Log-negativity capacity depends on the input resource

Abstract

A two-qutrit CPTP map separates assisted and separable-input log-negativity capacities.

Definition 1.1 (Free local ancillary systems).

$$\forall (dA:\mathbb{N}),\forall (dB:\mathbb{N}),\forall (r:\mathbb{N}),\forall (K:\mathrm{Fin}\left(r\right)\to \mathrm{Matrix}\left((\mathrm{Fin}\left(dA\right)\times \mathrm{Fin}\left(dB\right)),(\mathrm{Fin}\left(dA\right)\times \mathrm{Fin}\left(dB\right)),\mathbb{C}\right)),\forall (a:\mathbb{N}),\forall (b:\mathbb{N}),\forall (rho:\mathrm{Matrix}\left(((\mathrm{Fin}\left(dA\right)\times \mathrm{Fin}\left(a\right))\times (\mathrm{Fin}\left(dB\right)\times \mathrm{Fin}\left(b\right))),((\mathrm{Fin}\left(dA\right)\times \mathrm{Fin}\left(a\right))\times (\mathrm{Fin}\left(dB\right)\times \mathrm{Fin}\left(b\right))),\mathbb{C}\right)),\mathrm{extend}\left(K,rho\right)={D5.S3.Quantum.Foundation.FiniteKrausChannel.PhyslibLeaf.MatrixMap.of}_{kraus}\left(\lambda (k:\mathrm{Fin}\left(r\right))\mapsto \mathrm{reindex}\left(\mathrm{prodProdProdComm}\left(\mathrm{Fin}\left(dA\right),\mathrm{Fin}\left(dB\right),\mathrm{Fin}\left(a\right),\mathrm{Fin}\left(b\right)\right),\mathrm{prodProdProdComm}\left(\mathrm{Fin}\left(dA\right),\mathrm{Fin}\left(dB\right),\mathrm{Fin}\left(a\right),\mathrm{Fin}\left(b\right)\right),\mathrm{kronecker}\left(K\left(k\right),(1:\mathrm{Matrix}\left((\mathrm{Fin}\left(a\right)\times \mathrm{Fin}\left(b\right)),(\mathrm{Fin}\left(a\right)\times \mathrm{Fin}\left(b\right)),\mathbb{C}\right))\right)\right),\lambda (k:\mathrm{Fin}\left(r\right))\mapsto \mathrm{reindex}\left(\mathrm{prodProdProdComm}\left(\mathrm{Fin}\left(dA\right),\mathrm{Fin}\left(dB\right),\mathrm{Fin}\left(a\right),\mathrm{Fin}\left(b\right)\right),\mathrm{prodProdProdComm}\left(\mathrm{Fin}\left(dA\right),\mathrm{Fin}\left(dB\right),\mathrm{Fin}\left(a\right),\mathrm{Fin}\left(b\right)\right),\mathrm{kronecker}\left(K\left(k\right),(1:\mathrm{Matrix}\left((\mathrm{Fin}\left(a\right)\times \mathrm{Fin}\left(b\right)),(\mathrm{Fin}\left(a\right)\times \mathrm{Fin}\left(b\right)),\mathbb{C}\right))\right)\right)\right)\left(rho\right)$$

Formalization. D5/S3/Quantum/QuantumChannels/LogNegativityCapacityResourceDependence.extend (✓ std3).

Citation. Earl T. Campbell (2010). Optimal entangling capacity of dynamical processes. DOI: 10.1103/PhysRevA.82.042314. URL: https://arxiv.org/abs/1007.1445.

Commentary.

Section I, p. 2, verbatim: “Throughout, we assume that local ancilla (typically labelled as Hilbert space Ha′ ⊗ Hb′ ), local unitaries and classical communication are freely available as is appropriate in many communication and distributed computation scenarios.” The finite Kraus action is D5.S3.Quantum.Foundation.FiniteKrausChannel.PhyslibLeaf.MatrixMap.of_kraus applied to the two displayed families and rho. It uses identity Kraus factors on the ancillary pair, with Matrix.reindex ordering the cut as AA’ versus BB’.

