The residual-correlation sum of four qubits is not an entanglement monotone

Abstract

Bai, Yang and Wang (arXiv:quant-ph/0703098, Phys. Rev. A 76, 022336) conjecture that the sum of the residual correlations M = sum_k tau_k - 2 sum_{p<q} C_pq^2 of a four-qubit pure state, built from the one-qubit linear entropies and Wootters’ concurrences of the pairs, is an entanglement monotone. It is not: a two-outcome diagonal measurement on one qubit of the state (20|0001> + 2|1000> + 6|1011> + |1110>)/21 raises the average of M from 7552/194481 to 3528832/85766121.

Definition 1.1 (Linear entropy).

$$\forall k:\mathrm{Fin}\left(4\right),\forall \psi :(\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(2\right))\to \mathbb{C},\mathrm{linearEntropy}\left(k,\psi \right)=2\cdot (1-\mathrm{ReTr}\left(\mathrm{reducedState}\left(\{k\},\psi \right)\cdot \mathrm{reducedState}\left(\{k\},\psi \right)\right))$$

Formalization. D5/S3/Quantum/Entanglement/FourQubitResidualSumMonotoneRefutation.linearEntropy (✓ std3).

Citation. Yan-Kui Bai; Dong Yang; Z. D. Wang (2007). Multipartite quantum correlation and entanglement in four-qubit pure states. DOI: 10.1103/PhysRevA.76.022336. URL: https://arxiv.org/abs/quant-ph/0703098v2.

Commentary.

For a four-qubit vector psi, a function from the configurations Fin 4 -> Fin 2 to the complex numbers with the qubits A, B, C, D at the indices 0, 1, 2, 3, the reduced state of the qubit k is the existing reducedState, the partial trace of the projector onto psi over the other three qubits. The linear entropy of the qubit k is tau_k = 2 (1 - Tr rho_k^2).

Definition 1.2 (Concurrence of a pair).

$$\forall p:\mathrm{Fin}\left(4\right),\forall q:\mathrm{Fin}\left(4\right),\forall \psi :(\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(2\right))\to \mathbb{C},\forall \rho :\mathrm{Matrix}\left((\{p,q\}\to \mathrm{Fin}\left(2\right)),(\{p,q\}\to \mathrm{Fin}\left(2\right)),\mathbb{C}\right),\forall l:\mathrm{List}\left(\mathbb{R}\right),(\rho =\mathrm{reducedState}\left(\{p,q\},\psi \right))\Rightarrow ((l=\mathrm{sortDesc}\left(\mathrm{map}\left(\mathrm{re},\mathrm{roots}\left(\mathrm{charpoly}\left(\rho \cdot \mathrm{timeReversed}\left(\{p,q\},\rho \right)\right)\right)\right)\right))\Rightarrow (\mathrm{concurrence}\left(p,q,\psi \right)=\max (\sqrt{\mathrm{getD}\left(l,0,0\right)}-\sqrt{\mathrm{getD}\left(l,1,0\right)}-\sqrt{\mathrm{getD}\left(l,2,0\right)}-\sqrt{\mathrm{getD}\left(l,3,0\right)},0)))$$

Formalization. D5/S3/Quantum/Entanglement/FourQubitResidualSumMonotoneRefutation.concurrence (✓ std3).

Citation. Yan-Kui Bai; Dong Yang; Z. D. Wang (2007). Multipartite quantum correlation and entanglement in four-qubit pure states. DOI: 10.1103/PhysRevA.76.022336. URL: https://arxiv.org/abs/quant-ph/0703098v2.

Commentary.

For two qubits p and q, rho is the reduced state of the pair {p, q} (the existing reducedState) and the existing timeReversed applies sigma_y to each qubit of the pair after complex conjugation, so that rho timeReversed(rho) is the matrix rho_pq (sigma_y x sigma_y) rho_pq^* (sigma_y x sigma_y) of Wootters’ formula. The list l consists of the real parts of the roots of its characteristic polynomial, with multiplicity, sorted decreasingly; getD(l, i, 0) is its entry at position i, or 0 when the list is shorter. The concurrence is max(sqrt(l_0) - sqrt(l_1) - sqrt(l_2) - sqrt(l_3), 0); for p different from q the four entries are the eigenvalues lambda_1 >= lambda_2 >= lambda_3 >= lambda_4 of the paper.

Definition 1.3 (The sum of the residual correlations).

$$\forall \psi :(\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(2\right))\to \mathbb{C},\mathrm{residualSum}\left(\psi \right)=\sum _k\mathrm{linearEntropy}\left(k,\psi \right)-2\cdot \sum _{p<q}\mathrm{concurrence}{\left(p,q,\psi \right)}^2$$

Formalization. D5/S3/Quantum/Entanglement/FourQubitResidualSumMonotoneRefutation.residualSum (✓ std3).

