slug: gingrich-2002-sigma-abc-monotone bibkey: gingrich2002properties doi: 10.1103/PhysRevA.65.052302 url: https://arxiv.org/abs/quant-ph/0106042v2 triage: theorem motivation_gids:

• D5/S3/Quantum/Entanglement/GingrichKempeMonotoneRefutation.result

Gingrich’s proposed three-qubit monotone built from the Kempe invariant

Problem

R. M. Gingrich, Properties of entanglement monotones for three-qubit pure states, Phys. Rev. A 65, 052302 (2002), arXiv:quant-ph/0106042v2, Eq. (31) and the text after it:

${\sigma }_{ABC}=3-(I_1+I_2+I_3)I_4$ and numerical results suggest that it is an EM. … For the rest of the paper I will assume that ${\sigma }_{ABC}$ is an EM.

For $\psi =\sum t_{ijk}|ijk⟩$, the polynomial invariants are $P_{\sigma,\tau}=\sum t_{i_1j_1k_1}\cdots t_{i_nj_nk_n}, \bar t_{i_1j_{\sigma(1)}k_{\tau(1)}}\cdots\bar t_{i_nj_{\sigma(n)}k_{\tau(n)}}$ (Eq. (8)), with $I_1=P_{e,(12)}$, $I_2=P_{(12),e}$, $I_3=P_{(12),(12)}$ (the three one-qubit purities) and the Kempe invariant $I_4=P_{(123),(132)}$. An entanglement monotone (EM) in the paper’s sense satisfies $E(\psi )\ge {\sum }_k{p}_{k}E(A_k\psi /\sqrt{p_k})$ for every complete instrument $\{A_k\}$ on one party, $p_k=\Vert A_k\psi {\Vert }^2$.

Motivation

The paper proves that the monotones of a pure state determine its LU orbit, so for three qubits some monotone must depend on the Kempe invariant; the four known ones (the three one-versus-two tangles and the three-tangle) do not. ${\sigma }_{ABC}$ was proposed as that fifth monotone and used to bound conversion probabilities. The frozen declaration D5/S3/Quantum/Entanglement/GingrichKempeMonotoneRefutation.result shows that it is not a monotone.

Gap

Issue #12508 classifies the proposal as Tier 1 and records the checks made before any Lean:

• arXiv:quant-ph/0106042 v2 is the latest version and states the proposal with its numerical support (over 300,000 random states and operations) and the caveat that a small violating region might have been missed;
• Oreshkov–Brun, arXiv:quant-ph/0506181v6 (Phys. Rev. A 73, 042314 (2006)), §VI, state that “no rigorous proof of monotonicity was given” and construct a different, proved monotone depending on the Kempe invariant;
• Spedalieri, arXiv:quant-ph/0110179, §III, states that the bare Kempe invariant is not a monotone, which does not decide the combination $3-(I_1+I_2+I_3)I_4$; other papers use the name $\sigma$ for different functions (Emary–Beenakker, quant-ph/0311105; Lévay, quant-ph/0403060).

These are seat-reported and orchestrator-reported readings, not-found-in-searched-scope; they do not establish exhaustive worldwide novelty.

Route

1. $\psi =(5|000⟩+5|011⟩+2|110⟩)/(3\sqrt{6})$ and the instrument $K_0=\mathrm{diag}(3/5,0)$, $K_1=\mathrm{diag}(4/5,1)$ on qubit $A$, with $K_0^{†}{K}_0+K_1^{†}{K}_1=I$. Then $p_0=1/3$ with ${\varphi }_0=(|000⟩+|011⟩)/\sqrt{2}$, and $p_1=2/3$ with ${\varphi }_1=(2|000⟩+2|011⟩+|110⟩)/3$.
2. For $a|000⟩+b|011⟩+c|110⟩$, expanding Eq. (8) gives $I_1=(x+z{)}^2+y^2$, $I_2=x^2+(y+z{)}^2$, $I_3=(x+y{)}^2+z^2$ and $I_4=x^3+y^3+z^3+3xyz$, with $x=|a{|}^2$, $y=|b{|}^2$, $z=|c{|}^2$; diagonal operators on qubit $A$ and scalars keep this support.
3. Hence ${\sigma }_{ABC}(\psi )=8097475/3188646$, ${\sigma }_{ABC}({\varphi }_0)=5/2$ and ${\sigma }_{ABC}({\varphi }_1)=16792/6561$.
4. $p_0{\sigma }_{ABC}({\varphi }_0)+p_1{\sigma }_{ABC}({\varphi }_1)=8097813/3188646$ exceeds ${\sigma }_{ABC}(\psi )$ by $169/1594323$.