Definition 1.2 (Separable-input entangling capacity).

$$\forall (dA:\mathbb{N}),\forall (dB:\mathbb{N}),\forall (r:\mathbb{N}),\forall (K:\mathrm{Fin}\left(r\right)\to \mathrm{Matrix}\left((\mathrm{Fin}\left(dA\right)\times \mathrm{Fin}\left(dB\right)),(\mathrm{Fin}\left(dA\right)\times \mathrm{Fin}\left(dB\right)),\mathbb{C}\right)),\mathrm{capSep}\left(K\right)={\mathrm{iSup}}_{a:\mathbb{N}}({\mathrm{iSup}}_{b:\mathbb{N}}({\mathrm{iSup}}_{rho:\mathrm{Matrix}\left(((\mathrm{Fin}\left(dA\right)\times \mathrm{Fin}\left(a\right))\times (\mathrm{Fin}\left(dB\right)\times \mathrm{Fin}\left(b\right))),((\mathrm{Fin}\left(dA\right)\times \mathrm{Fin}\left(a\right))\times (\mathrm{Fin}\left(dB\right)\times \mathrm{Fin}\left(b\right))),\mathbb{C}\right)}({\mathrm{iSup}}_{((\mathrm{IsDensity}\left(rho\right))\wedge (\mathrm{IsSeparable}\left(rho\right)))}((\mathrm{logb}\left(2,\mathrm{traceNorm}\left(\mathrm{partialTranspose}\left(\mathrm{extend}\left(K,rho\right)\right)\right)\right):\mathbb{R}):EReal))))$$

Formalization. D5/S3/Quantum/QuantumChannels/LogNegativityCapacityResourceDependence.capSep (✓ std3).

Citation. Earl T. Campbell (2010). Optimal entangling capacity of dynamical processes. DOI: 10.1103/PhysRevA.82.042314. URL: https://arxiv.org/abs/1007.1445.

Commentary.

Section I, Eq. (1), p. 2, verbatim: “The maximum achievable entanglement for a dynamical map, Λ, when the initial state is separable, is given by:” The formula is E(Λ) = sup{E(Λ(σ))|σ ∈ S}. Eqs. (4)–(5) and the logarithmic rescaling use Real.logb 2 of the partial-transpose trace norm. The supremum in EReal includes every finite local ancilla dimension. IsDensity is positive semidefinite with trace one, and IsSeparable is a finite convex mixture of products of such matrices.

Definition 1.3 (Assisted entangling capacity).

$$\forall (dA:\mathbb{N}),\forall (dB:\mathbb{N}),\forall (r:\mathbb{N}),\forall (K:\mathrm{Fin}\left(r\right)\to \mathrm{Matrix}\left((\mathrm{Fin}\left(dA\right)\times \mathrm{Fin}\left(dB\right)),(\mathrm{Fin}\left(dA\right)\times \mathrm{Fin}\left(dB\right)),\mathbb{C}\right)),\mathrm{capAssisted}\left(K\right)={\mathrm{iSup}}_{a:\mathbb{N}}({\mathrm{iSup}}_{b:\mathbb{N}}({\mathrm{iSup}}_{rho:\mathrm{Matrix}\left(((\mathrm{Fin}\left(dA\right)\times \mathrm{Fin}\left(a\right))\times (\mathrm{Fin}\left(dB\right)\times \mathrm{Fin}\left(b\right))),((\mathrm{Fin}\left(dA\right)\times \mathrm{Fin}\left(a\right))\times (\mathrm{Fin}\left(dB\right)\times \mathrm{Fin}\left(b\right))),\mathbb{C}\right)}({\mathrm{iSup}}_{(\mathrm{IsDensity}\left(rho\right))}((\mathrm{logb}\left(2,\mathrm{traceNorm}\left(\mathrm{partialTranspose}\left(\mathrm{extend}\left(K,rho\right)\right)\right)\right)-\mathrm{logb}\left(2,\mathrm{traceNorm}\left(\mathrm{partialTranspose}\left(rho\right)\right)\right):\mathbb{R}):EReal))))$$

Formalization. D5/S3/Quantum/QuantumChannels/LogNegativityCapacityResourceDependence.capAssisted (✓ std3).

Citation. Earl T. Campbell (2010). Optimal entangling capacity of dynamical processes. DOI: 10.1103/PhysRevA.82.042314. URL: https://arxiv.org/abs/1007.1445.

Commentary.

