Quantum State Tomography: Optimal Methods and Bounds
Explore sample-optimal tomography of quantum states, minimizing loss and maximizing classical descriptions. Learn about the necessary/sufficient number of copies, boundary cases, and local asymptotic normality. Dive into optimal measurements, symmetry determinations, related work, and lower bounds in quantum state estimation.
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Sample-optimal tomography of quantum states J eongwan Haah (MIT) Aram Harrow (MIT) Zhengfeng J i (ISCAS/Waterloo) Xiaodi Wu (MIT -> Oregon) Nengkun Yu (Guelph/Waterloo) QIPC Leeds 2015.9.28 arXiv:1508.01797
state tomography Goal: minimize loss, i.e. max E dist( , ) M classical description quantum states
how many copies? Distance measures Trace distance: = || - ||1 / 2 Infidelity: = 1 F( , ) = 1 || 1 /2 1 /2 ||1 2 n = f( ,d, r) n = g( ,d, r) copies necessary/sufficient. States d dimensions Assume rank r. How does n scale with d, r, , ?
boundary case: d=2 Estimation protocol Measure in {|0 , |1 } basis Let q := (#0 s) / n. output suppose distribution of q p n 1 / 2 1 /
Local Asymptotic Normality Kahn-Guta 0804.3876 -dependent covariance matrix implies optimal n f(d) / = g(d) / 2 for unknown f, g Bloch ball
boundary case: constant error Intuition has d2 - 1 real parameters n d2 bounded rank: rd parameters n rd # of copies # of parameters? plausible, but not a proof.
boundary case: r=1 Symmetry determines optimal measurement Fidelity and Trace distance are equivalent: F2+ 2 = 1 Explicit formula for all moments of F [Chiribella, 1010.1875]
our results related work [O Donnell-Wright, 1508.01907] ongoing work (speculative) product measurements [Kueng, Rauhut, Terstiege 1410.6913] product measurements
lower bound rank-r d-dim states form a manifold of dimension rd distance =0.1 packing net has size exp(rd)[Szarek 81] state estimation can transmit O(rd) bits
Lower bound (for r=d) An ensemble {pi, i} can transmit at most = S( i pi i) - i pi S( i) bits per copy [Holevo 73] Given an -net 1, ..., M, choose pi=1 /M, i= i n Choose an /10 net of states of the form S( i) =log(d)-O( 2) O(n 2) O(log M) d2 (from last slide)
Upper bound inspiration 1. Use symmetry, cf. spectrum estimation [Keyl-Werner 01] and rank-1 case [Holevo 79] 2. Use pretty-good measurement (PGM) [Belavkin 75] [Hausladen-Wootters 94]
symmetries of (Cd)n U2Ud! U U U U (Cd)4 = Cd Cd Cd Cd (1324)2S4! Schur-Weyl duality P
spectrum estimation n= q ( ) Im q irrep of GL(d) For d=2, analogous to J (total angular momentum). In general, spec( ) Measuring causes no disturbance. Thm: [Keyl-Werner, quant-ph/0102027] m tr q ( ) exp(-n D( || spec( )) nd2 n O(d2log(d/ ) / 2) for spectrum estimation substantial improvements by O Donnell-Wright, 1501.05028
pretty-good measurement [Belavkin 75] [Hausladen-Wootters 94] Given an ensemble {pi, i}, define Mi = -1 /2 pi i -1 /2 with = i pi i Classical analogue Given underlying distribution p(i), and observed j p(j|i), guess i with probability p(i |j)using Bayes rule. Thm: [Barnum-Knill, quant-ph/0004088] Pr[PGM correct] Pr[optimal measurement is correct]2 Thm: [Harrow-Winter, quant-ph/0606131] Given a set of M states with pairwise infidelity , PGM requires O(log(M)/ ) copies to distinguish w.h.p.
putting it together 1. First estimate spectrum using Keyl-Werner. Measurement yields estimate . Do PGM with { = U U : U uniform} lemma: m 2 tr q (U U ) F( ,U U )2n nrd 2. ...a little more algebra... thm: Pr[guess | ] F( , )2n nO(rd)
things we dont know Efficiency? Not even known for pure states. Process tomography Other prior distributions / assumptions about Adaptive measurements Continuous-variable tomography
