Optimal quantum states depend on the Renyi orders

Abstract

A qutrit basis projector is Pareto optimal for the Shannon dual pair (1,1), but ceases to be optimal for the dual pair (3/5,3). Thus the optimal states for two projective measurements can depend on their Renyi orders.

Definition 1.1 (Standard-basis probabilities).

$$\forall d\in \mathbb{N},\forall rho\in \mathrm{DensityState}\left(\mathrm{Fin}\left(d\right)\right),\forall i\in \mathrm{Fin}\left(d\right),\mathrm{pX}\left(rho,i\right)=\mathrm{Re}\left(CStarMatrix.ofMatrix.symm\left(\mathrm{val}\left(rho\right)\right)\left(i,i\right)\right)$$

Formalization. D5/S3/Quantum/Information/RenyiOptimalStateDependenceRefutation.pX (✓ std3).

Citation. Kais Abdelkhalek; René Schwonnek; Hans Maassen; Fabian Furrer; Jörg Duhme; Philippe Raynal; Berthold-Georg Englert; Reinhard F. Werner (2015). Optimality of entropic uncertainty relations. DOI: 10.1142/S0219749915500458. URL: https://arxiv.org/abs/1509.00398v1.

Commentary.

The standard basis X gives the real diagonal entries of CStarMatrix.ofMatrix.symm(val(rho)); CStarMatrix.ofMatrix.symm is the inverse identity equivalence from CStarMatrix to Matrix, and val exposes the subtype value. Positivity and trace one make these a probability distribution. Indices run from 0 to d - 1.

Definition 1.2 (Probabilities in the columns of the overlap matrix).

$$\forall d\in \mathbb{N},\forall U\in \mathrm{unitaryGroup}\left(\mathrm{Fin}\left(d\right),\mathbb{C}\right),\forall rho\in \mathrm{DensityState}\left(\mathrm{Fin}\left(d\right)\right),\forall j\in \mathrm{Fin}\left(d\right),\mathrm{pY}\left(U,rho,j\right)=\mathrm{Re}\left((\mathrm{conjTranspose}\left(\mathrm{val}\left(U\right)\right)\cdot CStarMatrix.ofMatrix.symm\left(\mathrm{val}\left(rho\right)\right)\cdot \mathrm{val}\left(U\right))\left(j,j\right)\right)$$

Formalization. D5/S3/Quantum/Information/RenyiOptimalStateDependenceRefutation.pY (✓ std3).

Citation. Kais Abdelkhalek; René Schwonnek; Hans Maassen; Fabian Furrer; Jörg Duhme; Philippe Raynal; Berthold-Georg Englert; Reinhard F. Werner (2015). Optimality of entropic uncertainty relations. DOI: 10.1142/S0219749915500458. URL: https://arxiv.org/abs/1509.00398v1.

Commentary.

The overlap matrix has entries U_ij = <x_i|y_j>. Thus Y is the column basis of U, and its probabilities are the real diagonal entries of U* CStarMatrix.ofMatrix.symm(val(rho)) U. The star denotes conjugate transpose and val(U) exposes the matrix of the bundled unitary.

Definition 1.3 (Finite-order Renyi entropy).

$$\forall d\in \mathbb{N},\forall alpha\in \mathbb{R},\forall p\in \mathrm{Fin}\left(d\right)\to \mathbb{R},H\left(alpha,p\right)=\mathrm{if}(alpha=1)\mathrm{then}(\mathrm{shannonEntropy}\left(p\right))\mathrm{else}(\frac{\log \left({\sum }_{i:\mathrm{Fin}\left(d\right)}(\mathrm{if}(p\left(i\right)=0)\mathrm{then}(0)\mathrm{else}((p\left(i\right){)}^{alpha}))\right)}{1-alpha})$$

Formalization. D5/S3/Quantum/Information/RenyiOptimalStateDependenceRefutation.H (✓ std3).

Citation. Kais Abdelkhalek; René Schwonnek; Hans Maassen; Fabian Furrer; Jörg Duhme; Philippe Raynal; Berthold-Georg Englert; Reinhard F. Werner (2015). Optimality of entropic uncertainty relations. DOI: 10.1142/S0219749915500458. URL: https://arxiv.org/abs/1509.00398v1.

Commentary.

