Poisson Quantum Information

By taking a Poisson limit for a sequence of rare quantum objects, I derive simple formulas for the Uhlmann fidelity, the quantum Chernoff quantity, the relative entropy, and the Helstrom information. I also present analogous formulas in classical information theory for a Poisson model. An operator called the intensity operator emerges as the central quantity in the formalism to describe Poisson states. It behaves like a density operator but is unnormalized. The formulas in terms of the intensity operators not only resemble the general formulas in terms of the density operators, but also coincide with some existing definitions of divergences between unnormalized positive-semidefinite matrices. Furthermore, I show that the effects of certain channels on Poisson states can be described by simple maps for the intensity operators.


Introduction
The Poisson limit theorems, also called the laws of rare events or the laws of small numbers, underlie the ubiquity of Poisson statistics and enable a variety of simplifications in probability theory [1,2]. In this paper, I examine the consequences of taking a similar limit in quantum information theory and demonstrate that elegant formulas emerge under the limit. This body of work may be called Poisson quantum information, which has the potential to grow into a fruitful research topic on par with Gaussian quantum information [3,4].
The results put forth are a generalization and culmination of our earlier efforts concerning weak thermal light [5][6][7][8]. The results are especially relevant to the recent literature on the application of quantum information theory to partially coherent imaging [8][9][10][11][12][13][14][15][16][17], where there is some confusion regarding the correct formulas for the information quantities, leading to inconsistent results by different groups. This paper resolves the debate in support of Refs. [6,10] and extends the results, beyond the Helstrom and Fisher information quantities considered there. While special weakthermal-light models have already found success by enabling substantial simplifications in many previous studies in quantum optics [5,6,8,[18][19][20][21], the general theory here stands on its own and does not require the quantum state to be bosonic or exactly thermal, so it may be applied to particles and systems beyond photons, such as electrons and quasiparticles, whenever a quantum treatment of rare objects is needed.

Poisson states
Let P(H) be the set of positive-semidefinite operators on a Hilbert space H and P 1 (H) be the unit-trace subset P 1 (H) ≡ {τ ∈ P(H)| tr τ = 1}. Consider the Hilbert space (H 0 ⊕ H 1 ) ⊗M for M temporal modes, where H 0 is a 1-dimensional vacuum Hilbert space and H 1 is a d-dimensional Hilbert space for a quantum object. Let the density operator ρ M ∈ P 1 [(H 0 ⊕ H 1 ) ⊗M ] be ρ M = τ ⊗M , τ = 1 l=0 π l τ l , π 0 = 1 − , where τ 0 ∈ P 1 (H 0 ) is the vacuum state, τ 1 ∈ P 1 (H 1 ) is the density operator for the quantum object, and 0 ≤ ≤ 1 is the probability of having one object in each temporal mode. See, for example, Ref. [22] for the definitions of the mathematical concepts. In the context of optics, τ 1 may denote the one-photon density operator in d spatial and polarization modes. The formalism, in itself, is generic however-the object can be any elementary or composite quantum system. For example, to study intensity interferometry [23], where two-photon-coincidence events are postselected and one-photon events are ignored, τ 1 can be used to denote the two-photon density operator. Define the Poisson limit as (2.2) where N is the expected object number in total. In this limit, a central quantity in the ensuing theory is Γ ≡ N τ 1 ∈ P(H 1 ), (2.3) which I call the intensity operator. Similar to τ 1 , it is a positive-semidefinite operator on H 1 . Unlike τ 1 , however, its trace is not normalized, as the object number need not be conserved in a problem.
2 , . . . ) be the vectoral random variable from a multi-output objectcounting measurement of the kth temporal mode. Let its probability distribution be where I l is the projection operator into H l and E is a positive operator-valued measure (POVM) on H 1 that satisfies E j ∈ P(H 1 ) and j E j = I 1 . Under the Poisson limit, the integrated random variable M ≡ M k=1 N (k) has the Poisson distribution [1,24] where each intensity value is given by Appendix A states, in more rigorous terms, a Poisson limit theorem that gives Eqs. (2.7) and (2.8).
In view of the Poissonian properties, the quantum state given by Eqs. (2.1), together with the Poisson limit given by Eqs. (2.2), may be called a Poisson state, denoted as ρ without the subscript M . Although a rigorous treatment of Poisson states may require quantum stochastic calculus [25] or nonstandard analysis [26,27], in the following I take the more convenient approach of assuming Eqs. (2.1) and taking the Poisson limit at the end of a calculation. Appendix B discusses how Poisson states may be defined more rigorously in terms of Fock spaces.

