Asymptotic quantum estimation theory for the thermal states family

Masahito Hayashi^{1}

1Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan e-mail address: masahito@kusm.kyoto-u.ac.jp

Abstract

Concerning state estimation, we will compare two cases. In one case we cannot use the quantum correlations between samples. In the other case, we can use them. In addition, under the later case, we will propose a method which simultaneously measures the complex amplitude and the expected photon number for the thermal states.

Introduction

Quantum estimation is essentially different from classical estimation regarding the following two points. The first point is that we cannot simultaneously construct the optimal estimators corresponding to respective parameters because of non-commutativity between them. It has been a serious problem since the beginning of the quantum estimation [1, 2, 3]. The second point is that we can reduce the estimation error under the assumption that we can prepare independent and identical samples of the unknown quantum state. It was pointed by Nagaoka [4, 5] concerning the large deviation theory in one-parameter estimation. The purpose of this paper is to clear the second point concerning the mean square error (MSE).

Our situation is divided into the following two cases. In the first case, we estimate the unknown state by independently measuring every sample. In this case, we may decide the n-th POVM from n − 1 data which have been already given. In the second case, we estimate the unknown state by regarding n-sample system as a single composite system. In this case, we may use POVMs which are indivisible into every sample system. In order to construct these POVMs, we need to use quantum correlations between every sample. The former is called the non-quantum correlation case and the later the quantum correlation case. When the unknown state is a pure state, the errors of both are asymptotically equivalent in the first order [6]. Concerning the spin 1/2 system, see Hayashi [7].

In this paper, we formulate a general theory for the asymptotic quantum estimation. It is applied to the simultaneous estimation of the expected photon number and the complex amplitude for the quantum thermal state.

Asymptotic Estimation Theory

In this paper, we use a quantum state family S parameterized by finite parameters θ^{1}, ⋯, θ^{d}:

S := {ρ_{θ} ∈ S(H)|θ = (θ^{1}, …, θ^{d}) ∈ Θ ⊂ R^{d}},

where the set S(H) denotes the set of densities on H. For simplicity, we assume that ρ_{θ} is nondegenerate.

Non-quantum correlation case

The non-quantum correlation case is formulated as follows. A pair E_{n} = ({M_{k}}_{k=1}^{n}, θ̂_{n}) is called a recursive estimator where θ̂_{n} is a function estimating the unknown parameter from n data, and {M_{k}}_{k=1}^{n} is a sequence of POVMs M_{1}, M_{2}(ω_{1}), …, M_{n}(ω_{1}, …, ω_{n−1}) as follows: the n-th POVM M_{k}(ω_{1}, …, ω_{k−1}) is determined by k − 1 data which have been already given. A sequence {E_{n}}_{n=1}^{∞} of recursive estimators is called a recursive MSE consistent estimator if

lim_{n→∞} \underbrace{∫ ∫ ⋯ ∫}_{n} ∥θ̂_{n}(ω_{1}, ω_{2}, …, ω_{n}) − θ∥^{2} P_{ρ_{θ}}^{E_{n}}( dω_{1}, dω_{2}, …, dω_{n}) = 0, ∀θ ∈ Θ,

where

P_{ρ}^{E_{n}}( dω_{1}, dω_{2}, …, dω_{n}) = tr ρM_{1}( dω_{1}) tr ρM_{2}(ω_{1})( dω_{2}) … tr ρM_{n}(ω_{1}, …, ω_{n−1})( dω_{n}).

We define the non-quantum-correlational Cramér-Rao type bound C_{θ}^{NQC}(G) for a weighted matrix G(G is a d × d real positive symmetric matrix.) as:

C_{θ}^{NQC}(G) := inf {lim inf_{n→∞} Tr nGV_{θ}(E_{n})| {E_{n}}_{n=1}^{∞} is a recursive MSE consistent estimator } ,

where the MSE matrix V_{θ}(E_{n}) is given by:

V_{θ}^{i,j}(E_{n}) = \underbrace{∫ ⋯ ∫}_{n} (θ̂_{n}^{i}(ω_{1}, …, ω_{n}) − θ^{i}) (θ̂_{n}^{j}(ω_{1}, …, ω_{n}) − θ^{j}) P_{ρ_{θ}}^{E_{n}}( dω_{1}, …, dω_{n}).

