Cat pairs need not be optimal for metrology with noisy measurements

Abstract

Len, Gefen, Retzker and Kołodyński (arXiv:2109.01160, Nature Communications 13, 6971) conjecture that for every classical noise channel M_x = sum_i p(x|i) Pi_i applied independently to N probes, the pair of orthonormal states maximizing the noisy Fisher coefficient gamma can be taken of cat form cos(theta)|j>^N + sin(theta)|k>^N, -sin(theta)|j>^N + cos(theta)|k>^N. For a three-outcome qutrit detector with entries in (1/20)Z and two probes, the pair |10>, |21> attains a larger gamma than every cat pair.

Definition 1.1 (The single-probe noisy measurement).

$$\forall d:\mathbb{N},\forall X:Type,\forall p:X\to \mathrm{Fin}\left(d\right)\to \mathbb{R},\forall x:X,\mathrm{probeOp}\left(p,x\right)=\sum _{i\in \mathrm{Fin}\left(d\right)}p\left(x,i\right)\cdot \mathrm{single}\left(i,i,1\right)$$

Formalization. D5/S3/Quantum/Measurement/NoisyMeasurementCatOptimalityRefutation.probeOp (✓ std3).

Citation. Yink Loong Len, Tuvia Gefen, Alex Retzker, Jan Kołodyński (2022). Quantum metrology with imperfect measurements. DOI: 10.1038/s41467-022-33563-8. URL: https://arxiv.org/abs/2109.01160v2.

Commentary.

M_x = sum_i p(x|i) Pi_i, with Pi_i = |i><i| the matrix with a single 1 in position (i, i).

Definition 1.2 (Independent measurement of N probes).

$$\forall d:\mathbb{N},\forall X:Type,\forall p:X\to \mathrm{Fin}\left(d\right)\to \mathbb{R},\forall N:\mathbb{N},\forall xs:\mathrm{Fin}\left(N\right)\to X,\mathrm{multiProbeOp}\left(p,xs\right)=\mathrm{piKroneckerLinearEquiv}\left(\mathrm{tprod}\left((l\mapsto \mathrm{probeOp}\left(p,xs\left(l\right)\right))\right)\right)$$

Formalization. D5/S3/Quantum/Measurement/NoisyMeasurementCatOptimalityRefutation.multiProbeOp (✓ std3).

Citation. Yink Loong Len, Tuvia Gefen, Alex Retzker, Jan Kołodyński (2022). Quantum metrology with imperfect measurements. DOI: 10.1038/s41467-022-33563-8. URL: https://arxiv.org/abs/2109.01160v2.

Commentary.

Eq. (AMxvec): M_x = M_{x_1} (x) … (x) M_{x_N}. The frozen finite-family Kronecker equivalence piKroneckerLinearEquiv sends the tensor product of the single-probe matrices to a matrix on the words s : Fin N -> Fin d; its entry at (s, s’) is the product over l of the entries of M_{x_l} at (s(l), s’(l)).

Definition 1.3 (The noisy Fisher coefficient).

$$\forall d:\mathbb{N},\forall X:Type,[\mathrm{Fintype}\left(X\right)],\forall p:X\to \mathrm{Fin}\left(d\right)\to \mathbb{R},\forall N:\mathbb{N},\forall zeta,zetaperp,psi,psiperp:(\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(d\right))\to \mathbb{C},((psi=\frac{zeta+zetaperp}{\sqrt{2}})\wedge (psiperp=\frac{zeta-zetaperp}{\sqrt{2}}))\Rightarrow (\mathrm{gamma}\left(p,zeta,zetaperp\right)=\sum _{xs:\mathrm{Fin}\left(N\right)\to X}\mathrm{Re}\left(\frac{\frac{1}{4}\cdot (⟨psiperp,\mathrm{multiProbeOp}\left(p,xs\right)\cdot psi⟩+\overline{⟨psiperp,\mathrm{multiProbeOp}\left(p,xs\right)\cdot psi⟩}{)}^2}{⟨psi,\mathrm{multiProbeOp}\left(p,xs\right)\cdot psi⟩}\right))$$

Formalization. D5/S3/Quantum/Measurement/NoisyMeasurementCatOptimalityRefutation.gamma (✓ std3).

Citation. Yink Loong Len, Tuvia Gefen, Alex Retzker, Jan Kołodyński (2022). Quantum metrology with imperfect measurements. DOI: 10.1038/s41467-022-33563-8. URL: https://arxiv.org/abs/2109.01160v2.

Commentary.

Eq. (AgammaN): gamma = (1/4) sum over outcome words of [<psi_perp|V^dagger M_x V|psi> + c.c.]^2 / <psi|V^dagger M_x V|psi>, where V psi = (zeta + zeta_perp)/sqrt(2) and V psi_perp = (zeta - zeta_perp)/sqrt(2) invert the relations V (psi +- psi_perp)/sqrt(2) = zeta, zeta_perp of Eq. (AVPhichoice). Here <a, b> = sum_s conj(a(s)) b(s) and M psi is the matrix-vector product. The real part reads the source quotient, whose numerator and denominator are real.