Information quantities
To define information quantities, assume two Poisson states ρ and ρ . Assume that M and τ 0 are the same for the two states, while the other quantities may vary. The quantities for the second state are denoted with a prime; for example, the intensity operator is Γ and N ≡ M = tr Γ . Assume further that = O( ) (order at most in the Poisson limit), such that the Poisson limit applies to both states. My first proposition concerns the Uhlmann fidelity [28], the most notable application of which is to set useful bounds for quantum hypothesis testing [29].

Proposition 1. The Uhlmann fidelity
between two Poisson states is given by Taking the Poisson limit then leads to the proposition.
Equation (3.2) has a self-similar feature: it contains a fidelity expression, in terms of the intensity operators Γ and Γ , that has the same form as the general formula for the density operators. Another remarkable feature is that the d 2 B quantity, defined in the following expression coincides with the squared Bures-Wasserstein distance between unnormalized positive-semidefinite operators [28,30]. The next two propositions concern the quantum Chernoff quantity and the relative entropy. The Chernoff quantity is used in the quantum Chernoff bound for hypothesis testing [29,31,32], while the relative entropy is, of course, a fundamental quantity in quantum thermodynamics [33] and communication theory [4,29].

Proposition 2. The quantum Chernoff quantity
for two Poisson states is given by The quantum Chernoff distance [31,32] is then Proposition 3. The relative entropy between two Poisson states is given by If supp Γ ⊆ supp Γ , where supp denotes the support [4], then supp ρ ⊆ supp ρ , and D(ρ ρ ) = D(Γ Γ ) = ∞.
The proofs of Propositions 2 and 3 are delegated to Appendix C. In Eq. (3.8), the relative entropy is expressed in a more general form so that it is appropriate for unnormalized positive-definite operators as well [34][35][36]. Similar to Eq. (3.2), Eqs. (3.6) and (3.9) have a self-similar feature. It is also remarkable that the D s (Γ, Γ ) and D(Γ Γ ) quantities, defined in the following expressions coincide with the alpha-divergences between unnormalized positive-semidefinite matrices in the matrix-analysis literature [35,36]-D s (Γ, Γ ) here is identical to Eq. (4.179) in Ref. [36] if one sets s = (1 − α)/2 and D(Γ Γ ) here is identical to Eq. (4.166) in Ref. [36]. The classical Poisson model given by Eq. (2.7) leads to analogous formulas for the corresponding quantities in classical information theory. The proofs are trivial and omitted for brevity; see, for example, Ref. [37] for similar results. Assume two Poisson distributions and again denote the quantities for the second distribution with a prime.

Proposition 4. The Chernoff quantity
for two Poisson distributions is given by The Chernoff distance is then Equations (3.13) and (3.16) resemble Eqs. (3.6) and (3.9) and also possess a self-similar feature. Similar to the quantum case, the D s (Λ, Λ ) and D(Λ Λ ) quantities, defined in the following expressions

18)
coincide with the alpha-divergences between unnormalized positive distributions [38]. D 1/2 (Λ, Λ )/2, in particular, is the squared Hellinger distance [30]. A fundamental relation between the quantum and classical information quantities is monotonicity. It is interesting to note that it manifests for the Poisson states on two levels: on the level of ρ and on the level of Γ.  Proof. These bounds follow directly from the monotonicity relations [29]. Alternatively, one may take Γ = N τ 1 and Γ = N τ 1 , define p j ≡ tr E j τ 1 = Λ j /N and p j ≡ tr E j τ 1 = Λ j /N , and apply the monotonicity relations with respect to (τ 1 , τ 1 ) and (p, p ) in order to obtain which also lead to the bounds via Propositions 1-5.
Last but not the least, I present propositions concerning the Helstrom information and the Fisher information [18,29], which play crucial roles in parameter estimation. Let and τ 1 be functions of an unknown vectoral parameter θ ≡ (θ 1 , . . . , θ q ) ∈ Θ ⊆ R q . It follows that ρ M and Γ are also functions of θ.

