Two reduced states of one three-qutrit state both violate the CGLMP inequality

Abstract

Kumari, Ghose and Mann (arXiv:1704.06516, Phys. Rev. A 96, 012128) conjecture, from numerical studies, that the CGLMP inequality is monogamous for qutrits: for every three-qutrit state at most one of the reduced states rho_AB, rho_BC, rho_AC has B_CGLMP > 2, where B_CGLMP maximizes the CGLMP expression I_3 over their Fourier-phase family of measurements. It fails: for v = |002> + |011> + 2|020> + |100> + 2|112> - |121> + 2|210> + 2|222> and rho = v v^dagger / 20, both rho_AB and rho_AC reach I_3 = 1/2 + 14 sqrt 3 / 15 > 2.

Definition 1.1 (The Fourier transform).

$$\forall j:\mathrm{Fin}\left(3\right),\forall k:\mathrm{Fin}\left(3\right),dft\left(j,k\right)=\frac{omeg{a}^{\mathrm{val}\left(j\right)\cdot \mathrm{val}\left(k\right)}}{\sqrt{3}}$$

Formalization. D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.dft (✓ std3).

Citation. Meenu Kumari, Shohini Ghose, Robert B. Mann (2017). Sufficient condition for nonexistence of symmetric extension of qudits using Bell inequalities. DOI: 10.1103/PhysRevA.96.012128. URL: https://arxiv.org/abs/1704.06516v2.

Commentary.

U_FT is the three-dimensional discrete Fourier transform, with entries omega^{jk}/sqrt 3 where omega = exp(2 pi i/3) is the existing omega; its inverse U_FT^* is the conjugate transpose.

Definition 1.2 (The phase matrices).

$$\forall \phi :\mathrm{Fin}\left(3\right)\to \mathbb{R},\mathrm{phase}\left(\phi \right)=\mathrm{diagonal}\left((j\mapsto \exp (-(i\cdot \phi \left(j\right))))\right)$$

Formalization. D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.phase (✓ std3).

Citation. Meenu Kumari, Shohini Ghose, Robert B. Mann (2017). Sufficient condition for nonexistence of symmetric extension of qudits using Bell inequalities. DOI: 10.1103/PhysRevA.96.012128. URL: https://arxiv.org/abs/1704.06516v2.

Commentary.

U(phi) is the diagonal unitary with entries exp(-i phi(j)) for an angle vector phi : Fin 3 -> R.

Definition 1.3 (Joint outcome probabilities).

$$\forall \rho :\mathrm{Matrix}\left(\mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right),\mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right),\mathbb{C}\right),\forall A:\mathrm{Matrix}\left(\mathrm{Fin}\left(3\right),\mathrm{Fin}\left(3\right),\mathbb{C}\right),\forall B:\mathrm{Matrix}\left(\mathrm{Fin}\left(3\right),\mathrm{Fin}\left(3\right),\mathbb{C}\right),\forall j:\mathrm{Fin}\left(3\right),\forall k:\mathrm{Fin}\left(3\right),\mathrm{jointProb}\left(\rho ,A,B,j,k\right)=\mathrm{re}\left(\mathrm{tr}\left(\mathrm{kronecker}\left(\mathrm{single}\left(j,j,1\right),\mathrm{single}\left(k,k,1\right)\right)\cdot \mathrm{kronecker}\left(A,B\right)\cdot \rho \cdot \mathrm{kronecker}\left(A^H,B^H\right)\right)\right)$$

Formalization. D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.jointProb (✓ std3).

Citation. Meenu Kumari, Shohini Ghose, Robert B. Mann (2017). Sufficient condition for nonexistence of symmetric extension of qudits using Bell inequalities. DOI: 10.1103/PhysRevA.96.012128. URL: https://arxiv.org/abs/1704.06516v2.

Commentary.