Citation. Yan-Kui Bai; Dong Yang; Z. D. Wang (2007). Multipartite quantum correlation and entanglement in four-qubit pure states. DOI: 10.1103/PhysRevA.76.022336. URL: https://arxiv.org/abs/quant-ph/0703098v2.

Commentary.

Eq. (7) of the paper: M = sum_k tau_k - 2 sum_{p>q} C_pq^2 over the four qubits and the six pairs; each pair is counted once, here as p < q.

Definition 1.4 (The conjectured monotonicity).

$$(claim)\iff (\forall \psi :(\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(2\right))\to \mathbb{C},(\sum _w{\Vert \psi \left(w\right)\Vert }^2=1)\Rightarrow (\forall n:\mathbb{N},\forall K:\mathrm{Fin}\left(n\right)\to \mathrm{Matrix}\left(\mathrm{Fin}\left(2\right),\mathrm{Fin}\left(2\right),\mathbb{C}\right),(\sum _j{\left(K_j\right)}^H\cdot K_j=1)\Rightarrow (\forall p:\mathrm{Fin}\left(n\right)\to \mathbb{R},(\forall j:\mathrm{Fin}\left(n\right),p_j=\sum _w{\Vert (\mathrm{localOp}\left(0,K_j\right)\cdot \psi )\left(w\right)\Vert }^2)\Rightarrow (\sum _{j:p_j\ne 0}{p}_j\cdot \mathrm{residualSum}\left(\frac{1}{\sqrt{p_j}}\cdot (\mathrm{localOp}\left(0,K_j\right)\cdot \psi )\right)\le \mathrm{residualSum}\left(\psi \right)))))$$

Formalization. D5/S3/Quantum/Entanglement/FourQubitResidualSumMonotoneRefutation.claim (✓ std3).

Citation. Yan-Kui Bai; Dong Yang; Z. D. Wang (2007). Multipartite quantum correlation and entanglement in four-qubit pure states. DOI: 10.1103/PhysRevA.76.022336. URL: https://arxiv.org/abs/quant-ph/0703098v2.

Commentary.

An entanglement monotone does not increase on average under LOCC. A complete instrument K_0, …, K_{n-1} on qubit A, with sum_j K_j^dagger K_j = I, is a one-step LOCC protocol; K_j acts on qubit A (index 0) through the existing localOp, the product operator with factor K_j at qubit 0 and the identity elsewhere, applied to psi by matrix-vector multiplication: the outcome j occurs with probability p_j = ||K_j psi||^2 (the weights p of the display) and leaves the normalized state K_j psi / sqrt(p_j). The displayed statement asserts sum_j p_j M(K_j psi / sqrt(p_j)) <= M(psi) for every normalized four-qubit vector and every such instrument, omitting the outcomes with p_j = 0; it is a consequence of the conjecture that M is an entanglement monotone.

Definition 1.5 (Coordinates of a pair).

$$\forall p,q:\mathrm{Fin}\left(4\right),(p\ne q)\Rightarrow (\forall x:(\{p,q\}\to \mathrm{Fin}\left(2\right)),\mathrm{pairEquiv}\left(p,q,x\right)=(x\left(p\right),x\left(q\right)))$$

Formalization. D5/S3/Quantum/Entanglement/FourQubitResidualSumMonotoneRefutation.pairEquiv (✓ std3).

Source. Repository-derived.

Acknowledgement. Yan-Kui Bai; Dong Yang; Z. D. Wang (2007). Multipartite quantum correlation and entanglement in four-qubit pure states. DOI: 10.1103/PhysRevA.76.022336. URL: https://arxiv.org/abs/quant-ph/0703098v2.

Commentary.

For distinct qubits p and q, pairEquiv reads a configuration x of the pair {p, q} as the ordered pair (x(p), x(q)) in Fin 2 x Fin 2.

Definition 1.6 (Coordinates outside a pair).

$$\forall p,q,r,s:\mathrm{Fin}\left(4\right),(((\neg (r\in \{p,q\}))\wedge (\neg (s\in \{p,q\})))\wedge ((r\ne s)\wedge (\forall i:\mathrm{Fin}\left(4\right),(\neg (i\in \{p,q\}))\Rightarrow (i=r\vee i=s))))\Rightarrow (\forall z:(\mathrm{Outside}\left(\{p,q\}\right)\to \mathrm{Fin}\left(2\right)),\mathrm{outEquiv}\left(p,q,r,s,z\right)=(z\left(r\right),z\left(s\right)))$$

Formalization. D5/S3/Quantum/Entanglement/FourQubitResidualSumMonotoneRefutation.outEquiv (✓ std3).