Definition 1.4 (Repeated basis words).

$$\forall d,N:\mathbb{N},\forall j:\mathrm{Fin}\left(d\right),\forall s:(\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(d\right)),\mathrm{basisPower}\left(j\right)\left(s\right)=\prod _{l\in \mathrm{Fin}\left(N\right)}\mathrm{PiSingle}\left(j,1\right)\left(s\left(l\right)\right)$$

Formalization. D5/S3/Quantum/Measurement/NoisyMeasurementCatOptimalityRefutation.basisPower (✓ std3).

Citation. Yink Loong Len, Tuvia Gefen, Alex Retzker, Jan Kołodyński (2022). Quantum metrology with imperfect measurements. DOI: 10.1038/s41467-022-33563-8. URL: https://arxiv.org/abs/2109.01160v2.

Commentary.

|j>^N is the product of N copies of the basis vector |j> of the common eigenbasis.

Definition 1.5 (The cat state).

$$\forall d,N:\mathbb{N},\forall j,k:\mathrm{Fin}\left(d\right),\forall \theta :\mathbb{R},\mathrm{catState}\left(j,k,\theta \right)=\cos \left(\theta \right)\cdot \mathrm{basisPower}\left(j\right)+\sin \left(\theta \right)\cdot \mathrm{basisPower}\left(k\right)$$

Formalization. D5/S3/Quantum/Measurement/NoisyMeasurementCatOptimalityRefutation.catState (✓ std3).

Citation. Yink Loong Len, Tuvia Gefen, Alex Retzker, Jan Kołodyński (2022). Quantum metrology with imperfect measurements. DOI: 10.1038/s41467-022-33563-8. URL: https://arxiv.org/abs/2109.01160v2.

Commentary.

The first member of the conjectured optimal pair.

Definition 1.6 (Its orthogonal partner).

$$\forall d,N:\mathbb{N},\forall j,k:\mathrm{Fin}\left(d\right),\forall \theta :\mathbb{R},\mathrm{catPerp}\left(j,k,\theta \right)=-\sin \left(\theta \right)\cdot \mathrm{basisPower}\left(j\right)+\cos \left(\theta \right)\cdot \mathrm{basisPower}\left(k\right)$$

Formalization. D5/S3/Quantum/Measurement/NoisyMeasurementCatOptimalityRefutation.catPerp (✓ std3).

Citation. Yink Loong Len, Tuvia Gefen, Alex Retzker, Jan Kołodyński (2022). Quantum metrology with imperfect measurements. DOI: 10.1038/s41467-022-33563-8. URL: https://arxiv.org/abs/2109.01160v2.

Commentary.

The second member of the conjectured optimal pair.

Definition 1.7 (The conjectured optimality of cat pairs).

$$(claim)\iff (\forall d:\mathbb{N},(2\le d)\Rightarrow (\forall X:Type,[\mathrm{Fintype}\left(X\right)],\forall p:X\to \mathrm{Fin}\left(d\right)\to \mathbb{R},((\forall x:X,\forall i:\mathrm{Fin}\left(d\right),0<p\left(x,i\right))\wedge (\forall i:\mathrm{Fin}\left(d\right),\sum _{x\in X}p\left(x,i\right)=1))\Rightarrow (\forall N:\mathbb{N},(1\le N)\Rightarrow (\exists j,k:\mathrm{Fin}\left(d\right),(j\ne k)\wedge (\exists \theta :\mathbb{R},\forall zeta,zetaperp:(\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(d\right))\to \mathbb{C},(((⟨zeta,zeta⟩=1)\wedge (⟨zetaperp,zetaperp⟩=1))\wedge (⟨zetaperp,zeta⟩=0))\Rightarrow (\mathrm{gamma}\left(p,zeta,zetaperp\right)\le \mathrm{gamma}\left(p,\mathrm{catState}\left(j,k,\theta \right),\mathrm{catPerp}\left(j,k,\theta \right)\right)))))))$$

Formalization. D5/S3/Quantum/Measurement/NoisyMeasurementCatOptimalityRefutation.claim (✓ std3).

Citation. Yink Loong Len, Tuvia Gefen, Alex Retzker, Jan Kołodyński (2022). Quantum metrology with imperfect measurements. DOI: 10.1038/s41467-022-33563-8. URL: https://arxiv.org/abs/2109.01160v2.

Commentary.

For every d >= 2, every finite outcome set X, every classical noise channel p with all entries positive, so that a maximizing pair exists, and every N >= 1 there are labels j != k and an angle theta whose cat pair attains the largest gamma among all orthonormal pairs. Orthonormality uses <a, b> = sum_s conj(a(s)) b(s). This is the weakest reading of the conjecture; it does not require j and k to be found from a single probe.

