) = floor(n/2) + 1.” strata_touched:

• D5/S3/Quantum/SpinChains/WStateTIMPSBondDimension license: citation-only triage: anchor

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

Petr Klimov, Richik Sengupta, Jacob Biamonte, arXiv:2306.16456v2 [quant-ph, math.RA] (2023). Quotations are from the arXiv source.

The normalized $W$-state (Section 2; the align environment around the formula omitted):

The $W$-state of order $n$ is defined as $|W_n⟩=\frac{1}{\sqrt{n}}\sum _{(i_1,\dots ,i_n)\in \{0,1{\}}^n:\sum _{j=1}^n{i}_j=1}|i_1\dots i_n⟩$

The representations (Section 2):

TI MPS representation with PBC of bond dimension $d$ for the $W$-state are pair of complex matrices $A_0$ and $A_1$ such that

$\mathrm{Tr}(A_{i_1}{A}_{i_2}\dots A_{i_n})$ is $0$ for every $(i_1,\dots ,i_n)\in \{0,1{\}}^n$ with ${\sum }_j{i}_j\ne 1$ and $\frac{1}{\sqrt{n}}$ for every $(i_1,\dots ,i_n)\in \{0,1{\}}^n$ with ${\sum }_j{i}_j=1$.

After Theorem 1 (Section 3), which gives a representation of bond dimension $\lfloor n/2\rfloor +1$:

These experiments lead us to conjecture that the representation obtained in Theorem \ref{W state representation} could be optimal both in terms of asymptotics and the constant factor or in other words it is impossible to come up with a TI MPS representation with PBC with bond dimension smaller than \floor*{\frac{n}{2}}+1 for the $W$-state of order $n$.

Section 4 lists computed values of the minimal bond dimension $d(n)$, with $d(7)=4$, and asks:

Is the estimate obtained in Theorem \ref{W state representation} optimal? Or, in other words, is it true that d(n)=\floor*{\frac{n}{2}}+1 for the $W$-state? In particular, is it true that $d(6)=4?$

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2306.16456 (arXiv-issued; the arXiv record lists no journal reference).
• URL: https://arxiv.org/abs/2306.16456v2 (source W_eng_final.tex, md5 7ca3ea38231607a509cbbc09602101f3): the $W$-state (l. 180–184), the representations (l. 189–197), the conjecture (l. 414), the table of $d(n)$ (l. 750–768) and the problem (l. 772–774).