Source. Repository-derived.

Acknowledgement. Yan-Kui Bai; Dong Yang; Z. D. Wang (2007). Multipartite quantum correlation and entanglement in four-qubit pure states. DOI: 10.1103/PhysRevA.76.022336. URL: https://arxiv.org/abs/quant-ph/0703098v2.

Commentary.

When the qubits r and s are exactly the two qubits outside {p, q}, outEquiv reads a configuration z of those qubits as (z(r), z(s)).

Definition 1.7 (Coordinate of one qubit).

$$\forall k:\mathrm{Fin}\left(4\right),\forall x:(\{k\}\to \mathrm{Fin}\left(2\right)),\mathrm{singleEquiv}\left(k,x\right)=x\left(k\right)$$

Formalization. D5/S3/Quantum/Entanglement/FourQubitResidualSumMonotoneRefutation.singleEquiv (✓ std3).

Source. Repository-derived.

Acknowledgement. Yan-Kui Bai; Dong Yang; Z. D. Wang (2007). Multipartite quantum correlation and entanglement in four-qubit pure states. DOI: 10.1103/PhysRevA.76.022336. URL: https://arxiv.org/abs/quant-ph/0703098v2.

Commentary.

singleEquiv reads a configuration x of the single qubit k as its value x(k) in Fin 2.

Definition 1.8 (Coordinates outside one qubit).

$$\forall k,r,s,t:\mathrm{Fin}\left(4\right),(((((\neg (r\in \{k\}))\wedge (\neg (s\in \{k\})))\wedge (\neg (t\in \{k\})))\wedge (((r\ne s)\wedge (r\ne t))\wedge (s\ne t)))\wedge (\forall i:\mathrm{Fin}\left(4\right),(\neg (i\in \{k\}))\Rightarrow (\left(i=r\vee i=s\right)\vee i=t)))\Rightarrow (\forall z:(\mathrm{Outside}\left(\{k\}\right)\to \mathrm{Fin}\left(2\right)),out3Equiv\left(k,r,s,t,z\right)=(z\left(r\right),z\left(s\right),z\left(t\right)))$$

Formalization. D5/S3/Quantum/Entanglement/FourQubitResidualSumMonotoneRefutation.out3Equiv (✓ std3).

Source. Repository-derived.

Acknowledgement. Yan-Kui Bai; Dong Yang; Z. D. Wang (2007). Multipartite quantum correlation and entanglement in four-qubit pure states. DOI: 10.1103/PhysRevA.76.022336. URL: https://arxiv.org/abs/quant-ph/0703098v2.

Commentary.

When the qubits r, s and t are exactly the three qubits other than k, out3Equiv reads a configuration z of those qubits as (z(r), z(s), z(t)).

Definition 1.9 (Coordinates of four qubits).

$$\forall w:(\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(2\right)),\mathrm{fourEquiv}\left(w\right)=(w\left(0\right),w\left(1\right),w\left(2\right),w\left(3\right))$$

Formalization. D5/S3/Quantum/Entanglement/FourQubitResidualSumMonotoneRefutation.fourEquiv (✓ std3).

Source. Repository-derived.

Acknowledgement. Yan-Kui Bai; Dong Yang; Z. D. Wang (2007). Multipartite quantum correlation and entanglement in four-qubit pure states. DOI: 10.1103/PhysRevA.76.022336. URL: https://arxiv.org/abs/quant-ph/0703098v2.

Commentary.

fourEquiv reads a configuration w of the four qubits as (w(0), w(1), w(2), w(3)).

Definition 1.10 (The counterexample state).

$$\psi =\frac{1}{21}\cdot (20\cdot |0001⟩+2\cdot |1000⟩+6\cdot |1011⟩+|1110⟩)$$

Formalization. D5/S3/Quantum/Entanglement/FourQubitResidualSumMonotoneRefutation.psi (✓ std3).

Source. Repository-derived.

Acknowledgement. Yan-Kui Bai; Dong Yang; Z. D. Wang (2007). Multipartite quantum correlation and entanglement in four-qubit pure states. DOI: 10.1103/PhysRevA.76.022336. URL: https://arxiv.org/abs/quant-ph/0703098v2.

Commentary.

The state is (20|0001> + 2|1000> + 6|1011> + |1110>)/21, with norm one since 400 + 4 + 36 + 1 = 441.