Eq. (Probabilities): applying A to the first qutrit and B to the second and measuring in the computational basis, outcome (j, k) has probability tr(Pi_j (x) Pi_k (A (x) B) rho (A^dagger (x) B^dagger)), with Pi_j = single(j, j, 1) the projector onto |j>, kronecker the Kronecker product and H the conjugate transpose.

Definition 1.4 (Event probabilities).

$$\forall \rho :\mathrm{Matrix}\left(\mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right),\mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right),\mathbb{C}\right),\forall A:\mathrm{Matrix}\left(\mathrm{Fin}\left(3\right),\mathrm{Fin}\left(3\right),\mathbb{C}\right),\forall B:\mathrm{Matrix}\left(\mathrm{Fin}\left(3\right),\mathrm{Fin}\left(3\right),\mathbb{C}\right),\forall E:\mathrm{Fin}\left(3\right)\to \mathrm{Fin}\left(3\right)\to Prop,\mathrm{eventProb}\left(\rho ,A,B,E\right)=\sum _{a,b:E\left(a,b\right)}\mathrm{jointProb}\left(\rho ,A,B,a,b\right)$$

Formalization. D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.eventProb (✓ std3).

Citation. Meenu Kumari, Shohini Ghose, Robert B. Mann (2017). Sufficient condition for nonexistence of symmetric extension of qudits using Bell inequalities. DOI: 10.1103/PhysRevA.96.012128. URL: https://arxiv.org/abs/1704.06516v2.

Commentary.

The probability of an event E on the outcome pair is the sum of the joint probabilities of the outcome pairs (a, b) in E.

Definition 1.5 (The CGLMP expression).

$$\forall \rho :\mathrm{Matrix}\left(\mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right),\mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right),\mathbb{C}\right),\forall A_1,A_2,B_1,B_2:\mathrm{Matrix}\left(\mathrm{Fin}\left(3\right),\mathrm{Fin}\left(3\right),\mathbb{C}\right),cglmpI3\left(\rho ,A_1,A_2,B_1,B_2\right)=\mathrm{eventProb}\left(\rho ,A_1,B_1,((a,b)\mapsto a=b)\right)+\mathrm{eventProb}\left(\rho ,A_2,B_1,((a,b)\mapsto b=a+1)\right)+\mathrm{eventProb}\left(\rho ,A_2,B_2,((a,b)\mapsto a=b)\right)+\mathrm{eventProb}\left(\rho ,A_1,B_2,((a,b)\mapsto b=a)\right)-\mathrm{eventProb}\left(\rho ,A_1,B_1,((a,b)\mapsto a=b-1)\right)-\mathrm{eventProb}\left(\rho ,A_2,B_1,((a,b)\mapsto b=a)\right)-\mathrm{eventProb}\left(\rho ,A_2,B_2,((a,b)\mapsto a=b-1)\right)-\mathrm{eventProb}\left(\rho ,A_1,B_2,((a,b)\mapsto b=a-1)\right)$$

Formalization. D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.cglmpI3 (✓ std3).

Citation. Meenu Kumari, Shohini Ghose, Robert B. Mann (2017). Sufficient condition for nonexistence of symmetric extension of qudits using Bell inequalities. DOI: 10.1103/PhysRevA.96.012128. URL: https://arxiv.org/abs/1704.06516v2.

Commentary.

Eq. (CGLMP1): I_3 = P(A_1 = B_1) + P(B_1 = A_2 + 1) + P(A_2 = B_2) + P(B_2 = A_1) - P(A_1 = B_1 - 1) - P(B_1 = A_2) - P(A_2 = B_2 - 1) - P(B_2 = A_1 - 1), with outcomes added modulo 3 in Fin 3.

