Generalized concentratable entanglement is not monotone under enlarging the subsystem

Abstract

Liu, Knörzer, Wang and Tura (arXiv:2406.18517, Phys. Rev. Research 7, L032022) define the generalized concentratable entanglement C^{(K)}psi(s) = (1 - 2^{-|s|} sum{alpha in P(s)} Tr(rho_alpha^K)) / (K - 1) of an n-qubit pure state, with Tr(rho_emptyset^K) = 1, and conjecture that C^{(K)}_psi(s’) <= C^{(K)}_psi(s) whenever s’ is a subset of s, for every real K > 1. It fails at K = 5/2 for a four-qubit state with integer amplitudes: removing one qubit from {0, 1, 2} raises the value.

Definition 1.1 (The trace of a power of a reduced state).

$$\forall N:\mathbb{N},\forall K:\mathbb{R},\forall psi:(\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(2\right))\to \mathbb{C},(\mathrm{powerTrace}\left(K,psi,\varnothing \right)=1)\wedge (\forall alpha:\mathrm{Finset}\left(\mathrm{Fin}\left(N\right)\right),(alpha\ne \varnothing )\Rightarrow (\mathrm{powerTrace}\left(K,psi,alpha\right)=\mathrm{ReTr}\left(\mathrm{reducedState}{\left(alpha,psi\right)}^K\right)))$$

Formalization. D5/S3/Quantum/Entanglement/ConcentratableEntanglementSubsystemRefutation.powerTrace (✓ std3).

Citation. Xiaoyu Liu, Johannes Knörzer, Zherui Jerry Wang, Jordi Tura (2025). Generalized Concentratable Entanglement via Parallelized Permutation Tests. DOI: 10.1103/jtlj-qs3y. URL: https://arxiv.org/abs/2406.18517v1.

Commentary.

Tr(rho_alpha^K) is the real part of the trace of the K-th power, in the continuous functional calculus, of the existing reducedState of psi on the qubits alpha; the empty set contributes 1, as the paper takes Tr(rho_emptyset^K) = 1.

Definition 1.2 (Generalized concentratable entanglement).

$$\forall N:\mathbb{N},\forall K:\mathbb{R},\forall psi:(\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(2\right))\to \mathbb{C},\forall s:\mathrm{Finset}\left(\mathrm{Fin}\left(N\right)\right),\mathrm{gce}\left(K,psi,s\right)=\frac{1}{K-1}\cdot (1-\frac{1}{2^{|s|}}\cdot \sum _{alpha\subseteq s}\mathrm{powerTrace}\left(K,psi,alpha\right))$$

Formalization. D5/S3/Quantum/Entanglement/ConcentratableEntanglementSubsystemRefutation.gce (✓ std3).

Citation. Xiaoyu Liu, Johannes Knörzer, Zherui Jerry Wang, Jordi Tura (2025). Generalized Concentratable Entanglement via Parallelized Permutation Tests. DOI: 10.1103/jtlj-qs3y. URL: https://arxiv.org/abs/2406.18517v1.

Commentary.

Eq. (defeq): C^{(K)}psi(s) = (1 - 2^{-|s|} sum{alpha in P(s)} Tr(rho_alpha^K)) / (K - 1), the sum running over all subsets alpha of s.

Definition 1.3 (The conjectured monotonicity).

$$(claim)\iff (\forall N:\mathbb{N},\forall psi:(\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(2\right))\to \mathbb{C},(\sum _w{\Vert psi\left(w\right)\Vert }^2=1)\Rightarrow (\forall t,s:\mathrm{Finset}\left(\mathrm{Fin}\left(N\right)\right),(t\subseteq s)\Rightarrow (\forall K:\mathbb{R},(1<K)\Rightarrow (\mathrm{gce}\left(K,psi,t\right)\le \mathrm{gce}\left(K,psi,s\right)))))$$

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

Citation. Xiaoyu Liu, Johannes Knörzer, Zherui Jerry Wang, Jordi Tura (2025). Generalized Concentratable Entanglement via Parallelized Permutation Tests. DOI: 10.1103/jtlj-qs3y. URL: https://arxiv.org/abs/2406.18517v1.

Commentary.

Conjecture 1(1): for every number of qubits N, every normalized N-qubit vector psi, all subsets t of s of the qubits and every real K > 1, the value on t is at most the value on s.

Definition 1.4 (The amplitudes).

$$coeff=-1100\cdot |0000⟩-100\cdot |0101⟩-100\cdot |1010⟩+75\cdot |1100⟩-9\cdot |1111⟩$$

Formalization. D5/S3/Quantum/Entanglement/ConcentratableEntanglementSubsystemRefutation.coeff (✓ std3).

Source. Repository-derived.

Acknowledgement. Xiaoyu Liu, Johannes Knörzer, Zherui Jerry Wang, Jordi Tura (2025). Generalized Concentratable Entanglement via Parallelized Permutation Tests. DOI: 10.1103/jtlj-qs3y. URL: https://arxiv.org/abs/2406.18517v1.

Commentary.

The unnormalized amplitudes are -1100, -100, -100, 75 and -9 on the basis states |0000>, |0101>, |1010>, |1100> and |1111>; their squares sum to 1235706.

Definition 1.5 (The counterexample state).

$$\forall w:(\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(2\right)),psi\left(w\right)=\frac{coeff\left(w\right)}{\sqrt{1235706}}$$

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

Source. Repository-derived.