Definition 1.11 (The measurement on qubit A).

$$K_0=\mathrm{diag}\left(\frac{21}{29},0\right),K_1=\mathrm{diag}\left(\frac{20}{29},1\right)$$

Formalization. D5/S3/Quantum/Entanglement/FourQubitResidualSumMonotoneRefutation.instrument (✓ std3).

Source. Repository-derived.

Acknowledgement. Yan-Kui Bai; Dong Yang; Z. D. Wang (2007). Multipartite quantum correlation and entanglement in four-qubit pure states. DOI: 10.1103/PhysRevA.76.022336. URL: https://arxiv.org/abs/quant-ph/0703098v2.

Commentary.

The two outcomes are K_0 = diag(21/29, 0) and K_1 = diag(20/29, 1) on qubit A, with K_0^dagger K_0 + K_1^dagger K_1 = I.

Theorem 1.12 (The residual-correlation sum increases on average).

$$\neg claim$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Entanglement/FourQubitResidualSumMonotoneRefutation.result (✓ std3). ∎

Resolves. Problems/bai-2007-residual-sum-monotone (refuted) by D5/S3/Quantum/Entanglement/FourQubitResidualSumMonotoneRefutation.result.

Source. Repository-derived.

Acknowledgement. Yan-Kui Bai; Dong Yang; Z. D. Wang (2007). Multipartite quantum correlation and entanglement in four-qubit pure states. DOI: 10.1103/PhysRevA.76.022336. URL: https://arxiv.org/abs/quant-ph/0703098v2.

Commentary.

The outcome K_0 has probability 400/841 and leaves the product state |0001>, so its M is zero. The outcome K_1 has probability 441/841 and leaves (400|0001> + 58|1000> + 174|1011> + 29|1110>)/441. Every two-qubit reduced state of the three states is a real X matrix, with diagonal u, v, r, s in the basis 00, 01, 10, 11, an entry w between 00 and 11 and an entry z between 01 and 10; after reindexing the reduced state and sigma_y x sigma_y to Fin 2 x Fin 2, which keeps the characteristic polynomial, the characteristic polynomial factors as (X^2 - 2(us + w^2)X + (us - w^2)^2)(X^2 - 2(vr + z^2)X + (vr - z^2)^2), with roots (sqrt(us) + |w|)^2, (sqrt(us) - |w|)^2, (sqrt(vr) + |z|)^2 and (sqrt(vr) - |z|)^2, and sorting them gives the concurrence. The concurrences of the pairs AB, AC, AD, BC, BD, CD are (0, 240, 80, 4, 12, 0)/441 for the state and (0, 139200, 46400, 3364, 10092, 0)/194481 for the second outcome; with the linear entropies, M of the state is 7552/194481 and M of the second outcome is 2967747712/37822859361. The average of M after the measurement is 3528832/85766121, which exceeds 7552/194481 by 198400/85766121.

References

• Truth anchor: D5/S3/Quantum/Entanglement/FourQubitResidualSumMonotoneRefutation.claim
• Truth anchor: D5/S3/Quantum/Entanglement/FourQubitResidualSumMonotoneRefutation.concurrence
• Truth anchor: D5/S3/Quantum/Entanglement/FourQubitResidualSumMonotoneRefutation.fourEquiv
• Truth anchor: D5/S3/Quantum/Entanglement/FourQubitResidualSumMonotoneRefutation.instrument
• Truth anchor: D5/S3/Quantum/Entanglement/FourQubitResidualSumMonotoneRefutation.linearEntropy
• Truth anchor: D5/S3/Quantum/Entanglement/FourQubitResidualSumMonotoneRefutation.out3Equiv
• Truth anchor: D5/S3/Quantum/Entanglement/FourQubitResidualSumMonotoneRefutation.outEquiv
• Truth anchor: D5/S3/Quantum/Entanglement/FourQubitResidualSumMonotoneRefutation.pairEquiv
• Truth anchor: D5/S3/Quantum/Entanglement/FourQubitResidualSumMonotoneRefutation.psi
• Truth anchor: D5/S3/Quantum/Entanglement/FourQubitResidualSumMonotoneRefutation.residualSum
• Truth anchor: D5/S3/Quantum/Entanglement/FourQubitResidualSumMonotoneRefutation.result
• Truth anchor: D5/S3/Quantum/Entanglement/FourQubitResidualSumMonotoneRefutation.singleEquiv
• Dependency: D5/S3/Quantum/Entanglement/PurityTimeReversalOverlapMinimum
• Dependency: D5/S3/Quantum/Information/StabilizerPairLocalUnitaryInequivalence