Asymptotic equivalence of the global and local minimax cost

I will compare the minimal cost obtainable within two paradigms. In the first one, allN resources (quantum gates) are used in a fully optimal way (which I will analyze within minimax formalism).

In the second, there is a restriction that only n gates may be used in a single trial, while the experiment is repeated a large number of times k (which may be effectively described within QFI formalism).

While the first one demands defining a finite-sized set, in which the procedure is designed to work wellΘ, the second is strictly focused on a single pointθ₀. To make a reasonable comparison, in a single parameter case inCh. 4, I have chosen Θ = [θ0−δ/2, θ0+δ/2]and I have written the exact bounds for finite region size δ, showing, that its impact decreases with increasingN.

Here I use an analogous idea, but I will not investigate the rate of convergence. Instead, I focus only on asymptotic results. A simple approach, which captures the essential problem, has been proposed in [Hayashi [2011]] (originally for single parameter case), so-called local asymptotic minimax cost.

Let us define δ−neighborhood of θ0 as Θ(θ0, δ) = {θ : ∀_iθi ∈ [θ0i−δ/2,θ0i +δ/2]}. Consider the sequence of triples: initial states, N unitary controls and measurements, for each possible N, namely (ρ^N,{V_i^N},{M_˜^N

θ }). Then for N-gates protocol the output state is given as ρ^N_θ = V_N^N◦(E_θ⊗11)◦...V₁^N ◦(E_θ⊗11)◦ρ^N. By local asymptotic minimax cost around pointθ0 with cost matrix C_θ0 we define:

inf

{(ρ^N,{V_i^N},{M_˜^N

θ })}lim

δ→0 lim

N→∞N² sup

θ∈Θ(θ₀,δ)

Z

dθ˜Tr(ρ^N_θM_θ˜^N)tr(C_θ0(˜θ−θ)(˜θ−θ)^T), (6.3) Note that the order of taking limits limδ→0limN→∞ is crucial here – for the opposite order, the trivial constant estimator ˜θ = θ₀ would lead to zero cost, while the current form corresponds to the analysis fromCh. 4, but without the necessity of performing calculations for finiteδ/N. Now, I will generalize the reasoning from [Hayashi[2011]] (originally performed for U(1)estimation) for our case.

Theorem. Given a channel E_g(ρ) = UgρUg^†, being a unitary representation of the group g ∈ G.

Let θ = [θ₁, ..., θ_p] be the local parametrization of the group element around some pointg_θ0. Let C(g_θ, g_θ˜) be the cost invariant for acting the group, and C_θ is its Hessian with respect to variable θ˜, at point θ (we also assume some basic regularity properties, which will be given later). Then,

for the most general adaptive scheme, the local asymptotic minimax cost is the same as the global asymptotic minimax cost (while the second one is a covariant problem):

∀_θ0 inf

{(ρ^N,{V_i^N},{M_˜^N

θ })}lim

δ→0 lim

N→∞N² sup

θ∈Θ(θ0,δ)

Z

dθTr(M˜ ˜θ^Nρ^N_θ)tr(Cθ0(˜θ−θ)(˜θ−θ)^T)

= lim

N→∞N² inf

(ρ^N,{V_i^N},{M_g˜^N})

sup

g∈G

Z

d˜gTr(ρ^N_g M_˜g^N)C(g,˜g)

! (6.4)

Proof. We start with the observation that the asymptotic minimax cost for g∈G_δ ⊂G is for sure smaller or equal to the asymptotic minimax cost forg∈G:

inf

{(ρ^N,{V_i^N},{M_˜^N

θ})}lim

δ→0 lim

N→∞N² sup

g∈G_δ

Z

d˜gTr(MN,˜gρN,g)C(g,˜g)

≤ lim

N→∞N² inf

(ρ^N,{V_i^N},{M_˜g^N})

sup

g∈G

Z

d˜gTr(ρ^N_g M_˜g^N)C(g,˜g)

!

, (6.5) whereG_δ⊂G isδ−neighborhood of the group neutral elemente. I start by showing that they are in fact equal.

Similarly, as in the single parameter case, I introduce the notation for the minimax cost with a finite δ, N:

minimax(G_δ, N) = inf

(ρ^N,{V_i^N},{M_˜g^N})

sup

g∈G_δ

Z

d˜gTr(ρ^N_g M_g˜^N)C(g,g).˜ (6.6) In this notation, RHS ofEq. (6.5)is simplylimN→∞N²minimax(G, N), while LHS may be bounded from below bylimδ→0limN→∞N²minimax(G_δ, N)(as takinginf outside of limit may only increase the value of an objective function). It is also clear that

N→∞lim N²minimax(Gδ, N)≤ lim

N→∞N²minimax(G, N). (6.7) What remains to be proven is the weak inequality in the opposite direction:

N→∞lim N²minimax(Gδ, N)≥^? lim

N→∞N²minimax(G, N). (6.8) I do it by the following reasoning.

