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The exact critical concurrence for universal quantum teleportation

Abstract

For every pure two-qubit input sqrt(a)|00> + sqrt(1 - a)|11> with 1/2 < a < 1, a qubit channel acting on Bob can make the shared state useful for universal quantum teleportation exactly when its initial concurrence exceeds sqrt(5 - 2 sqrt(3))/3. Necessity holds for every finite Kraus family, and a non-unital amplitude-damping channel attains sufficiency.

Definition 1.1 (The initial pure vector).

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

Citation. Arkaprabha Ghosal, Debarshi Das, Subhashish Banerjee (2021). Characterizing qubit channels in the context of quantum teleportation. DOI: 10.1103/PhysRevA.103.052422. URL: https://arxiv.org/abs/2102.02054v2.

Commentary.

Section V uses the vector sqrt(a)|00> + sqrt(1 - a)|11>. Alice is the first qubit and Bob the second. Its only nonzero coordinates for 0 < a < 1 are sqrt(a) at (0, 0) and sqrt(1 - a) at (1, 1), viewed as complex numbers. The definition is given for every real a using Real.sqrt.

Definition 1.2 (The initial two-qubit operator).

Formalization. D5/S3/Quantum/Entanglement/UniversalTeleportationCriticalConcurrence.initial (✓ std3).

Citation. Arkaprabha Ghosal, Debarshi Das, Subhashish Banerjee (2021). Characterizing qubit channels in the context of quantum teleportation. DOI: 10.1103/PhysRevA.103.052422. URL: https://arxiv.org/abs/2102.02054v2.

Commentary.

The initial operator is D5.S3.Quantum.PureState.PureStateHandshake.rankOneDensity applied to psi(a), the outer product |psi(a)><psi(a)| with pointwise complex conjugation. For 0 <= a <= 1 this is a rank-one density matrix.

Definition 1.3 (Failure to preserve the identity).

Formalization. D5/S3/Quantum/Entanglement/UniversalTeleportationCriticalConcurrence.NonUnital (✓ std3).

Citation. Arkaprabha Ghosal, Debarshi Das, Subhashish Banerjee (2021). Characterizing qubit channels in the context of quantum teleportation. DOI: 10.1103/PhysRevA.103.052422. URL: https://arxiv.org/abs/2102.02054v2.

Commentary.

The Kraus family is non-unital when the sum of K_k K_k adjoint differs from the identity. Thus its action on the identity does not return the identity.

Definition 1.4 (Sending Bob’s qubit through the channel).

Formalization. D5/S3/Quantum/Entanglement/UniversalTeleportationCriticalConcurrence.out (✓ std3).

Citation. Arkaprabha Ghosal, Debarshi Das, Subhashish Banerjee (2021). Characterizing qubit channels in the context of quantum teleportation. DOI: 10.1103/PhysRevA.103.052422. URL: https://arxiv.org/abs/2102.02054v2.

Commentary.

Bob’s channel acts on the second tensor factor. D5.S3.Quantum.Foundation.FiniteKrausChannel.PhyslibLeaf.MatrixMap.of_kraus is applied to two copies of the family k maps to I tensor K_k and then to the initial operator. Its action is the sum of conjugations by that family, with Matrix.kronecker denoting the tensor product of matrices.

Definition 1.5 (Usefulness for universal quantum teleportation).

Formalization. D5/S3/Quantum/Entanglement/UniversalTeleportationCriticalConcurrence.UsefulUQT (✓ std3).

Citation. Arkaprabha Ghosal, Debarshi Das, Subhashish Banerjee (2021). Characterizing qubit channels in the context of quantum teleportation. DOI: 10.1103/PhysRevA.103.052422. URL: https://arxiv.org/abs/2102.02054v2.

Commentary.

There are local unitaries U and V and a real t > 1/3 for which the conjugated operator has correlation matrix -t I, with zero off-diagonal entries. The correlation matrix is D5.S3.Quantum.Entanglement.SteeringEllipsoidFullyEntangledFraction.corr, whose entries are the real parts of tr(rho (sigma_i tensor sigma_j)). For two-qubit states this is the source’s Section II.C criterion |t11| = |t22| = |t33| > 1/3 in the canonical form of Section II.B, Eq. (3), with lambda_k = -1. The displayed definition permits any complex two-qubit operator rho.