Definition 1.6 (The CGLMP expression at given angles).

$$\forall \rho :\mathrm{Matrix}\left(\mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right),\mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right),\mathbb{C}\right),\forall ({\theta }_1,{\theta }_2):(\mathrm{Fin}\left(2\right)\to \mathrm{Fin}\left(3\right)\to \mathbb{R})\times (\mathrm{Fin}\left(2\right)\to \mathrm{Fin}\left(3\right)\to \mathbb{R}),\mathrm{cglmpAt}\left(\rho ,({\theta }_1,{\theta }_2)\right)=cglmpI3\left(\rho ,dft\cdot \mathrm{phase}\left(\left({\theta }_1\right)\left(0\right)\right),dft\cdot \mathrm{phase}\left(\left({\theta }_1\right)\left(1\right)\right),df{t}^H\cdot \mathrm{phase}\left(\left({\theta }_2\right)\left(0\right)\right),df{t}^H\cdot \mathrm{phase}\left(\left({\theta }_2\right)\left(1\right)\right)\right)$$

Formalization. D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.cglmpAt (✓ std3).

Citation. Meenu Kumari, Shohini Ghose, Robert B. Mann (2017). Sufficient condition for nonexistence of symmetric extension of qudits using Bell inequalities. DOI: 10.1103/PhysRevA.96.012128. URL: https://arxiv.org/abs/1704.06516v2.

Commentary.

cglmpAt(rho, theta) is I_3 of rho at the twelve angles theta = (theta_1, theta_2), with A_k = U_FT U(phi_k) for phi_k = theta_1(k) and B_l = U_FT^* U(phi’_l) for phi’_l = theta_2(l). Eq. (CGLMP2) takes B_CGLMP(rho) to be its maximum over theta.

Definition 1.7 (The CGLMP value of a state).

$$\forall \rho :\mathrm{Matrix}\left(\mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right),\mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right),\mathbb{C}\right),\forall b:\mathbb{R},(\mathrm{IsCglmpValue}\left(\rho ,b\right))\iff ((\exists \theta :(\mathrm{Fin}\left(2\right)\to \mathrm{Fin}\left(3\right)\to \mathbb{R})\times (\mathrm{Fin}\left(2\right)\to \mathrm{Fin}\left(3\right)\to \mathbb{R}),\mathrm{cglmpAt}\left(\rho ,\theta \right)=b)\wedge (\forall \theta :(\mathrm{Fin}\left(2\right)\to \mathrm{Fin}\left(3\right)\to \mathbb{R})\times (\mathrm{Fin}\left(2\right)\to \mathrm{Fin}\left(3\right)\to \mathbb{R}),\mathrm{cglmpAt}\left(\rho ,\theta \right)\le b))$$

Formalization. D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.IsCglmpValue (✓ std3).

Citation. Meenu Kumari, Shohini Ghose, Robert B. Mann (2017). Sufficient condition for nonexistence of symmetric extension of qudits using Bell inequalities. DOI: 10.1103/PhysRevA.96.012128. URL: https://arxiv.org/abs/1704.06516v2.

Commentary.

Eq. (CGLMP2): b is B_CGLMP(rho) when b is the greatest value of cglmpAt(rho, theta) over all angles theta, that is, the maximum of I_3 over the twelve angles.

Definition 1.8 (The reduced state on AB).

$$\forall \rho :\mathrm{Matrix}\left(\mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right),\mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right),\mathbb{C}\right),\mathrm{rhoAB}\left(\rho \right)=\mathrm{partialTraceRight}\left(\mathrm{submatrix}\left(\rho ,(((a,b),c)\mapsto (a,b,c)),(((a,b),c)\mapsto (a,b,c))\right)\right)$$

Formalization. D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.rhoAB (✓ std3).

Citation. Meenu Kumari, Shohini Ghose, Robert B. Mann (2017). Sufficient condition for nonexistence of symmetric extension of qudits using Bell inequalities. DOI: 10.1103/PhysRevA.96.012128. URL: https://arxiv.org/abs/1704.06516v2.

Commentary.

For a three-qutrit matrix indexed by (a, b, c), rho_AB traces out C: the existing partialTraceRight applied to rho re-indexed by ((a, b), c) -> (a, b, c).