We have the following equation:

C_{θ}^{NQC}(G) = inf {Tr G (J_{θ}^{M} )^{−1}| M is a POVM on H} ,

where J_{θ}^{M} denotes the Fisher information matrix at θ of {tr ρ_{θ}M( dω)|θ ∈ Θ}. It is derived by Jensen’s inequality [8]. Under some regular condition, we show that there exists a recursive MSE consistent estimator {E_{n}}_{n=1}^{∞} such that [8]:

n Tr GV_{θ}(E_{n}) → C_{θ}^{NQC}(G) as n → ∞, ∀θ ∈ Θ.

According Holevo[3], we have J_{θ}^{M} ≤ J̃_{θ}, where J̃_{θ} is the RLD Fisher information matrix defined as: J̃_{θ;i,j} := tr(L̃_{θ;i})^{∗}ρ_{θ}L̃_{θ;j}, L̃_{θ;i} := (ρ_{θ})^{−1} \frac{∂ρ_{θ}}{∂θ^{i}} . Therefore we have the following inequality

C_{θ}^{NQC}(G) ≥ C_{θ}^{R}(G),

where

C_{θ}^{R}(G) := inf {tr GV |V is a d × d real symmetric matrix V ≥ J̃_{θ}^{−1}}
= Tr G Re J̃_{θ}^{−1} + Tr |\sqrt{G} Im J̃_{θ}^{−1}\sqrt{G}| .

Quantum correlation case

Next, we formulate the quantum correlation case. For this purpose, we consider a quantum counterpart of independent and identically distributed condition. If H_{1}, …, H_{n} are n Hilbert spaces which correspond to the physical systems, then their composite system is represented by the tensor Hilbert space.

H^{(n)} := H_{1} ⊗ ⋯ ⊗ H_{n}.

Thus, a state on the composite system is denoted by a density operator ρ^{(n)} on H^{(n)}. In particular if n element systems (H_{1}, …, H_{n}) of the composite system H^{(n)} are independent of each other, there exists a density ρ_{k} on H_{k} such that

ρ^{(n)} = ρ_{1} ⊗ ⋯ ⊗ ρ_{n}, on H^{(n)}.

The condition:

H_{1} = ⋯ = H_{n} = H, ρ_{1} = ⋯ = ρ_{n} = ρ

corresponds to the independent and identically distributed condition in the classical case. Therefore, we consider the parameter estimation problem for the family {ρ_{θ}^{(n)} on H|θ ∈ Θ} which is called the n-i.i.d. extended family.

In this case, we use a sequence {M^{n}}_{n=1}^{∞} of POVMs where M^{n} is a POVM on H whose measurable set is R^{d} as an estimator. A sequence {M^{n}}_{n=1}^{∞} is called an MSE consistent estimator if {M^{n}}_{n=1}^{∞} satisfies

lim_{n→∞} ∥θ̂ − θ∥^{2} tr ρ_{θ}^{(n)}M^{n}( dθ̂) = 0, ∀θ ∈ Θ.

A recursive MSE consistent estimator can be regarded as an MSE consistent estimator because a recursive estimator E_{n} = ({M_{k}}_{k=1}^{n}, θ̂_{n}) is regarded as a POVM M(E_{n}) as follows:

M(E_{n})(B) := ∫_{θ̂_{n}^{−1}(B)} ⨂_{k=1}^{n} M_{k}(ω_{1}, …, ω_{k−1})( dω_{k}), ∀B ⊂ R^{d} on H^{(n)}.