Proposition 7. Define the q × q Helstrom information matrix as
where a • b ≡ (ab + ba)/2 denotes the Jordan product and σ µ is a symmetric logarthmic derivative (SLD) of ρ M , defined as a Hermitian-operator solution to For a Poisson state, K is given by where S µ is an SLD of Γ, viz., a Hermitian-operator solution to Proof. Given the tensor product in Eqs. (2.1), it is known that [29] K(ρ M ) = M K(τ ). (3.29) Given the direct sum in Eqs. (2.1), it can also be shown that Taking the Poisson limit leads to the proposition.
Proposition 8 (well known; see, for example, Refs. [1,2]). The Fisher information matrix for a Poisson distribution is given by Again, with the Poisson limit, the Helstrom and Fisher information quantities observe a selfsimilar feature-they are given by the same formulas as the general ones, except that the unnormalized Γ or Λ is substituted into each.
The information quantities obey a monotonicity relation, which follows from the monotonicity of Helstrom information in two ways.
Proof. The monotonicity [29] holds on two levels: K(ρ) ≥ J(P M ), and also K(Γ) ≥ J(Λ), which can be proved independently by expressing Γ and Λ in terms of N , τ 1 , and p = Λ/N and applying the monotonicity relation with respect to τ 1 and p.

Poisson states versus Fock states
It is important to emphasize that the Poisson theory is in general different from the usual theory for the Fock state τ ⊗L 1 . For example, from Eq. (3.2), the fidelity for the Poisson states can be expressed in terms of τ 1 and τ 1 as As Γ and Γ are not normalized, their fidelity is bounded as When F (τ 1 , τ 1 ) = 1 and F (Γ, Γ ) = √ N N , F (ρ, ρ ) in the Poisson theory is still less than 1 if N = N , because the different expected object numbers still lead to distinguishability. On the other hand, if F (τ 1 , τ 1 ) = 0 and F (Γ, Γ ) = 0, F (ρ, ρ ) is still positive, because both states contain the identical τ 0 . Another example is the Helstrom information which is different from It is not difficult to prove that, on the per-object basis, as N may also depend on θ and the total object number may give extra information.
In the context of optics, the thermal state in an ultraviolet limit can be shown to approach the Poisson state, with the mutual coherence matrix in statistical optics [23] becoming a matrix representation of Γ [5,6,8]. Alternative analyses of the exact thermal state have yielded results that are consistent with the Poisson theory [19,[39][40][41]. In the context of partially coherent imaging, Propositions 7 and 8 are consistent with our treatment in Ref. [10], although many other studies on this topic [9,[11][12][13]15] compute the Helstrom information using K(τ 1 ) only and may underestimate the amount of information. The Poisson model is more realistic than the Fock model because the former can account for the effects of inefficiency and loss, which are unavoidable for sources imaged with a finite aperture [17]. The use of K(τ 1 ) is justified only if N does not depend on θ, in which case K(Γ) = N K(τ 1 ). I illustrate these concepts with a concrete example. [8][9][10][11][12][13][14][15][16][17]. The intensity operator can be modeled as [10]

Example 1. Consider the imaging of two equally bright partially coherent optical sources via a diffraction-limited system in one dimension
is the expected photon number from one isolated source, γ ∈ C is the degree of coherence with |γ| ≤ 1, ψ : R → C is the point-spread function of the imaging system, |x is the Dirac position eigenket that obeys x|x = δ(x − x ), and θ ∈ R ≥0 is the separation between the two sources in Airy units. The expected total photon number is When γ = 0 (incoherent sources), N = 2N 0 is independent of θ, but when γ = 0, N may depend on θ, as the waves from the two sources can interfere, thus enhancing or suppressing the radiation energy. For example, when γ = 1 (fully coherent and in-phase sources) and θ = 0, N = 4N 0 , which is consistent with the elementary fact that n identical in-phase sources with zero separations should radiate at a power ∝ n 2 [42].
To compute the Helstrom information K(Γ), I assume the Gaussian point-spread function and use the steps detailed in Appendix D to compute Eqs. (3.27) and (3.28) numerically. The results are plotted in Fig. 1 (3.27) and (3.28), without returning to the rare-photon model or treating and τ 1 separately in the calculations. As also noticed in Ref. [17], the Helstrom information plotted in Fig. 1 appears to be identical to the Fisher information with the Hermite-Gaussian measurements plotted in Ref. [10, Fig. 1] for all values of γ, suggesting that the measurements are optimal for any γ and not just for the γ = 0 case proven in Ref. [6].