Falsifier

The kernel-checked result is the negation of the statement that for every three-qubit vector $\psi$ with ${\sum }_w|\psi (w){|}^2=1$ and every finite family $K_0,\dots ,K_{n-1}$ of $2\times 2$ complex matrices with ${\sum }_k{K}_k^{†}{K}_k=I$, acting on qubit $A$ through the frozen StabilizerPairLocalUnitaryInequivalence.localOp, the sum over $k$ of $p_k\mathrm{Re}{\sigma }_{ABC}(K_k\psi /\sqrt{p_k})$ (omitting the outcomes with $p_k=0$) is at most $\mathrm{Re}{\sigma }_{ABC}(\psi )$. The invariants are written as in Eq. (8), with $t_{ijk}=\psi (i,j,k)$, permutations in Equiv.Perm (Fin n) and finRotate 3 for $(123)$. The paper’s ${\sigma }_{ABC}$ is real valued, and $I_1,\dots ,I_4$ are real for every state ($I_4$ because conjugation exchanges $P_{(123),(132)}$ and $P_{(132),(123)}$, which coincide after relabeling the copies), so comparing real parts is the paper’s inequality. A complete instrument on one qubit is an admissible operation in the paper’s definition, so the failure refutes the proposal.

Evidence

Exact recomputation (Python rational arithmetic, issue #12508): the four invariants of $\psi$, ${\varphi }_0$ and ${\varphi }_1$ evaluated directly from Eq. (8), the outcome probabilities and the gap $169/1594323$; the same computation gives the gap $166036/352218537$ for the search seat’s first example $(4|000⟩+4|011⟩+|110⟩)/\sqrt{33}$ with $K_0=\mathrm{diag}(\sqrt{3}/2,0)$, $K_1=\mathrm{diag}(1/2,1)$. The literature seat recomputed both independently.

The canonical source is D5/S3/Quantum/Entanglement/GingrichKempeMonotoneRefutation.lean. Its public declarations are polyInvariant, sigmaABC, claim, psi, instrument and result; the operator on qubit $A$ is localOp of D5/S3/Quantum/Information/StabilizerPairLocalUnitaryInequivalence. The frozen module state has statement identity sha256:f88ba93e1f57e98e6ba5c8b4e46a75f8368f6997bab2d7b64e67c8ac942f3fc6. The result declaration has statement identity sha256:43b83312ba5b4895fe7531d1e06a7f9f5df119805d5ee5ecd3e9aeb20587c517. The Freeze event is sha256:ae01db2fdbce829658ccbecafe87ca4dcceb1f77f08d7af52bfebaeb019c65c3; its project-level frozen prerequisite is the Freeze event of D5/S3/Quantum/Information/StabilizerPairLocalUnitaryInequivalence. The proof uses only the standard axioms propext, Classical.choice and Quot.sound; no sorry, native_decide, or new axiom.

Triage

Tier 1 proposal of a 2002 journal article; resolution Refuted by D5/S3/Quantum/Entanglement/GingrichKempeMonotoneRefutation.result. proof_shape: bind-only (the settlement of the named proposal is the new content; no escape witness); admission_basis: open-problem-resolution (issue #12508). Utility kind certified-instance, basis refutes the module’s claim.

