Quantum information

Results in Quantum information

• Selected result:

  No universal finite collapse of the upper PPT entanglement-cost hierarchy

  Answered negativelyFinite collapse of the Lami–Mele–Regula upper PPT entanglement-cost hierarchy

  Mathematical physics · Quantum information · standalone Lean package

  For every q ≥ 1 a full-rank product-Bell-diagonal state in local dimension 2^((2q+4)4^q) has κ_{q+1}(ρ) < κ_q(ρ), so no fixed finite level of the upper PPT entanglement-cost hierarchy is exact for all states.

  Manuscript: No universal finite collapse of the upper PPT entanglement-cost hierarchy · Lean proof: No universal finite collapse of the upper PPT entanglement-cost hierarchy · Verification: No universal finite collapse of the upper PPT entanglement-cost hierarchy

• Selected result:

  Universal entanglement embezzlement with inverse-quadratic logarithmic infidelity

  Answered positivelyLeung–Wang question on the rate of general universal entanglement embezzling families

  Mathematical physics · Quantum information · standalone Lean package

  For each L ≥ 1 one target-independent catalyst of local dimension n = 2^(4L) lets local basis permutations reach root fidelity at least 1 − 125m/L² = 1 − 2000m/(log₂ n)² for every Schmidt probability vector on m entries.

  Manuscript: Universal entanglement embezzlement with inverse-quadratic logarithmic infidelity · Lean proof: Universal entanglement embezzlement with inverse-quadratic logarithmic infidelity · Verification: Universal entanglement embezzlement with inverse-quadratic logarithmic infidelity

• Strong minimum-rank multiplicativity from unextendible product constraints

  Determined exactlySchmidt number of tensor powers of the Tiles UPB bound entangled state

  Mathematical physics · Quantum information · standalone Lean package

  If a complex matrix subspace S is annihilated by an unextendible family of product functionals, every nonzero element of S ⊗ T has rank at least 2m(T); for the Tiles bound entangled state ρ, SN(ρ^⊗N) = 2^N for all N ≥ 1.

  Manuscript: Strong minimum-rank multiplicativity from unextendible product constraints · Lean proof: Strong minimum-rank multiplicativity from unextendible product constraints · Verification: Strong minimum-rank multiplicativity from unextendible product constraints

• Exact conditional contraction for Pauli channels and depolarization in every dimension

  Determined exactlyExact conditional hockey-stick contraction coefficients of Pauli and depolarizing channels

  Mathematical physics · Quantum information · standalone Lean package

  For every γ ≥ 1 the conditional hockey-stick contraction coefficient is 1 − 8ab/(a+b) for a Pauli channel with smallest weights a ≤ b, and s[1+(d²−1)s]/[1−(d²−3)s] for depolarization λ = −s, 0 ≤ s ≤ 1/(d²−1), in dimension d.

  Manuscript: Exact conditional contraction for Pauli channels and depolarization in every dimension · Lean proof: Exact conditional contraction for Pauli channels and depolarization in every dimension · Verification: Exact conditional contraction for Pauli channels and depolarization in every dimension

• Exact three-setting correlation bounds at fixed steering centre

  Determined exactlyMaximum CJWR three-setting steering value at fixed steering-ellipsoid centre (two qubits)

  Mathematical physics · Quantum information · standalone Lean package

  For two-qubit states with steering-ellipsoid centre length c, the exact supremum of F₃² = ‖T‖²_F is 1 + (2−3c)²/(2−2c−c²) for c ≤ 2/3 and 1 for c > 2/3, so F₃ > 1 is possible exactly when c < 2/3.

  Manuscript: Exact three-setting correlation bounds at fixed steering centre · Lean proof: Exact three-setting correlation bounds at fixed steering centre · Verification: Exact three-setting correlation bounds at fixed steering centre

• A sum-of-squares proof of the four-qubit Higuchi–Sudbery entropy bound

  ProvedHiguchi–Sudbery problem: maximal average two-qubit marginal entropy of four qubits

  Mathematical physics · Quantum information · standalone Lean package

  For every pure four-qubit state the average von Neumann entropy of the three two-qubit marginals ρ_AB, ρ_AC, ρ_AD is at most log 2 + (1/2) log 3 (natural logarithms), and the Higuchi–Sudbery state attains it.

  Manuscript: A sum-of-squares proof of the four-qubit Higuchi–Sudbery entropy bound · Lean proof: A sum-of-squares proof of the four-qubit Higuchi–Sudbery entropy bound · Verification: A sum-of-squares proof of the four-qubit Higuchi–Sudbery entropy bound

• The three-state root-fidelity entropy bound

  ProvedFannes–de Melo–Roga–Życzkowski conjecture: root-fidelity matrix entropy bounds Holevo χ

  Mathematical physics · Quantum information · standalone Lean package

  For any three density matrices ρᵢ of finite dimension and probabilities pᵢ, the Holevo quantity S(Σ pᵢρᵢ) − Σ pᵢS(ρᵢ) is at most the entropy of the 3 × 3 matrix C_ij = √(pᵢpⱼ) ‖√ρᵢ √ρⱼ‖₁.

  Manuscript: The three-state root-fidelity entropy bound · Lean proof: The three-state root-fidelity entropy bound · Verification: The three-state root-fidelity entropy bound

• A pure-state matched-basis information inequality violation in dimension five

  CounterexamplePure-state ECQC: matched-basis mutual-information inequality with mutually unbiased bases

  Mathematical physics · Quantum information · standalone Lean package

  A normalized pure state in local dimension 5 with six fixed mutually unbiased bases, used identically by both parties, has five-setting classical information sum 5 log 5 versus quantum mutual information 2 log 5 (ratio 5/2).

  Manuscript: A pure-state matched-basis information inequality violation in dimension five · Lean proof: A pure-state matched-basis information inequality violation in dimension five · Verification: A pure-state matched-basis information inequality violation in dimension five