Poisson channels
For the Poisson theory to remain useful for problems involving quantum channels, the channels should preserve the Poissonianity of a state. I call such channels Poisson channels. A tracepreserving completely positive (TPCP) map Φ on ρ M is Poisson if where τ l ∈ P 1 (H l ), H 0 is 1-dimensional, and = O( ), such that the output is still a Poisson state in the Poisson limit. Let ρ(Γ) denote the Poisson state with intensity operator Γ in the sense of Eqs. (2.1)-(2.3). The input-output relation in the Poisson limit can be abbreviated as Proof. These monotonicity relations follow from the monotonicity of the information quantities with respect to the density operators in Eqs. To construct examples ofΦ in the following, I assume further that Φ is local, in the sense of where φ : P(H 0 ⊕ H 1 ) → P(H 0 ⊕ H 1 ) is a TPCP map that gives Under the local assumption,Φ is affine, as shown by Proposition 11 below, although Example 3 later demonstrates thatΦ need not be trace-preserving and Example 4 demonstrates thatΦ need not be linear.

Proposition 11. Given Eqs. (5.7) and (5.8),Φ is affine.
Proof. Let the Kraus form [22,29] of φ be where 0 denotes a zero operator and A αlm is an operator that maps H m to H l , it can be shown that For the = O( ) requirement to be satisfied, it is necessary for the map to satisfy Multiplying both sides of Eq. (5.11) by M and taking the Poisson limit, the output intensity operator becomesΦ , which does not depend on Γ. Equation (5.14) is affine with respect to Γ (though not necessarily linear).
where g is the density matrix of τ 1 with respect to the basis above.

16)
is a TPCP map. In the Poisson limit,Φ The trace-preserving nature of φ 1 means that the channel preserves the object number. More specific examples include the unitary map which models a lossless linear device via a unitary operator U , and the map which models a 100%-efficient object-counting measurement via the POVM E that gives Eqs. (2.7) and (2.8). Propositions 6 and 9 are hence special cases of Proposition 10.
As the object number need not be conserved,Φ need not be trace-preserving. The next examples ofΦ are noteworthy departures from the usual TPCP maps in quantum information theory.

Example 3 (loss). Let H l = H l and
where 0 ≤ η j ≤ 1 is the transmission coefficient for each mode. Then

32) implies that thisΦ is not trace-preserving and not even linear, as a tracepreserving or linear map on the zero operator must give the zero operator.
Setting , the spontaneous-emission probability per temporal mode, to be O( ) ensures that the occurrence of objects remains rare and N = N + N remains finite under the Poisson limit. In reality, however, a channel with a high spontaneous-emission probability may make the final state non-Poisson and may also involve other processes such as stimulated emission, so one should double-check the accuracy of the Poisson approximation before applying the Poisson theory to a real spontaneous-emission channel. Equation (5.31) is a quantum analog of the fact that the sum of two independent Poisson processes is also a Poisson process. It may be useful for modeling dark counts or background noise [43][44][45]. Another version of this fact is the following.

Example 5 (composition). The tensor product of two Poisson states may be expressed as
To derive this relation, consider Ignore all O( 2 ) terms in Eq. (5.34), including the two-object component τ 1 ⊗ τ 1 . The output intensity operator in the Poisson limit becomes Since τ 0 = |0 0| and τ 0 = |0 0 | are 1-dimensional, any tensor product of an operator with τ 0 or τ 0 is isomorphic to the original operator, and Eq. (5.35) can be abbreviated as Γ ⊕ Γ .
Example 6 (marginalization). Let Γ be an intensity operator on H 1 ⊕ H 1 and τ be a density operator on denote the partial trace with respect to H 0 ⊕ H 1 . A Kraus form is where I 1 is the projection operator into H 1 given by Eq. (5.29).

Conclusion
In conclusion, I have shown that the Poisson limit leads to elegant results in quantum information theory, with the intensity operator emerging as the central quantity. The familiar appearances of the formulas mean that one may borrow existing results from general information theory and apply them to the intensity operators in the study of Poisson states. Although the unnormalized nature of the operators may require extra care, one can still take advantage of many known results concerning unnormalized positive-semidefinite matrices [22,35,36,38], as the formulas here coincide with many of them.
There are many potential generalizations and extensions. It should be possible to define the Poisson limit more generally for a tensor product of non-identical states and an infinite-dimensional Γ, in analogy with the more general Poisson limit theorems [1,2]. The mathematical rigor of the limit may be improved through quantum stochastic calculus [25] or nonstandard analysis [26,27]. The mathematical theory of Poisson channels may be refined further.
In terms of applications, some specialized aspects of the Poisson theory have already found success in the study of weak thermal light for optical sensing and imaging [8,18], but a consolidation of the results under the umbrella of Poisson quantum information may reveal new insights. Given the importance of the classical Poisson theory in diverse areas [1,2], the quantum Poisson theory is envisioned to see wider applications in quantum technologies beyond optics, wherever a sequence of rare objects may be encountered.