Definition 1.9 (The reduced state on BC).

$$\forall \rho :\mathrm{Matrix}\left(\mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right),\mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right),\mathbb{C}\right),\mathrm{rhoBC}\left(\rho \right)=\mathrm{partialTraceLeft}\left(\rho \right)$$

Formalization. D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.rhoBC (✓ std3).

Citation. Meenu Kumari, Shohini Ghose, Robert B. Mann (2017). Sufficient condition for nonexistence of symmetric extension of qudits using Bell inequalities. DOI: 10.1103/PhysRevA.96.012128. URL: https://arxiv.org/abs/1704.06516v2.

Commentary.

rho_BC traces out A with the existing partialTraceLeft.

Definition 1.10 (The reduced state on AC).

$$\forall \rho :\mathrm{Matrix}\left(\mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right),\mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right),\mathbb{C}\right),\mathrm{rhoAC}\left(\rho \right)=\mathrm{partialTraceRight}\left(\mathrm{submatrix}\left(\rho ,(((a,c),b)\mapsto (a,b,c)),(((a,c),b)\mapsto (a,b,c))\right)\right)$$

Formalization. D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.rhoAC (✓ std3).

Citation. Meenu Kumari, Shohini Ghose, Robert B. Mann (2017). Sufficient condition for nonexistence of symmetric extension of qudits using Bell inequalities. DOI: 10.1103/PhysRevA.96.012128. URL: https://arxiv.org/abs/1704.06516v2.

Commentary.

rho_AC traces out B: the existing partialTraceRight applied to rho re-indexed by ((a, c), b) -> (a, b, c), so that A is measured with A_k and C with B_l.

Definition 1.11 (The conjectured monogamy).

$$(claim)\iff (\forall \rho :\mathrm{Matrix}\left(\mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right),\mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right)\times \mathrm{Fin}\left(3\right),\mathbb{C}\right),((\mathrm{PosSemidef}\left(\rho \right))\wedge (\mathrm{tr}\left(\rho \right)=1))\Rightarrow (\forall b_{AB},b_{BC},b_{AC}:\mathbb{R},(((\mathrm{IsCglmpValue}\left(\mathrm{rhoAB}\left(\rho \right),b_{AB}\right))\wedge (\mathrm{IsCglmpValue}\left(\mathrm{rhoBC}\left(\rho \right),b_{BC}\right)))\wedge (\mathrm{IsCglmpValue}\left(\mathrm{rhoAC}\left(\rho \right),b_{AC}\right)))\Rightarrow ((2<b_{AB})\Rightarrow ((b_{BC}\le 2)\wedge (b_{AC}\le 2)))))$$

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

Citation. Meenu Kumari, Shohini Ghose, Robert B. Mann (2017). Sufficient condition for nonexistence of symmetric extension of qudits using Bell inequalities. DOI: 10.1103/PhysRevA.96.012128. URL: https://arxiv.org/abs/1704.06516v2.

Commentary.

Eq. (35): for every three-qutrit state rho_ABC, that is every positive semidefinite 27 x 27 matrix of trace one, and for the maxima b_AB, b_BC, b_AC of I_3 on rho_AB, rho_BC, rho_AC, b_AB > 2 implies b_BC <= 2 and b_AC <= 2.

Definition 1.12 (The counterexample vector).

$$stateVec=|002⟩+|011⟩+2\cdot |020⟩+|100⟩+2\cdot |112⟩+-|121⟩+2\cdot |210⟩+2\cdot |222⟩$$

Formalization. D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.stateVec (✓ std3).

Source. Repository-derived.

Acknowledgement. Meenu Kumari, Shohini Ghose, Robert B. Mann (2017). Sufficient condition for nonexistence of symmetric extension of qudits using Bell inequalities. DOI: 10.1103/PhysRevA.96.012128. URL: https://arxiv.org/abs/1704.06516v2.

Commentary.

The unnormalized vector has squared norm 1 + 1 + 4 + 1 + 4 + 1 + 4 + 4 = 20; every basis state |abc> in its support has a - b - c = 1 modulo 3.