Section I, Eq. (2), p. 2, verbatim: “However, when prior entanglement is available, the maximum increase of entanglement is:” The formula is Eᴰ(Λ) = sup{E(Λ(σ)) − E(σ)|σ ∈ D}. Both logarithms have base two. No restriction to positive partial transpose is imposed on these inputs.

Definition 1.4 (The all-CPTP resource-independence question).

$$claim\iff \forall (dA:\mathbb{N}),\forall (dB:\mathbb{N}),\forall (r:\mathbb{N}),\forall (K:\mathrm{Fin}\left(r\right)\to \mathrm{Matrix}\left((\mathrm{Fin}\left(dA\right)\times \mathrm{Fin}\left(dB\right)),(\mathrm{Fin}\left(dA\right)\times \mathrm{Fin}\left(dB\right)),\mathbb{C}\right)),(\sum _{k:\mathrm{Fin}\left(r\right)}(\mathrm{conjTranspose}\left(K\left(k\right)\right)\cdot K\left(k\right))=(1:\mathrm{Matrix}\left((\mathrm{Fin}\left(dA\right)\times \mathrm{Fin}\left(dB\right)),(\mathrm{Fin}\left(dA\right)\times \mathrm{Fin}\left(dB\right)),\mathbb{C}\right)))\Rightarrow \mathrm{capAssisted}\left(K\right)=\mathrm{capSep}\left(K\right)$$

Formalization. D5/S3/Quantum/QuantumChannels/LogNegativityCapacityResourceDependence.claim (✓ std3).

Citation. Earl T. Campbell (2010). Optimal entangling capacity of dynamical processes. DOI: 10.1103/PhysRevA.82.042314. URL: https://arxiv.org/abs/1007.1445.

Commentary.

Section VII, p. 7, verbatim: “The most obvious open question is whether log-negativity EC is resource independent for all unitaries, or indeed all CPTP maps, or whether this is a singular feature of 2-qubit gates.” The assertion here is its all-CPTP clause. The dimensions dA and dB and the finite Kraus count r range over natural numbers. Completeness of the Kraus family states trace preservation. The two capacities include every finite local ancillary system.

Definition 1.5 (The symmetric Bell vector).

$$(symKet:(\mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right))\to \mathbb{C})=\mathrm{single}\left(((0,1):(\mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right))),(1:\mathbb{C})\right)+\mathrm{single}\left(((1,0):(\mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right))),(1:\mathbb{C})\right)$$

Formalization. D5/S3/Quantum/QuantumChannels/LogNegativityCapacityResourceDependence.symKet (✓ std3).

Source. Repository-derived.

Commentary.

The unnormalized vector has amplitude one at (0,1) and (1,0), and zero elsewhere in the two-qutrit space. Pi.single denotes a function supported at one index.

Definition 1.6 (The antisymmetric Bell vector).

$$(antiKet:(\mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right))\to \mathbb{C})=\mathrm{single}\left(((0,1):(\mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right))),(1:\mathbb{C})\right)-\mathrm{single}\left(((1,0):(\mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right))),(1:\mathbb{C})\right)$$

Formalization. D5/S3/Quantum/QuantumChannels/LogNegativityCapacityResourceDependence.antiKet (✓ std3).

Source. Repository-derived.

Commentary.

The unnormalized vector has amplitude one at (0,1), minus one at (1,0), and zero elsewhere in the two-qutrit space. The two Bell vectors give the off-diagonal terms in the positive and negative parts of the Bell state’s partial transpose.

Theorem 1.7 (Separable states have positive partial transpose).

$$\forall (A:Type),\forall (B:Type),\forall (finiteA:\mathrm{Fintype}\left(A\right)),\forall (finiteB:\mathrm{Fintype}\left(B\right)),\forall (eqA:\mathrm{DecidableEq}\left(A\right)),\forall (eqB:\mathrm{DecidableEq}\left(B\right)),\forall (sigma:\mathrm{Matrix}\left((A\times B),(A\times B),\mathbb{C}\right)),(\mathrm{IsSeparable}\left(sigma\right))\Rightarrow \mathrm{PosSemidef}\left(\lambda (i:(A\times B))\mapsto \lambda (j:(A\times B))\mapsto sigma\left((\mathrm{fst}\left(i\right),\mathrm{snd}\left(j\right)),(\mathrm{fst}\left(j\right),\mathrm{snd}\left(i\right))\right)\right)$$

Proof. Machine-checked in Lean as D5/S3/Quantum/QuantumChannels/LogNegativityCapacityResourceDependence.separable_ppt (✓ std3). ∎

Source. Repository-derived.

