Minimax estimation of qubit states with Bures risk

Abstract

The central problem of quantum statistics is to devise measurement schemes for the estimation of an unknown state, given an ensemble of n independent identically prepared systems. For locally quadratic loss functions, the risk of standard procedures has the usual scaling of 1/n. However, it has been noticed that for fidelity based metrics such as the Bures distance, the risk of conventional (non-adaptive) qubit tomography schemes scales as 1/ √ n for states close to the boundary of the Bloch sphere. Several proposed estimators appear to improve this scaling, and our goal is to analyse the problem from the perspective of the maximum risk over all states.We propose qubit estimation strategies based on separate and adaptive measurements, that achieve 1/n scalings for the maximum Bures risk. The estimator involving local measurements uses a fixed fraction of the available resource n to estimate the Bloch vector direction; the length of the Bloch vector is then estimated from the remaining copies by measuring in the estimator eigenbasis. The estimator based on collective measurements uses local asymptotic normality techniques which allows us derive upper and lower bounds to its maximum Bures risk. We also discuss how to construct a minimax optimal estimator in this setup. Finally, we consider quantum relative entropy and show that the risk of the estimator based on collective measurements achieves a rate O(n −1 log n) under this loss function. Furthermore, we show that no estimator can achieve faster rates, in particular the 'standard' rate n −1 . arXiv:1708.04941v2 [quant-ph]

Multi-parameter estimation beyond quantum Fisher information

“…Although QLAN results can be proved for restricted models around such states (e.g. pure state models), the general asymptotic analysis for boundary states needs to be dealt with separately (see [127] for the qubits minimax theory) and will not be discussed here.…”

A comparative study of estimation methods in quantum tomography

“…More recently, quantum tomography has become a crucial validation tool current quantum technology applications [38,58,29,67]. The experimental challenges have stimulated research in new directions such as compressed sensing [35,54,27,70,20,5,3], estimation of permutationally invariant states [73], adaptive and selflearning tomography [55,34,60,26,39,63], incomplete tomography [74], minimax bounds [25,4], Bayesian estimation [15,33], and confidence regions [7,18,71,24,53]. Since 'full tomography' becomes impossible for systems composed of even a f < l a t e x i t s h a 1 _ b a s e 6 4 = " 7 j z U X Q l i 6…”

“…More recently, quantum tomography has become a crucial validation tool current quantum technology applications [38,58,29,67]. The experimental challenges have stimulated research in new directions such as compressed sensing [35,54,27,70,20,5,3], estimation of permutationally invariant states [73], adaptive and selflearning tomography [55,34,60,26,39,63], incomplete tomography [74], minimax bounds [25,4], Bayesian estimation [15,33], and confidence regions [7,18,71,24,53]. Since 'full tomography' becomes impossible for systems composed of even a f < l a t e x i t s h a 1 _ b a s e 6 4 = " 7 j z U X Q l i 6 6 y S g D b 8 y D n C 2 L x + t k s = " > A A A B + X i c b V C 7 T s M w F L 0 p r 1 J e A U Y W i x a J q U q 6 A F s l F s Y i 0 Y f U R p X j O q 1 V x 4 5 s p 1 I V 9 U 9 Y G E C I l T 9 h 4 2 9 w 2 g z Q c i T L R + f c K x + f M O F M G 8 / 7 d k p b 2 z u 7 e + X 9 y s H h 0 f G J e 3 r W 0 T J V h L a J 5 F L 1 Q q w p Z 4 K 2 D T O c 9 h J F c R x y 2 g 2 n 9 7 n f n V G l m R R P Z p 7 Q I M Z j w S J G s L H S 0 H V r g 1 D y k Z 7 H 9 s q i R W 3 o V r 2 6 t w T a J H 5 B q l C g N X S / B i N J 0 p g K Q z j W u u 9 7 i Q k y r A w j n C 4 q g 1 T T B J M p H t O + p Q L H V A f Z M v k C X V l l h C K p 7 B E G L d X f G x m O d Z 7 N T s b Y T P S 6 l 4 v / e f 3 U R L d B x k S S G i r I 6 q E o 5 c h I l N e A R k x R Y v j c E k w U s 1 k R m W C F i b F l V W w J / v q X N 0 m n U f e 9 u v / Y q D b v i j r K c A G X c A 0 + 3 E A T H q A F b S A w g 2 d 4 h T c n c 1 6 c d + d j N V p y i p 1 z + A P n 8 w d r q 5 N 5 < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 7 j z U X Q l i 6 6 y S g D b 8 y D n C 2 L x + t k s = " > A A A B + X i c b V C 7 T s M w F L 0 p r 1 J e A U Y W i x a J q U q 6 A F s l F s Y i 0 Y f U R p X j O q 1 V x 4 5 s p 1 I V 9 U 9 Y G E C I l T 9 h 4 2 9 w 2 g z Q c i T L R + f c K x + f M O F M G 8 / 7 d k p b 2 z u 7 e + X 9 y s H h 0 f G J e 3 r W 0 T J V h L a J 5 F L 1 Q q w p Z 4 K 2 D T O c 9 h J F c R x y 2 g 2 n 9 7 n f n V G l m R R...…”

“…More recently, quantum tomography has become a crucial validation tool current quant um technology applications [29,38,58,66]. The experimental challenges have stimulated research in new directions such as compressed sensing [3,5,20,27,35,54,69], estimation of permutationally invariant states [72], adaptive and selflearning tomography [26,34,39,55,60,62], incomplete tomography [73], minimax bounds [4,25], Bayesian estimation [15,33], and confidence regions [7,18,24,53,70]. Since 'full tomography' becomes impossible for systems composed of even a moderate number of qubits, research has focused on the estimation of states which belong to smaller dimensional models which nevertheless capture relevant physical properties, such as low rank states [17,36,37,48,49] and matrix product states [12,20,50].…”

“…The mean squared Frobenius error E ρ − ρ 2 2 of the estimators for 4 qubits rank-r states of equal eigenvalues with k = 100 random bases and different repetition numbers. approximated by the sum of Hellinger distance and a quadratic form in the parameters of the unitary connecting the eigenbases of the two states[4] …”