Definition 1.13 (The counterexample state).

$$rho=\frac{1}{20}\cdot \mathrm{vecMulVec}\left(stateVec,\mathrm{star}\left(stateVec\right)\right)$$

Formalization. D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.rho (✓ std3).

Source. Repository-derived.

Acknowledgement. Meenu Kumari, Shohini Ghose, Robert B. Mann (2017). Sufficient condition for nonexistence of symmetric extension of qudits using Bell inequalities. DOI: 10.1103/PhysRevA.96.012128. URL: https://arxiv.org/abs/1704.06516v2.

Commentary.

The state is the rank-one density matrix v v^dagger / 20.

Definition 1.14 (The angle unit).

$$\forall n:\mathbb{N},\mathrm{ang}\left(n\right)=\frac{n\cdot \pi }{6}$$

Formalization. D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.ang (✓ std3).

Source. Repository-derived.

Acknowledgement. Meenu Kumari, Shohini Ghose, Robert B. Mann (2017). Sufficient condition for nonexistence of symmetric extension of qudits using Bell inequalities. DOI: 10.1103/PhysRevA.96.012128. URL: https://arxiv.org/abs/1704.06516v2.

Commentary.

Angles are integer multiples of pi/6, so every phase is a twelfth root of unity.

Theorem 1.15 (Both reduced states violate the inequality).

$$\neg claim$$

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

Resolves. Problems/kumari-2017-cglmp-monogamy (refuted) by D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.result.

Source. Repository-derived.

Acknowledgement. Meenu Kumari, Shohini Ghose, Robert B. Mann (2017). Sufficient condition for nonexistence of symmetric extension of qudits using Bell inequalities. DOI: 10.1103/PhysRevA.96.012128. URL: https://arxiv.org/abs/1704.06516v2.

Commentary.

Each joint probability of a reduced state of v v^dagger / 20 is one twentieth of a sum of squared amplitudes, which are sums of twelfth roots of unity divided by 3. With angles in units of pi/6, phi_1 = (0, 2, 7), phi_2 = (0, 2, 1), phi’_1 = (0, 8, 4), phi’_2 = (0, 10, 2) on rho_AB and phi_1 = (0, 6, 9), phi_2 = (0, 6, 3), phi’_1 = (0, 10, 2), phi’_2 = (0, 0, 6) on rho_AC give, in the order of Eq. (CGLMP1), the event probabilities 23/60 + sqrt 3/5, 23/60 + sqrt 3/5, 1/2, 1/2, 23/60 - sqrt 3/5, 23/60 - sqrt 3/5, 1/4 - sqrt 3/15, 1/4 - sqrt 3/15, so I_3 = 1/2 + 14 sqrt 3/15 > 2 for both. For every two-qutrit matrix, I_3 is continuous in the angles and unchanged when an angle moves by 2 pi, so it attains its maximum on the compact box [0, 2 pi]^24; each maximum of rho_AB and rho_AC is therefore at least the stated value. The state is positive semidefinite with trace one.

References

• Truth anchor: D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.IsCglmpValue
• Truth anchor: D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.ang
• Truth anchor: D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.cglmpAt
• Truth anchor: D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.cglmpI3
• Truth anchor: D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.claim
• Truth anchor: D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.dft
• Truth anchor: D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.eventProb
• Truth anchor: D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.jointProb
• Truth anchor: D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.phase
• Truth anchor: D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.result
• Truth anchor: D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.rho
• Truth anchor: D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.rhoAB
• Truth anchor: D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.rhoAC
• Truth anchor: D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.rhoBC
• Truth anchor: D5/S3/Quantum/Entanglement/CglmpMonogamyRefutation.stateVec
• Dependency: D5/S3/Quantum/Entanglement/QutritThresholdSharing
• Dependency: D5/S3/Quantum/Information/PartialTraceMutualInformation
• Dependency: D5/S3/Quantum/Recovery/FiniteLocalLatitudeGeometry