The minimal translation-invariant MPS bond dimension of the W-state

P. Klimov, R. Sengupta and J. Biamonte, On Translation-Invariant Matrix Product States and advances in MPS representations of the $W$-state, arXiv:2306.16456v2, Section 3, conjecture after their Theorem 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 $n$.

A TI MPS representation with PBC of bond dimension $d$ of the normalized $W$-state of order $n$ is a pair of complex $d\times d$ matrices $A_0$, $A_1$ with $\operatorname{Tr}(A_{i_1}\cdots A_{i_n})=1/\sqrt n$ when exactly one $i_j$ is $1$ and $0$ otherwise (Section 2). Problem 4 of Section 4 asks whether $d(n)=\lfloor n/2\rfloor+1$, and the table of Section 4 lists $d(7)=4$.

The $W$-state is translation invariant, but a translation-invariant MPS with periodic boundary conditions needs a bond dimension that grows with $n$: the known lower bound is $\Omega(n^{1/(3+\delta)})$ for every $\delta>0$, and Theorem 1 of the paper gives the upper bound $\lfloor n/2\rfloor+1$. The conjecture says that this upper bound is exact. The frozen declaration D5/S3/Quantum/SpinChains/WStateTIMPSBondDimension.result refutes it.

Issue #12026 classifies the conjecture as Tier 1 and records the bounded literature check before any Lean. It found:

• arXiv v2 is the latest version, with no journal version;
• the five citing works listed by Semantic Scholar (arXiv:2408.04729, 2408.13033, 2410.19541, 2412.19635, 2503.16327) give no representation below $\lfloor n/2\rfloor+1$; 2410.19541 calls the minimal TI MPS an open question, and 2503.16327 cites the paper as evidence for $\lfloor L/2\rfloor+1$;
• arXiv:1809.08185, arXiv:2108.00031 and quant-ph/0608197 give only border bond dimension, lower bounds and the $O(n)$ representation.

These are orchestrator-reported literature readings, not-found-in-searched-scope; they do not establish exhaustive worldwide novelty or exclude an independent answer.

1. Take the $3\times3$ matrices $A_0=E_{12}+E_{21}+E_{23}$ and $A_1=E_{31}/\sqrt7$, with $E_{ij}$ the matrix units, and write $B=\sqrt7A_1=E_{31}$.
2. $A_0^3=A_0$, and $(A_0^r)_{13}$ is $1$ for even $r\ge2$ and $0$ otherwise.
3. A word with $k\ge1$ letters $1$ has trace $7^{-k/2}\prod_j(A_0^{r_j})_{13}$, where the cyclic gaps $r_j\ge0$ between consecutive letters $1$ add up to $7-k$.
4. For $k=1$ the gap is $6$ and the trace is $1/\sqrt7$. For $k\ge2$ all gaps even and at least $2$ needs $3k\le7$, so $k=2$, and then the gaps add up to $5$, which is odd: the trace is $0$. For $k=0$, $\operatorname{Tr}A_0^7=\operatorname{Tr}A_0=0$.
5. So $d(7)\le3<4=\lfloor7/2\rfloor+1$. In Lean the 128 word traces of the integer matrices are computed by the kernel and the factor $1/\sqrt7$ is pulled out of each word.

The kernel-checked result excludes the bound $d\ge\lfloor n/2\rfloor+1$ for $n=7$, hence the conjecture, the answer "yes" to Problem 4, and the table entry $d(7)=4$. Changing the definition of a representation (the normalization $1/\sqrt n$, the cyclic trace, or the order of the matrix product) would change the question.

Exact SymPy arithmetic over all 128 words and an independent NumPy computation give the traces above (0 failures). The same matrices give 10 and 17 failures for $n=6$ and $n=8$, and perturbing one entry by $0.01$ gives 7–21 failures. As a positive control, the paper's Theorem 1 construction for $n=5,6,7$ passes the same checker (issue #12026).