Section II, equation (6), p. 4 defines Renyi entropy by log(sum_i p_i^alpha)/(1-alpha) away from order one, and by the Shannon entropy at order one. Here shannonEntropy(p) = sum_i -p_i log(p_i) is the frozen finite Shannon entropy. All logarithms are natural: the paper states, verbatim, ‘The logarithms can be taken in any base (as long as it is always the same base).’ The explicit zero-mass branch implements 0^alpha = 0 for the positive orders in the conjecture. Orders are finite real numbers; infinity is outside this encoding.

Definition 1.4 (Entropy-coordinate order).

$$\forall d\in \mathbb{N},\forall U\in \mathrm{unitaryGroup}\left(\mathrm{Fin}\left(d\right),\mathbb{C}\right),\forall alpha\in \mathbb{R},\forall beta\in \mathbb{R},\forall rho\in \mathrm{DensityState}\left(\mathrm{Fin}\left(d\right)\right),\forall sigma\in \mathrm{DensityState}\left(\mathrm{Fin}\left(d\right)\right),\mathrm{Below}\left(U,alpha,beta,rho,sigma\right)\iff ((H\left(alpha,\mathrm{pX}\left(rho\right)\right)\le H\left(alpha,\mathrm{pX}\left(sigma\right)\right))\wedge (H\left(beta,\mathrm{pY}\left(U,rho\right)\right)\le H\left(beta,\mathrm{pY}\left(U,sigma\right)\right)))$$

Formalization. D5/S3/Quantum/Information/RenyiOptimalStateDependenceRefutation.Below (✓ std3).

Citation. Kais Abdelkhalek; René Schwonnek; Hans Maassen; Fabian Furrer; Jörg Duhme; Philippe Raynal; Berthold-Georg Englert; Reinhard F. Werner (2015). Optimality of entropic uncertainty relations. DOI: 10.1142/S0219749915500458. URL: https://arxiv.org/abs/1509.00398v1.

Commentary.

Section II, p. 4: ‘For any choice we can define the order relation ⊑ on the state space, so that ρ⊑ρ′ stands for “f₁(ρ)≤f₁(ρ′) and f₂(ρ)≤f₂(ρ′)”.’ Here f(rho) = (H(alpha,pX(rho)), H(beta,pY(U,rho))); Below(U,alpha,beta,rho,sigma) encodes rho ⊑ sigma.

Definition 1.5 (Pareto optimality over every density state).

$$\forall d\in \mathbb{N},\forall U\in \mathrm{unitaryGroup}\left(\mathrm{Fin}\left(d\right),\mathbb{C}\right),\forall alpha\in \mathbb{R},\forall beta\in \mathbb{R},\forall rho\in \mathrm{DensityState}\left(\mathrm{Fin}\left(d\right)\right),\mathrm{Optimal}\left(U,alpha,beta,rho\right)\iff (\forall sigma\in \mathrm{DensityState}\left(\mathrm{Fin}\left(d\right)\right),(\mathrm{Below}\left(U,alpha,beta,sigma,rho\right))\Rightarrow (\mathrm{Below}\left(U,alpha,beta,rho,sigma\right)))$$

Formalization. D5/S3/Quantum/Information/RenyiOptimalStateDependenceRefutation.Optimal (✓ std3).

Citation. Kais Abdelkhalek; René Schwonnek; Hans Maassen; Fabian Furrer; Jörg Duhme; Philippe Raynal; Berthold-Georg Englert; Reinhard F. Werner (2015). Optimality of entropic uncertainty relations. DOI: 10.1142/S0219749915500458. URL: https://arxiv.org/abs/1509.00398v1.

Commentary.

Section II, p. 4, verbatim: ‘We call a state ρ optimal if ρ′⊑ρ implies ρ⊑ρ′, and hence f(ρ)=f(ρ′).’ The quantified competitor sigma ranges over all complex density states in the same dimension, with the same U and entropy orders.

Definition 1.6 (Independence conjecture).

$$claim\iff (\forall d\in \mathbb{N},\forall U\in \mathrm{unitaryGroup}\left(\mathrm{Fin}\left(d\right),\mathbb{C}\right),\forall alpha\in \mathbb{R},\forall beta\in \mathbb{R},\forall alphaPrime\in \mathbb{R},\forall betaPrime\in \mathbb{R},(\frac{1}{2}<alpha)\Rightarrow ((\frac{1}{2}<beta)\Rightarrow ((\frac{1}{2}<alphaPrime)\Rightarrow ((\frac{1}{2}<betaPrime)\Rightarrow ((\frac{1}{alpha}+\frac{1}{beta}=2)\Rightarrow ((\frac{1}{alphaPrime}+\frac{1}{betaPrime}=2)\Rightarrow (\forall rho\in \mathrm{DensityState}\left(\mathrm{Fin}\left(d\right)\right),(\mathrm{Optimal}\left(U,alpha,beta,rho\right))\Rightarrow (\mathrm{Optimal}\left(U,alphaPrime,betaPrime,rho\right)))))))))$$

Formalization. D5/S3/Quantum/Information/RenyiOptimalStateDependenceRefutation.claim (✓ std3).