Definition 1.8 (The detector).

$$((detector\left(0\right)=(\frac{1}{20},\frac{14}{20},\frac{2}{20}))\wedge (detector\left(1\right)=(\frac{15}{20},\frac{2}{20},\frac{17}{20})))\wedge (detector\left(2\right)=(\frac{4}{20},\frac{4}{20},\frac{1}{20}))$$

Formalization. D5/S3/Quantum/Measurement/NoisyMeasurementCatOptimalityRefutation.detector (✓ std3).

Source. Repository-derived.

Acknowledgement. Yink Loong Len, Tuvia Gefen, Alex Retzker, Jan Kołodyński (2022). Quantum metrology with imperfect measurements. DOI: 10.1038/s41467-022-33563-8. URL: https://arxiv.org/abs/2109.01160v2.

Commentary.

Row x lists p(x|0), p(x|1), p(x|2); every column sums to 1 and every entry is at least 1/20.

Definition 1.9 (The first witness state).

$$witness=|10⟩$$

Formalization. D5/S3/Quantum/Measurement/NoisyMeasurementCatOptimalityRefutation.witness (✓ std3).

Source. Repository-derived.

Acknowledgement. Yink Loong Len, Tuvia Gefen, Alex Retzker, Jan Kołodyński (2022). Quantum metrology with imperfect measurements. DOI: 10.1038/s41467-022-33563-8. URL: https://arxiv.org/abs/2109.01160v2.

Commentary.

The basis word 10 of two qutrits.

Definition 1.10 (The second witness state).

$$witnessPerp=|21⟩$$

Formalization. D5/S3/Quantum/Measurement/NoisyMeasurementCatOptimalityRefutation.witnessPerp (✓ std3).

Source. Repository-derived.

Acknowledgement. Yink Loong Len, Tuvia Gefen, Alex Retzker, Jan Kołodyński (2022). Quantum metrology with imperfect measurements. DOI: 10.1038/s41467-022-33563-8. URL: https://arxiv.org/abs/2109.01160v2.

Commentary.

The basis word 21 of two qutrits.

Theorem 1.11 (A crossed pair beats every cat pair).

$$\neg claim$$

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

Resolves. Problems/len-2022-noisy-metrology-cat-optimality (refuted) by D5/S3/Quantum/Measurement/NoisyMeasurementCatOptimalityRefutation.result.

Source. Repository-derived.

Acknowledgement. Yink Loong Len, Tuvia Gefen, Alex Retzker, Jan Kołodyński (2022). Quantum metrology with imperfect measurements. DOI: 10.1038/s41467-022-33563-8. URL: https://arxiv.org/abs/2109.01160v2.

Commentary.

Take d = 3, X = Fin 3, the detector above and N = 2. Write m_x(s) = p(x_1|s_1) p(x_2|s_2). For real orthonormal vectors supported on two words u != v the coefficient is gamma = sum_x t (1 - t) (m_x(u) - m_x(v))^2 / (t m_x(u) + (1 - t) m_x(v)), where t is the squared weight of u in (zeta + zeta_perp)/sqrt(2). The witness pair has u = 10, v = 21 and t = 1/2, and gamma = 6643859399/9075312000 > 73/100. Every cat pair with j != k has u = jj, v = kk and t = (cos(theta) - sin(theta))^2 / 2. Each summand is D - ab/D - (2t - 1)(a - b) with D = ta + (1 - t)b, and ab/D >= ab (2/D_0 - D/D_0^2) for D_0 = t_0 a + (1 - t_0) b, so gamma is bounded by an affine function of t. Evaluating it at t = 0 and t = 1 with t_0 = 2061/4000, 191/500, 1939/4000, 363/800, 309/500 or 437/800 for the six ordered pairs (j, k) gives gamma < 73/100. So no cat pair maximizes gamma.

References

• Truth anchor: D5/S3/Quantum/Measurement/NoisyMeasurementCatOptimalityRefutation.basisPower
• Truth anchor: D5/S3/Quantum/Measurement/NoisyMeasurementCatOptimalityRefutation.catPerp
• Truth anchor: D5/S3/Quantum/Measurement/NoisyMeasurementCatOptimalityRefutation.catState
• Truth anchor: D5/S3/Quantum/Measurement/NoisyMeasurementCatOptimalityRefutation.claim
• Truth anchor: D5/S3/Quantum/Measurement/NoisyMeasurementCatOptimalityRefutation.detector
• Truth anchor: D5/S3/Quantum/Measurement/NoisyMeasurementCatOptimalityRefutation.gamma
• Truth anchor: D5/S3/Quantum/Measurement/NoisyMeasurementCatOptimalityRefutation.multiProbeOp
• Truth anchor: D5/S3/Quantum/Measurement/NoisyMeasurementCatOptimalityRefutation.probeOp
• Truth anchor: D5/S3/Quantum/Measurement/NoisyMeasurementCatOptimalityRefutation.result
• Truth anchor: D5/S3/Quantum/Measurement/NoisyMeasurementCatOptimalityRefutation.witness
• Truth anchor: D5/S3/Quantum/Measurement/NoisyMeasurementCatOptimalityRefutation.witnessPerp
• Dependency: D5/S3/ObserverMemory/PrimePowerTensorTower