bibkey: gingrich2002properties authors: Robert M. Gingrich year: 2002 title: “Properties of entanglement monotones for three-qubit pure states” doi: 10.1103/PhysRevA.65.052302 url: https://arxiv.org/abs/quant-ph/0106042v2 claim: “For a three-qubit pure state sum t_ijk |ijk>, the polynomial invariants are P_{sigma,tau} = sum t_{i_1 j_1 k_1} … t_{i_n j_n k_n} conj(t_{i_1 j_sigma(1) k_tau(1)}) … conj(t_{i_n j_sigma(n) k_tau(n)}) over permutations sigma, tau of n elements (Eq. (8)), with I_1 = P_{e,(12)}, I_2 = P_{(12),e}, I_3 = P_{(12),(12)} and the Kempe invariant I_4 = P_{(123),(132)}. The paper proposes sigma_ABC = 3 - (I_1 + I_2 + I_3) I_4 as a fifth entanglement monotone, a function E with E(psi) >= sum_k p_k E(A_k psi / sqrt(p_k)) for every complete local instrument A_k on one party, states that numerical results suggest it is one, and assumes it for the rest of the paper.” strata_touched:

• D5/S3/Quantum/Entanglement/GingrichKempeMonotoneRefutation license: citation-only triage: anchor

Properties of entanglement monotones for three-qubit pure states

Robert M. Gingrich, Phys. Rev. A 65, 052302 (2002); arXiv:quant-ph/0106042v2. Quotations are from the arXiv v2 source, with its macros kept as printed (\sstate is $|\psi ⟩$).

The definition of an entanglement monotone (Eq. (5)), for Kraus operators with ${\sum }_k{A}_k^{(i)†}{A}_k^{(i)}={\mathcal{I}}_i$:

A (non-increasing) EM is a real valued function E \left( \sstate \right) such that E \left( \sstate \right) \ge \sum_{k} p_k E \left( \frac{I_{1} \otimes \ldots \otimes A_{k}^{(i)} \otimes \ldots \otimes I_{n} \sstate}{\sqrt{p_{k}}} \right) for any state \sstate, operation $A_k^{(i)}$, and space $i$ where p_k = \| I_{1} \otimes \ldots \otimes A_{k}^{(i)} \otimes \ldots \otimes I_{n} \sstate \|^2 .

The polynomial invariants (Eq. (8)):

P_{\sigma,\tau} \left( \sstate \right) = \sum t_{i_1 j_1 k_1} \ldots t_{i_n j_n k_n} \bar{t}_{i_1 j_{\sigma (1)} k_{\tau (1)}} \ldots \bar{t}_{i_n j_{\sigma (n)} k_{\tau (n)}} where $\sigma$ and $\tau$ are permutations on $n$ elements, repeated indices are summed and $\bar{t}$ stands for the complex conjugate of $t$

The invariants used:

$I_1=P_{e,(12)}$, $I_2=P_{(12),e}$, $I_3=P_{(12),(12)}$, $I_4=P_{(123),(132)}$

The proposal (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. After generating over 300,000 random states and applying a random operation to each of them the inequality in equation (\ref{moninequality}) was never violated by ${\sigma }_{ABC}$.

For the rest of the paper I will assume that ${\sigma }_{ABC}$ is an EM.

The paper proves that five independent continuous monotones must exist and uses ${\sigma }_{ABC}$ as the candidate depending on $I_4$, in a bound on conversion probabilities (§III) and in the discussion of further monotones (§IV–§V). Oreshkov and Brun, arXiv:quant-ph/0506181v6 (Phys. Rev. A 73, 042314 (2006)), §VI, record that “no rigorous proof of monotonicity was given” for ${\sigma }_{ABC}$ and construct a different, proved monotone.

The encoding takes three-qubit vectors as functions $(\mathrm{Fin}3\to \mathrm{Fin}2)\to \mathbb{C}$, qubits $A,B,C$ at the indices $0,1,2$ and $t_{ijk}=\psi (i,j,k)$; the permutations are Equiv.Perm (Fin n), with finRotate 3 for $(123)$; and a $2\times 2$ matrix $K$ acts on qubit $A$ as the existing localOp 0 K. Since the paper’s ${\sigma }_{ABC}$ is real valued, the statement compares real parts.

Verified locator

• DOI: https://doi.org/10.1103/PhysRevA.65.052302 (Phys. Rev. A 65, 052302 (2002)).
• URL: https://arxiv.org/abs/quant-ph/0106042v2 (v2, 2001-07-04, the latest version; source entpaper2.tex, md5 16f2785916409ec7b0283ede7f655d19): the definition of an entanglement monotone (l. 160–179), Eq. (8) (l. 202–216), the invariants $I_1,\dots ,I_4$ (l. 236–239), the proposal and its numerical support (l. 589–603) and the bound on conversion probabilities that uses it (l. 604–627).