Citation. Kais Abdelkhalek; René Schwonnek; Hans Maassen; Fabian Furrer; Jörg Duhme; Philippe Raynal; Berthold-Georg Englert; Reinhard F. Werner (2015). Optimality of entropic uncertainty relations. DOI: 10.1142/S0219749915500458. URL: https://arxiv.org/abs/1509.00398v1.

Commentary.

Conjecture V.8, section V.E, p. 21, ‘(Independence of the optimal states of (α,β))’: ‘If ρ is an optimal state for any unitary operator and any α,β>½ satisfying the duality relation (2), then ρ is also an optimal state for all other dual pairs.’ The optimality definition is, verbatim (section II, p. 4): ‘We call a state ρ optimal if ρ′⊑ρ implies ρ⊑ρ′, and hence f(ρ)=f(ρ′).’ Encoding: d is a natural dimension, U is any complex unitary, rho is any density state, alpha and beta are the first finite dual pair, and alphaPrime and betaPrime are the second. Both pairs obey 1/alpha + 1/beta = 2 and every order exceeds 1/2. The source excludes the extremal pair {1/2,infinity}. Zero-based Fin(d) indices relabel its d basis outcomes.

Theorem 1.7 (A qutrit refutes order independence).

$$\neg claim$$

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

Resolves. Problems/abdelkhalek-et-al-2015-optimal-state-independence-refutation (refuted) by D5/S3/Quantum/Information/RenyiOptimalStateDependenceRefutation.result.

Source. Repository-derived.

Acknowledgement. Kais Abdelkhalek; René Schwonnek; Hans Maassen; Fabian Furrer; Jörg Duhme; Philippe Raynal; Berthold-Georg Englert; Reinhard F. Werner (2015). Optimality of entropic uncertainty relations. DOI: 10.1142/S0219749915500458. URL: https://arxiv.org/abs/1509.00398v1.

Commentary.

Use the real orthogonal overlap matrix with rows (sqrt(2)/2,sqrt(2)/2,0), (sqrt(2)/4,-sqrt(2)/4,sqrt(3)/2), and (sqrt(2)sqrt(3)/4,-sqrt(2)sqrt(3)/4,-1/2). The X basis projectors are OrthogonalRecordEntropy.pointerState specialized to Fin(3). The Y laws of these projectors are p0 = (1/2,1/2,0), p1 = (1/8,1/8,3/4), and p2 = (3/8,3/8,1/4). At orders (1,1), zero X entropy forces any dominating density state to be an X basis projector: the Shannon zero-entropy characterization forces a point mass, and positivity eliminates the off-diagonal entries. The Y entropies are log(2), (9/4)log(2)-(3/4)log(3), and (11/4)log(2)-(3/4)log(3); 27 < 32 makes the last two strictly larger than the first. Hence the first projector is optimal. At orders (3/5,3), the second projector has the same zero X entropy and Y entropy -(1/2)log(109/256) < log(2), since 1/4 < 109/256. It strictly dominates the first projector, so the latter is not optimal. Both pairs satisfy duality and have orders strictly above 1/2.

References

• Truth anchor: D5/S3/Quantum/Information/RenyiOptimalStateDependenceRefutation.Below
• Truth anchor: D5/S3/Quantum/Information/RenyiOptimalStateDependenceRefutation.H
• Truth anchor: D5/S3/Quantum/Information/RenyiOptimalStateDependenceRefutation.Optimal
• Truth anchor: D5/S3/Quantum/Information/RenyiOptimalStateDependenceRefutation.claim
• Truth anchor: D5/S3/Quantum/Information/RenyiOptimalStateDependenceRefutation.pX
• Truth anchor: D5/S3/Quantum/Information/RenyiOptimalStateDependenceRefutation.pY
• Truth anchor: D5/S3/Quantum/Information/RenyiOptimalStateDependenceRefutation.result
• Dependency: D5/S3/Entropy/EntropyEquality
• Dependency: D5/S3/Quantum/Foundation/FiniteStateChannel
• Dependency: D5/S3/Quantum/Information/OrthogonalRecordEntropy