Commentary.

For arbitrary finite types A and B, a finite convex mixture of product density matrices remains positive semidefinite after transposing the B indices. The matrix entry at (a,b),(c,d) is sigma((a,d),(c,b)). Positivity follows by applying partial transpose to each product and adding the nonnegative weighted terms.

Theorem 1.8 (A CPTP separation).

$$\neg (\forall (dA:\mathbb{N}),\forall (dB:\mathbb{N}),\forall (r:\mathbb{N}),\forall (K:\mathrm{Fin}\left(r\right)\to \mathrm{Matrix}\left((\mathrm{Fin}\left(dA\right)\times \mathrm{Fin}\left(dB\right)),(\mathrm{Fin}\left(dA\right)\times \mathrm{Fin}\left(dB\right)),\mathbb{C}\right)),(\sum _{k:\mathrm{Fin}\left(r\right)}(\mathrm{conjTranspose}\left(K\left(k\right)\right)\cdot K\left(k\right))=(1:\mathrm{Matrix}\left((\mathrm{Fin}\left(dA\right)\times \mathrm{Fin}\left(dB\right)),(\mathrm{Fin}\left(dA\right)\times \mathrm{Fin}\left(dB\right)),\mathbb{C}\right)))\Rightarrow \mathrm{capAssisted}\left(K\right)=\mathrm{capSep}\left(K\right))$$

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

Resolves. Problems/campbell-2010-log-negativity-resource-independence (refuted) by D5/S3/Quantum/QuantumChannels/LogNegativityCapacityResourceDependence.result.

Source. Repository-derived.

Acknowledgement. Earl T. Campbell (2010). Optimal entangling capacity of dynamical processes. DOI: 10.1103/PhysRevA.82.042314. URL: https://arxiv.org/abs/1007.1445.

Acknowledgement. Charles H. Bennett, David P. DiVincenzo, Tal Mor, Peter W. Shor, John A. Smolin, Barbara M. Terhal (1999). Unextendible Product Bases and Bound Entanglement. DOI: 10.1103/PhysRevLett.82.5385. URL: https://arxiv.org/abs/quant-ph/9808030.

Commentary.

The Tiles complementary projector Q is positive under partial transpose. The input Q/4 therefore has zero log-negativity and gives output partial-transpose trace norm two. For a separable input with arbitrary finite local ancillas, the conditional ancillary matrices are positive under partial transpose; a rational product-overlap estimate bounds the output norm by 2 minus 1/6561. The assisted capacity is at least one, whereas the separable-input capacity is at most log base two of 2 minus 1/6561, strictly less than one. This does not decide the question for unitary channels.

References

• Truth anchor: D5/S3/Quantum/QuantumChannels/LogNegativityCapacityResourceDependence.antiKet
• Truth anchor: D5/S3/Quantum/QuantumChannels/LogNegativityCapacityResourceDependence.capAssisted
• Truth anchor: D5/S3/Quantum/QuantumChannels/LogNegativityCapacityResourceDependence.capSep
• Truth anchor: D5/S3/Quantum/QuantumChannels/LogNegativityCapacityResourceDependence.claim
• Truth anchor: D5/S3/Quantum/QuantumChannels/LogNegativityCapacityResourceDependence.extend
• Truth anchor: D5/S3/Quantum/QuantumChannels/LogNegativityCapacityResourceDependence.result
• Truth anchor: D5/S3/Quantum/QuantumChannels/LogNegativityCapacityResourceDependence.separable_ppt
• Truth anchor: D5/S3/Quantum/QuantumChannels/LogNegativityCapacityResourceDependence.symKet
• Dependency: D5/S3/Quantum/Entanglement/AbsolutePPT/QuadbitMaximumPurity
• Dependency: D5/S3/Quantum/Entanglement/GHZMeasureBiseparableBound
• Dependency: D5/S3/Quantum/Information/SLDSpectralEnergy
• Dependency: D5/S3/Quantum/Information/SeparableStateLocalUnitaryStabilizerObstruction
• Dependency: D5/S3/Quantum/QuantumChannels/IptPptSquaredEquivalence
• Dependency: D5/S3/QuantumChannels/SchurChannelMixedUnitaryDistance