Acknowledgement. Xiaoyu Liu, Johannes Knörzer, Zherui Jerry Wang, Jordi Tura (2025). Generalized Concentratable Entanglement via Parallelized Permutation Tests. DOI: 10.1103/jtlj-qs3y. URL: https://arxiv.org/abs/2406.18517v1.

Commentary.

The state is the amplitude vector divided by sqrt(1235706).

Definition 1.6 (Coordinates of three qubits).

$$\forall x:(\{0,1,2\}\to \mathrm{Fin}\left(2\right)),\mathrm{tripleEquiv}\left(x\right)=(x\left(0\right),x\left(1\right),x\left(2\right))$$

Formalization. D5/S3/Quantum/Entanglement/ConcentratableEntanglementSubsystemRefutation.tripleEquiv (✓ std3).

Source. Repository-derived.

Acknowledgement. Xiaoyu Liu, Johannes Knörzer, Zherui Jerry Wang, Jordi Tura (2025). Generalized Concentratable Entanglement via Parallelized Permutation Tests. DOI: 10.1103/jtlj-qs3y. URL: https://arxiv.org/abs/2406.18517v1.

Commentary.

tripleEquiv reads a configuration x of the qubits {0, 1, 2} as (x(0), x(1), x(2)).

Definition 1.7 (Coordinate outside three qubits).

$$\forall z:(\mathrm{Outside}\left(\{0,1,2\}\right)\to \mathrm{Fin}\left(2\right)),out1Equiv\left(z\right)=z\left(3\right)$$

Formalization. D5/S3/Quantum/Entanglement/ConcentratableEntanglementSubsystemRefutation.out1Equiv (✓ std3).

Source. Repository-derived.

Acknowledgement. Xiaoyu Liu, Johannes Knörzer, Zherui Jerry Wang, Jordi Tura (2025). Generalized Concentratable Entanglement via Parallelized Permutation Tests. DOI: 10.1103/jtlj-qs3y. URL: https://arxiv.org/abs/2406.18517v1.

Commentary.

out1Equiv reads a configuration z of the qubit outside {0, 1, 2} as its value z(3).

Theorem 1.8 (Enlarging the subsystem can lower the value).

$$\neg claim$$

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

Resolves. Problems/liu-2025-gce-subsystem-monotonicity (refuted) by D5/S3/Quantum/Entanglement/ConcentratableEntanglementSubsystemRefutation.result.

Source. Repository-derived.

Acknowledgement. Xiaoyu Liu, Johannes Knörzer, Zherui Jerry Wang, Jordi Tura (2025). Generalized Concentratable Entanglement via Parallelized Permutation Tests. DOI: 10.1103/jtlj-qs3y. URL: https://arxiv.org/abs/2406.18517v1.

Commentary.

Take N = 4, K = 5/2, t = {0, 1} and s = {0, 1, 2}. The reduced states of psi, scaled by 1235706, are: on one qubit diag(1220000, 15706) for the qubits 0 and 1 and diag(1225625, 10081) for the qubit 2; on {0, 1}, {0, 2} and {1, 2} a direct sum of diagonal entries and one 2 x 2 block, with block determinants 9900^2, 100^2 and 10000^2; on {0, 1, 2} the rank-two matrix u u^T + w w^T with u and w orthogonal of squared norms 1225625 and 10081. Write Q = 1235706 rho for each of them. Each Q has an explicit positive semidefinite square root S, with S^2 = Q: square roots of the diagonal entries, (M + sqrt(det M) I) / sqrt(tr M + 2 sqrt(det M)) on a 2 x 2 block M, and u u^T / |u| + w w^T / |w|. Hence rho^{5/2} = (S / sqrt(1235706))^5 and Tr(rho^{5/2}) = Tr(Q^2 S) / 1235706^{5/2}. With T_alpha = Tr(rho_alpha^{5/2}), 8 (K - 1) (C(t) - C(s)) = T_2 + T_{02} + T_{12} + T_{012} - 1 - T_0 - T_1 - T_{01}; rational bounds for eight square roots show that it is about 5.77 * 10^{-7} > 0, so C^{(5/2)}({0, 1}) > C^{(5/2)}({0, 1, 2}).

References

• Truth anchor: D5/S3/Quantum/Entanglement/ConcentratableEntanglementSubsystemRefutation.claim
• Truth anchor: D5/S3/Quantum/Entanglement/ConcentratableEntanglementSubsystemRefutation.coeff
• Truth anchor: D5/S3/Quantum/Entanglement/ConcentratableEntanglementSubsystemRefutation.gce
• Truth anchor: D5/S3/Quantum/Entanglement/ConcentratableEntanglementSubsystemRefutation.out1Equiv
• Truth anchor: D5/S3/Quantum/Entanglement/ConcentratableEntanglementSubsystemRefutation.powerTrace
• Truth anchor: D5/S3/Quantum/Entanglement/ConcentratableEntanglementSubsystemRefutation.psi
• Truth anchor: D5/S3/Quantum/Entanglement/ConcentratableEntanglementSubsystemRefutation.result
• Truth anchor: D5/S3/Quantum/Entanglement/ConcentratableEntanglementSubsystemRefutation.tripleEquiv
• Dependency: D5/S3/Quantum/Entanglement/FourQubitResidualSumMonotoneRefutation