We define the quantum-correlational Cramér-Rao type bound C_{θ}^{QC}(G) for a weighted matrix G as:

C_{θ}^{QC}(G) := inf {lim inf_{n→∞} n Tr GV_{θ}(M^{n})| {M^{n}}_{n=1}^{∞} is an MSE consistent estimator }

where the MSE matrix V_{θ}(M^{n}) is given by:

V_{θ}^{i,j}(M^{n}) = ∫_{R^{d}}(θ̂^{i} − θ^{i})(θ̂^{j} − θ^{j}) tr ρ_{θ}^{(n)}M^{n}( dθ̂).

We have the following equation

C_{θ}^{QC}(G) = lim inf_{n→∞} nC_{θ}^{n}(G),

where C_{θ}^{n}(G) denotes the non-quantum-correlational Cramér-Rao type bound for the n-i.i.d. extended family [8]. From the definition of the n-i.i.d. extended family, we have C_{θ}^{NQC}(G) ≥ nC_{θ}^{n}(G). Therefore, we have the first inequality of (1).

C_{θ}^{NQC}(G) ≥ C_{θ}^{QC}(G) ≥ C_{θ}^{R}(G).  (1)

It shows the second inequality of (1) that J_{θ}^{M^{n}} ≤ nJ̃_{θ} for any POVM M^{n} on H^{(n)}. Therefore the difference between C_{θ}^{NQC}(G) and C_{θ}^{QC}(G) means the difference of the quantum correlation case from the non-quantum correlation case. Under some regular condition, we can show that there exists an MSE consistent estimator {M^{n}}_{n=1}^{∞} such that [8]:

n Tr GV_{θ}(M^{n}) → C_{θ}^{QC}(G) as n → ∞, ∀θ ∈ Θ.

Quantum thermal states family

Now we consider the estimation for the the complex amplitude ζ and expected photon number N for the quantum thermal states family defined as:

S := \{ρ_{ζ,N} := \frac{1}{πN} ∫_{C} exp (−\frac{|ζ − α|^{2}}{N} ) |α⟩⟨α| d^{2}α| ζ ∈ C, N > 0\} .

Estimation of complex amplitude ζ

In the case of that photon number N is known, we estimate the tow unknown parameters ζ = (θ^{1} + iθ^{2})/\sqrt{2}. This estimation problem is investigated by Yuen, Lax and Holevo [2, 3]. In this case they calculated the inverse J̃_{θ}^{−1} of the RLD Fisher information matrix as:

J̃_{θ}^{−1} = ( \begin{matrix}N + \frac{1}{2} & \frac{i}{2} \\ −\frac{i}{2} & N + \frac{1}{2} \end{matrix} ) .

They calculated the non-quantum-correlational Cramér-Rao type bound C_{θ}^{NQC}(G) as follows:

C_{θ}^{NQC}(G) = C_{θ}^{R}(G) = 2 (N + \frac{1}{2}) g_{1} + \sqrt{g_{1}^{2} − g_{2}^{2} − g_{3}^{2}},  (2)

where the weighted matrix G is parameterized as:

G = ( \begin{matrix}g_{1} + g_{2} & g_{3} \\ g_{3} & g_{1} − g_{2} \end{matrix} ) .

From (1) and (2), we have the following equations.

C_{θ}^{NQC}(G) = C_{θ}^{QC}(G) = C_{θ}^{R}(G).  (3)

In this case, the optimal estimator is the squeezed heterodyne.

Simultaneous estimation of complex amplitude ζ and expected photon number N

Next we consider the case of that both of the expected photon number N and the complex amplitude ζ are unknown. In this case, we estimate three unknown parameters θ^{1}, θ^{2} and θ^{3} = N. The first equation of (3) isn’t held. Therefore, the squeezed heterodyne isn’t optimal. The inverse J̃_{θ}^{−1} of the RLD Fisher information matrix is calculates as:

J̃_{θ}^{−1} = ( \begin{matrix}N + \frac{1}{2} & \frac{i}{2} & 0 \\ −\frac{i}{2} & N + \frac{1}{2} & 0 \\ 0 & 0 & N(N + 1) \end{matrix} ) .