What the settlement shows

• Proved by result: the proposal fails: some normalized three-qubit state and some complete instrument on qubit $A$ raise the average of ${\sigma }_{ABC}$.
• Proved inside the proof of result: the closed forms of $I_1,\dots ,I_4$ on every vector supported on $|000\rangle,|011\rangle, |110\rangle$;forthewitness$\psi$ with $K_0=\mathrm{diag}(3/5,0)$, $K_1=\mathrm{diag}(4/5,1)$, the values ${\sigma }_{ABC}(\psi )$, ${\sigma }_{ABC}({\varphi }_0)$, ${\sigma }_{ABC}({\varphi }_1)$ above and the increase $169/1594323$.
• Mechanism (literature seat, 席位自报; checked by the orchestrator, not stated in Lean): on ${\psi }_u=\sqrt{(1-u)/2}(|000⟩+|011⟩)+\sqrt{u}|110⟩$ the closed forms give $\sigma_{ABC}(\psi_u)=F(u)=\tfrac52+\tfrac u2+ \tfrac{3u^2}4-\tfrac{9u^3}2+\tfrac{21u^4}4-\tfrac{9u^5}2$, and for $0<u<v<1$ a diagonal instrument on $A$ splits ${\psi }_u$ into ${\psi }_0$ and ${\psi }_v$ with probabilities $1-u/v$ and $u/v$. Monotonicity would make $F$ concave along these splittings, but $F^{\prime \prime }(0)=3/2>0$; the witness is $u=2/27$, $v=1/9$. The family is locally equivalent (by $X$ on $B$) to W-class states, the region the paper’s random sampling did not resolve.
• Follows from it (orchestrator argument, not stated in Lean): for the fixed instrument both outcome probabilities are positive, so the gap is continuous in $\psi$ and stays positive on an open set of states; replacing $K_0,K_1$ by $\mathrm{diag}(3/5,\epsilon )$, $\mathrm{diag}(4/5,\sqrt{1-{\epsilon }^2})$ with small $\epsilon >0$ gives invertible Kraus operators with the same strict increase. The failure is not confined to a set of measure zero, contrary to the possibility the paper allows (l. 599–602).
• Effect on the paper’s other conclusions (checked by reading the v2 source):
  • Refuted: the assumption that ${\sigma }_{ABC}$ is an EM (l. 589–599) and the proposal in the abstract (l. 22–23) of a form for the monotone depending on the Kempe invariant; the statement that ${\tau }_{(AB)C},{\tau }_{(AC)B},{\tau }_{(BC)A},{\tau }_{ABC},{\sigma }_{ABC}$ are five independent continuous EMs (§IV, l. 633–636).
  • Needs another justification: the use of the ratio ${\sigma }_{ABC}(\psi )/{\sigma }_{ABC}(\varphi )$ as an upper bound on conversion probabilities (§III, l. 604–627); the functions ${\upsilon }^{\pm }$ built from the $\tau$ and ${\sigma }_{ABC}$ (l. 659–662); and the minimum over $\mathcal E={\tau_{(AB)C},\tau_{(AC)B},\tau_{(BC)A},\tau_{ABC}, \sigma_{ABC},\upsilon^\pm}$ in §V (l. 692–694, l. 800–812), where ${\sigma }_{ABC}$ can be replaced by a proved monotone depending on the Kempe invariant, such as the one of Oreshkov–Brun. A failure of average monotonicity does not by itself exhibit a pair of states violating the ratio bound: in the witness each term $p_k{\sigma }_{ABC}({\varphi }_k)$ is below ${\sigma }_{ABC}(\psi )$.
  • Unaffected: Theorem 1 (the EMs determine the LU orbit, l. 532–579) and its consequence that five independent continuous EMs, one of them depending on $I_4$, exist (l. 580–588); the four $\tau$ monotones; the existence arguments for EMs sensitive to $I_6$ and for an EM distinguishing the GHZ and W classes (§IV); Theorem 2 on $f$-type functions (§V).
• Not formalized here: the existence question the paper raises, an EM depending on the Kempe invariant, is answered by the proved monotone of Oreshkov–Brun (their Eq. (79)).
• Open here: the extent of the violating region of ${\sigma }_{ABC}$ beyond the W-class family.

ASSUMED-UNVERIFIED

The literature checks do not establish worldwide priority, and a complete forward-citation graph was not obtained; the citing-work readings are seat-reported. The Lean kernel verifies the encoded statement and its axiom closure; its correspondence to the paper, including the permutation conventions and the restriction to instruments on qubit $A$, is checked by reading the source and the definitions.