The canonical source is D5/S3/Quantum/SpinChains/WStateTIMPSBondDimension.lean. Its public declarations are IsWStateTIMPS, claim and result. The frozen module state has statement identity sha256:fd1e75a90d7c9c176ed83ca1295f07558a4caf391781f06e1d7c9f3c285e68c0. The result declaration has statement identity sha256:8debb2e851e522c233797783f07fc7d6b5e8597be4fe6289c23b239431fd636d. The Freeze event is sha256:ac8c966e2d620df874db853c6f0de88ba4ae76757fabf4a4c4bc2f531cbbf038. It has no project-level frozen prerequisites (pinned Mathlib only). The proof uses only the standard axioms propext, Classical.choice and Quot.sound; no sorry, native_decide, or new axiom.

Tier 1 published conjecture; resolution Refuted by D5/S3/Quantum/SpinChains/WStateTIMPSBondDimension.result. proof_shape: bind-only (evaluation at an explicit pair of matrices); admission_basis: open-problem-resolution (issue #12026). Utility certified-instance, refuting claim.

What the refutation shows

• Proved in this module: the normalized $W$-state of order $7$ has a translation-invariant MPS representation with periodic boundary conditions of bond dimension $3<\lfloor7/2\rfloor+1$.
• Where the conjecture fails: in the proof of Theorem 1, $(A_0^m){d1}$ vanishes for $m\le d-2$ and is nonzero for $d-1\le m\le2d-2$ (display (d1), l. 373–379), and a word of weight $k\ge2$ is killed only by its shortest cyclic gap, which is below $\lfloor n/2\rfloor$ (l. 407). That mechanism needs $d$ of order $n/2$. The witness keeps the paper's shape of a scaled matrix unit ($A_1$ a multiple of a matrix unit: $E{1d}$ in the paper, $E_{31}$ here) but takes a periodic $A_0$: the entry $(A_0^r)_{13}$ is nonzero only on an arithmetic progression of lengths, and the words of weight $\ge2$ are killed by parity instead of by short gaps.
• Computed, not formalized (orchestrator): with $A_1$ a multiple of $E_{d1}$ and a nonnegative integer $A_0$, exact traces over all words give $d(10)\le4$, $d(11)\le4$ and $d(13)\le4$, against $\lfloor n/2\rfloor+1=6$, $6$, $7$. An exhaustive search over $0/1$ matrices $A_0$ with this $A_1$ finds no representation for $(n,d)=(4,2),(5,2),(6,2),(6,3),(8,3),(9,3),(10,3),(6,4),(8,4),(9,4),(12,4)$. This restricted search does not settle the table entries $d(8)=5$ and $d(9)=5$; they are open here.
• Model derivation, not checked in Lean (GPT Pro seat): for $D\ge3$ and $n=D(D-1)+1$, the $(D-1)$-cycle on the vertices $1,\dots,D-1$ with the extra edge $D-1\to D$, and $A_1$ a multiple of $E_{D1}$, give $d(W_n)\le D$. A word of weight $k\ge2$ would need $k\equiv1\pmod{D-1}$, hence $k\ge D$, against $n<D^2$. So $d(n)/n\to0$ along this subsequence, and the "asymptotics" part of the conjecture fails along it. The orchestrator recomputed $D=4$ ($n=13$) over all $2^{13}$ words. Whether $d(n)=O(\sqrt n)$ for all $n$ is open here.
• Unchanged: Theorem 1 (the upper bound $\lfloor n/2\rfloor+1$), the lower bound $\Omega(n^{1/(3+\delta)})$, the paper's general constructions and its algorithm for $d(\psi)$ do not depend on the conjecture. The paper's question whether $d(6)=4$, and its question whether $d(n)$ is monotone, remain open here.

The bounded literature check does not establish worldwide priority or absence of an independent counterexample. The Lean kernel verifies the encoded statement and its axiom closure; correspondence to the external paper, including the reading of the cases display as the definition of a representation and the hypothesis $n\ge2$ as a weakening, is checked by reading the source and the definitions.