Therefore we can calculated C_{θ}^{R}(G) as:

C_{θ}^{R}(G) = g_{0}N(N + 1) + 2 (N + \frac{1}{2}) g_{1} + \sqrt{g_{1}^{2} − g_{2}^{2} − g_{3}^{2}},

if the weighted matrix G can be parameterized as:

G = ( \begin{matrix}g_{1} + g_{2} & g_{3} & 0 \\ g_{3} & g_{1} − g_{2} & 0 \\ 0 & 0 & g_{0} \end{matrix} ) .  (4)

If the weighted matrix G can be parameterized as (4), we obtain the following equations:

C_{θ}^{NQC}(G) > C_{θ}^{QC}(G) = C_{θ}^{R}(G).  (5)

A proof for C_{θ}^{NQC}(G) > C_{θ}^{QC}(G) is omitted. The inequality C_{θ}^{NQC}(G) > C_{θ}^{QC}(G) means that we cannot the simultaneous measurement of the photon number counting and heterodyne for a single sample.

Construction of an MSE consistent estimator {M^{n}}_{n=1}^{∞} attaining C_{θ}^{R}(I)

Now, for a weighted matrix I, we construct an MSE consistent estimator {M^{n}}_{n=1}^{∞} such that

lim_{n→∞} n tr V_{θ}(M^{n}) = C_{θ}^{R}(I), ∀θ ∈ Θ.

It is sufficient for C_{θ}^{QC}(I) = C_{θ}^{R}(I) to construct such an MSE consistent estimator. Every POVM M^{n} is constructed in the following step:

(1) Evolve the unknown state \underbrace{ρ_{ζ,N} ⊗ ⋯ ⊗ ρ_{ζ,N}}_{n} as:

\underbrace{ρ_{ζ,N} ⊗ ⋯ ⊗ ρ_{ζ,N}}_{n} → U_{n} \underbrace{ρ_{ζ,N} ⊗ ⋯ ⊗ ρ_{ζ,N}}_{n} U_{n}^{∗} = ρ_{\sqrt{n}ζ,N} ⊗ \underbrace{ρ_{0,N} ⊗ ⋯ ⊗ ρ_{0,N}}_{n−1} on H^{(n)},

where

U_{n} = exp ϕ_{n−1}(a_{n}^{∗}a_{1} − a_{1}^{∗}a_{n}) ⋯ exp ϕ_{2}(a_{3}^{∗}a_{1} − a_{1}^{∗}a_{3}) exp ϕ_{1}(a_{2}^{∗}a_{1} − a_{1}^{∗}a_{2}) on H^{(n)}
ϕ_{i} = arctan \frac{1}{\sqrt{i}}, i = 1, 2, …, n − 1.

a_{i} denotes the annihilation operator on H_{i}.

(2) Measure the first sample ρ_{\sqrt{n}ζ,N} by the heterodyne, then we get the estimate ζ̂ of the complex amplitude.

(3) Measure the others by the photon counting, then we obtain n − 1 data which obey the probability distribution P^{N}(k):

P^{N}(k) = \frac{1}{N + 1} (\frac{N}{N + 1})^{k} , k = 0, 1 ….

(4) We obtain the estimate N̂ of the expected photon number N by the maximum likelihood estimator of the probability distribution P^{N}(k_{1}), …, P^{N}(k_{n−1}).

Conclusion

We formulate an asymptotic quantum estimation theory. This theory is applied to the simultaneous measurement of the photon number counting and the heterodyne for thermal states. It is a future study to realize the MSE consistent estimator proposed in this paper in an actual physical system.

Acknowledgments

This work was supported by the Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists No. 9404. The author would like to thank Dr. K. Matsumoto for useful discussions of these topics.

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