The separable numerical range of one two-qubit observable

Abstract

For a single two-qubit observable A, the length of the separable numerical range is at least half the length of the numerical range, and the projector onto (|00> + |11>)/sqrt(2) attains one half. This proves Conjecture 8 of T. Simnacher, J. Czartowski, K. Szymański and K. Życzkowski (arXiv:2107.04365): the minimal volume ratio mu_{2,1} equals 1/2.

Definition 1.1 (Separable states).

$$\forall \rho :{\mathbb{C}}^{4\times 4},\rho \in \mathrm{separableStates}\iff (\mathrm{separableCone}\left(\rho \right)\wedge \mathrm{Tr}\left(\rho \right)=1)$$

Formalization. D5/S3/Quantum/Entanglement/SeparableNumericalRangeRatio.separableStates (✓ std3).

Citation. Timo Simnacher; Jakub Czartowski; Konrad Szymański; Karol Życzkowski (2021). Confident entanglement detection via the separable numerical range. DOI: 10.1103/PhysRevA.104.042420. URL: https://arxiv.org/abs/2107.04365v1.

Commentary.

A separable two-qubit state is a state that is a finite sum of Kronecker products of positive semidefinite 2 x 2 matrices; equivalently, a convex combination of product states.

Definition 1.2 (Restricted numerical ranges).

$$\forall X\subseteq {\mathbb{C}}^{4\times 4},\forall A:{\mathbb{C}}^{4\times 4},\mathrm{numericalRange}\left(X,A\right)=\{\mathrm{Re}\left(\mathrm{Tr}\left(\rho A\right)\right)\mid \rho \in X\}$$

Formalization. D5/S3/Quantum/Entanglement/SeparableNumericalRangeRatio.numericalRange (✓ std3).

Citation. Timo Simnacher; Jakub Czartowski; Konrad Szymański; Karol Życzkowski (2021). Confident entanglement detection via the separable numerical range. DOI: 10.1103/PhysRevA.104.042420. URL: https://arxiv.org/abs/2107.04365v1.

Commentary.

For a set X of states and a matrix A, the restricted numerical range L_X(A) is the set of real parts of Tr(rho A) over rho in X. For Hermitian A and a state rho the trace is real.

Definition 1.3 (The conjecture).

$$claim\iff (\mathrm{IsLeast}\left(\{\frac{\mathrm{vol}\left(\mathrm{numericalRange}\left(\mathrm{separableStates},A\right)\right)}{\mathrm{vol}\left(\mathrm{numericalRange}\left(\{\rho \mid \rho \in {\mathbb{C}}^{4\times 4},\mathrm{IsDensity}\left(\rho \right)\},A\right)\right)}\mid A\in {\mathbb{C}}^{4\times 4},A^{\ast }=A\wedge \mathrm{vol}\left(\mathrm{numericalRange}\left(\{\rho \mid \rho \in {\mathbb{C}}^{4\times 4},\mathrm{IsDensity}\left(\rho \right)\},A\right)\right)\ne 0\},\frac{1}{2}\right))$$

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

Citation. Timo Simnacher; Jakub Czartowski; Konrad Szymański; Karol Życzkowski (2021). Confident entanglement detection via the separable numerical range. DOI: 10.1103/PhysRevA.104.042420. URL: https://arxiv.org/abs/2107.04365v1.

Commentary.

One half is the least value of vol L_Sep(A) / vol L(A) over Hermitian two-qubit matrices A with vol L(A) not zero, where vol is Lebesgue measure on the real line, L(A) is the numerical range over all two-qubit density matrices (IsDensity: positive semidefinite with trace 1) and L_Sep(A) the numerical range over separable states. For scalar A both ranges are points, so the ratio is defined exactly when A is not scalar. The least value is attained, so it is the minimum of the paper’s Definition 2 for n = 2, d = 2 and k = 1.

Theorem 1.4 (The minimal ratio is one half).

$$claim$$

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

Resolves. Problems/simnacher-2021-two-qubit-separable-numerical-range-ratio (proved) by D5/S3/Quantum/Entanglement/SeparableNumericalRangeRatio.result.

Source. Repository-derived.

Acknowledgement. Timo Simnacher; Jakub Czartowski; Konrad Szymański; Karol Życzkowski (2021). Confident entanglement detection via the separable numerical range. DOI: 10.1103/PhysRevA.104.042420. URL: https://arxiv.org/abs/2107.04365v1.

Commentary.

In the magic basis a two-qubit vector u is a product vector exactly when the sum of the squares of its four coordinates vanishes. After a phase, a unit vector z has z^T z = C with 0 <= C <= 1. For a real unit vector r orthogonal to the real part of z, the vectors z + i t r with t = -eta +- sqrt(eta^2 + C), eta = (Im z) . r, are product vectors, and a convex combination of their projectors equals zz^* + C rr^T; so zz^* + C rr^T is separable. For two unit vectors u and v, two orthonormal real vectors r and s orthogonal to the real parts of both give one separable noise N = (rr^T + ss^T)/2 with trace 1, and with c = max(C_u, C_v) the matrices (P_u + cN)/(1 + c) and (P_v + cN)/(1 + c) are separable states whose difference is (P_u - P_v)/(1 + c). Hence the expectation values of A at two pure states differ by at most 2 times the length of L_Sep(A); by the spectral decomposition of states the same holds for any two states, so vol L(A) <= 2 vol L_Sep(A), using that L_Sep(A) is an interval. For the Bell projector, L(A) contains [0, 1] and every product state has expectation (1/2) times the sum over a, b of sigma_ab tau_ab, which lies in [0, 1/2]; so the ratio is at most 1/2, and therefore equal to 1/2.

References

• Truth anchor: D5/S3/Quantum/Entanglement/SeparableNumericalRangeRatio.claim
• Truth anchor: D5/S3/Quantum/Entanglement/SeparableNumericalRangeRatio.numericalRange
• Truth anchor: D5/S3/Quantum/Entanglement/SeparableNumericalRangeRatio.result
• Truth anchor: D5/S3/Quantum/Entanglement/SeparableNumericalRangeRatio.separableStates
• Dependency: D5/S3/Quantum/Entanglement/StructuredNegativityCoincidenceRefutation