Definition 1.6 (The critical concurrence).

Formalization. D5/S3/Quantum/Entanglement/UniversalTeleportationCriticalConcurrence.Cstar (✓ std3).

Citation. Arkaprabha Ghosal, Debarshi Das, Subhashish Banerjee (2021). Characterizing qubit channels in the context of quantum teleportation. DOI: 10.1103/PhysRevA.103.052422. URL: https://arxiv.org/abs/2102.02054v2.

Commentary.

The exact threshold is sqrt(5 - 2 sqrt(3))/3 = sqrt((2/3)(4 + sqrt(3))(1 - (4 + sqrt(3))/6)), approximately 0.4131. The latter radical is the sufficient threshold after Eq. (50) in Section V.2; the source rounds it to 0.41.

Definition 1.7 (Concurrence of the input vector).

Formalization. D5/S3/Quantum/Entanglement/UniversalTeleportationCriticalConcurrence.conc (✓ std3).

Citation. Arkaprabha Ghosal, Debarshi Das, Subhashish Banerjee (2021). Characterizing qubit channels in the context of quantum teleportation. DOI: 10.1103/PhysRevA.103.052422. URL: https://arxiv.org/abs/2102.02054v2.

Commentary.

Section V gives the initial pure-state concurrence as 2 sqrt(a(1 - a)). The function is defined for every real a; on 1/2 < a < 1 its value is strictly between zero and one.

Definition 1.8 (The exact threshold statement).

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

Citation. Arkaprabha Ghosal, Debarshi Das, Subhashish Banerjee (2021). Characterizing qubit channels in the context of quantum teleportation. DOI: 10.1103/PhysRevA.103.052422. URL: https://arxiv.org/abs/2102.02054v2.

Commentary.

For each 1/2 < a < 1, every finite Kraus channel producing a useful universal-teleportation state requires concurrence strictly above Cstar. Conversely, above Cstar there is a non-unital finite Kraus channel producing such a state. This gives the precise critical value sought in Section V.2, Conjecture 1, using the source’s exact radical.

Theorem 1.9 (Necessity and non-unital attainment).

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

Resolves. Problems/ghosal-das-banerjee-2021-critical-concurrence-universal-teleportation (proved) by D5/S3/Quantum/Entanglement/UniversalTeleportationCriticalConcurrence.result.

Source. Repository-derived.

Acknowledgement. Arkaprabha Ghosal, Debarshi Das, Subhashish Banerjee (2021). Characterizing qubit channels in the context of quantum teleportation. DOI: 10.1103/PhysRevA.103.052422. URL: https://arxiv.org/abs/2102.02054v2.

Commentary.

Put r = 2a - 1, so the squared initial concurrence is 1 - r squared. Bob’s trace-preserving channel leaves Alice’s marginal unchanged, and local unitaries preserve the length r of Alice’s Bloch vector. In a frame with correlation matrix -t I, let n be the unit direction of Alice’s Bloch vector and beta Bob’s Bloch component along n. Positivity of the corner on the two negative spin projections gives r + beta <= 1 - t. Cauchy-Schwarz for the positive trace form, applied to the opposite spin projections and their product with the swap matrix, gives (r - beta) squared <= (1 - t)(1 + 3t). Eliminating beta yields r squared + rt + t squared <= r + t. Since t > 1/3, this forces r < (1 + sqrt(3))/3 and hence concurrence strictly greater than Cstar.

For sufficiency take t = (1 - r + sqrt((1 - r)(1 + 3r)))/2 and gamma = (1 - t)/(1 + r). Above the concurrence threshold one has 1/3 < t < 1 and gamma > 0. The amplitude-damping Kraus pair K_0 = diag(sqrt(1 - gamma), 1), K_1 = sqrt(gamma)|1><0| is trace preserving and non-unital. The identity t squared = (1 - r)(t + r) gives output correlation matrix diag(t, -t, t); conjugating Bob by sigma_y changes it to -t I. Thus the exact threshold is sqrt(5 - 2 sqrt(3))/3, with strict inequality required for usefulness.

References