Having in total N gates, at first, I perform √

N independent single-gate experiments to find an approximated valuegest(for example, by using ML estimator). From the central limit theorem, the probability that the true value does not belong toδ−size neighborhood of the indication of estimator g /∈g_estG_δ (whereg_estG_δ is the set G_δ shifted by the action of g_est), decreases exponentially with a

number of measurementsperr(√

N)∝e⁻

√ N. Then I spend remainingN−√

N gates to perform optimal estimation strategy forg∈g_estG_δ. Since such a two-step strategy is, in general, only suboptimal, I have:

minimax(G, N)≤perr(

√

N)cmax+ (1−perr(

√

N))minimax(gestGδ, N −√

N), (6.9) where cmax = maxg,˜gC(g,g)˜ . Moreover, due to the symmetry of the whole problem, the RHS does not depend on g_est. After application of limN→∞N²· to the both sites, and the use of limN→∞ (N−√

N)²

N² = 1, we obtain:

Nlim→∞N²minimax(G, N)≤ lim

N→∞N²(Gδ, N −√

N) = lim

N→∞N²(Gδ, N), (6.10) which, together withEq. (6.7) gives:

N→∞lim N²minimax(Gδ, N) = lim

N→∞N²minimax(G, N). (6.11) Moreover, even if it was formulated forG_δ around neutral element e, due to covariant properties of the problem, it remains valid also for anyg_θ0G_δ.

Now I will argue that replacingC(g_θ, g˜θ)by its Hessian inθ0does not change the minimal obtainable value. To show that we need to justify that for the protocol minimizing LHS of Eq. (6.4), the probability of the gross errorθ˜−θ is negligible. I do it in the following way.

For any protocols sequence{(ρ^N,{V_i^N},{M_˜^N

θ })}, for any finiteδfwe introduce following correction.

If the result of measurement satisfies θ˜ ∈ Θ(θ₀, δ_f), we leave it unchanged. However, if any θ˜_i ∈/ [θ_0i−δ/2, θ_0i+δ/2], we change these result fromθ˜_itoθ_0i. For any sequence, such a change may only decrease the value of LHS of Eq. (6.4), so for the optimal one, it does not change. Therefore, for the protocols locally optimal around θ₀ (in the sense ofEq. (6.3)), for any finiteδ_f, the probability of getting result outside of Θ(θ0, δf) is negligible.

On the other hand, for sufficiently regular cost function:

∀_ϵ∃_δf∀_{θ,˜θ∈Θ(θ}

0,δf)|C(g_θ, g_˜θ)−(θ−θ)˜ ^TC_θ0(θ−θ)| ≤˜ ϵ·(θ−θ)˜ ^TC_θ0(θ−θ)T˜ =ϵ·tr(C_θ0(θ−˜θ)(θ−˜θ)^T) (6.12) and from that:

∀_ϵ∃_δf 1

1 +ϵC(θ,θ)˜ ≤tr(C_θ0(θ−θ)(θ˜ −˜θ)^T)≤ 1

1−ϵC(θ,θ).˜ (6.13) The reasoning may be repeated for anyϵ, which proves the statement. □

However, fromSec. 5.3, we know that for the covariant problem, the parallel covariant one is optimal

among all (adaptive or not) protocols. We may finally state:

∀_θ0 inf

{(ρ^N,{V_i^N},{M_˜^N

θ })}

δ→0lim lim

N→∞N² sup

θ∈Θ_δ

Z

dθ˜Tr(M_θ˜^Nρ^N_θ)tr(C(˜θ−θ)(˜θ−θ)^T)

= lim

N→∞N² inf

ρ^N,M_e^N

Z

d˜gTr((U_˜g^†⊗11)^⊗NM_e(U_˜g⊗11)^⊗Nρ)C(e,g),˜ (6.14)

Further, I will be interested mainly in the examples where the Hessian is the identity matrix, so the figure of merit is the sum of MSEs tr(C(θ−˜θ)(θ−θ)˜ ^T) =Pp

i=1∆²θ˜_i. I will analyze the optimal achievable precision for the cases where:

• all N resources are used optimally (analyzed within minimax formalism) or there is an addi- tional constraint for the maximal amount of resources used in single trialn, which is repeated many times k→ ∞ (analyzed with the use of quantum Cramér-Rao bound).

• all parameters are measured jointly (JNT) in a fully optimal way, or from the very beginning, the resources are divided between parameters, which are measured separately (SEP).

These give four options, for which the sum of variances Pp

i=1∆²θ˜_i will be labeled as ∆d²θ˜_JNT,

∆²θ˜^CR_JNT, ∆d²θ˜_SEP, and ∆²θ˜^CR_SEP respectively (see also Fig. 6.1 for a graphical explanation on the example). In a further discussion, I will be interested only in asymptotic values (N → ∞ or k→ ∞), which will be denoted by ≃.

No documento Heisenberg Limit beyond Quantum Fisher Information (páginas 69-72)