A Poisson limit theorem
To state the Poisson limit theorem rigorously, I first rephrase the measurement model in terms of measure theory. Consider a multi-output object-counting measurement of the kth temporal mode. Let Y denote the set of outputs and B denote its σ-algebra. For example, for an ideal direct-imaging measurement, each (x, y) ∈ Y = R 2 is a possible position of the object on the image plane, while each B ∈ B is a region of the image plane. Let X k ∈ {0, 1} be the total object number detected by the measurement. Assume where E denotes the expectation, [statement] is the Iverson bracket defined as and E[statement] is the probability that the statement is true. Conditioned on X k = 1, let Y k ∈ Y be the output that detects the object. Assume where E : B → P(H 1 ) is a POVM on H 1 . The object count in a set of outputs can then be modeled by the random measure N (k) : B → {0, 1}, defined as X k and Y k can be taken as independent random variables. Integrated over M temporal modes, the random measure becomes is the detected object number in total and the set {Z 1 , . . . , Z L M } is simply a relabeling of {Y k |X k = 1}, which is a set of L M independent and identically distributed (i.i.d.) random elements. As X is Bernoulli, L M is binomial. In the Poisson limit, L M becomes Poisson, denoted by L, and M M → M is called the Kac empirical point process [24].
is a Poisson process.
The intensity measure Λ : B → R ≥0 of the Poisson process is given by In the main text, the abbreviations N

B Poisson states on Fock spaces
(H 0 ⊕ H 1 ) ⊗M is called a toy Fock space in Refs. [25,46]. The Poisson limit of ρ M on the toy Fock space should remain well defined if one is willing to adopt nonstandard analysis, taking M to be an unlimited natural number and = N/M to be infinitesimal [26,27]. In the context of ordinary calculus, one may think of as O(dt), where dt is an infinitesimal interval in time or a similar continuous degree of freedom. In the Poisson channel theory, Eq. (5.12) implies that each Kraus operator A α10 should scale as O( , which is why quantum stochastic calculus may be needed to define the operators rigorously in the Poisson limit. A more standard way to deal with an infinite number of temporal modes in quantum stochastic calculus is to abandon the toy Fock space and use instead the Fock space F(L 2 [0, 1] ⊗ H 1 ), where F is defined as [25] and the L 2 [0, 1] space, modeling the temporal modes for one object, is defined as See Refs. [25,26,46] for discussions of the relation between the toy Fock space and F(L 2 [0, 1]⊗H 1 ).
It is an open problem, outside the scope of this paper, how Poisson states may be expressed on If the objects are collected in one place and their arrival times are ignored, one can assume a simpler Fock space where Π denotes an appropriate map, L M is the binomial random variable for the total object number, and τ ⊗0 1 = τ 0 is assumed. This representation may be useful because any function of density operators that satisfies a monotonicity relation with respect to TPCP maps can be computed using Π(ρ M ) instead of ρ M , by virtue of the following proposition. Many other results in this paper can also be rederived from Eq. (B.8). Equations (2.1) are more intuitive from the physics point of view, however, and are therefore used in the main text.

C Proofs of Propositions 2, 3, and 12
Proof of Proposition 2. To derive Eq. (3.6), take similar steps to those in the proof of Proposition 1 to obtain and then take the Poisson limit. Note that Eq. (3.7) takes the infimum of the Poisson limit, but to prove that it is the same as the Poisson limit of the infimum, one may need to prove the uniform convergence of C s (ρ M , ρ M ) to C s (ρ, ρ ) over 0 ≤ s ≤ 1, beyond the pointwise convergence just proved. This complication suggests that the Poisson limit is an intricate mathematical issue, and the Fock-space representation given by Eq. (B.8), the Poisson limit of which has already been taken, may be a better starting point for rigorous proofs. In particular, Eq. (B.8) leads directly to Eqs. (3.6) and (3.7) without any ambiguity.
Proof of Proposition 12. To prove that Π is TPCP, I construct a physical procedure that takes ρ M as input and produces Π(ρ M ) as output.
1. Designate the input system as system A and an auxiliary system with Hilbert space (H 0 ⊕ H 1 ) ⊗M and initial state τ ⊗M (c) If X k = 1, put the object detected in the kth temporal mode of system A (in conditional state I 1 τ I 1 /(tr I 1 τ I 1 )) into an empty temporal mode of system B (any mode in state τ 0 ).  With the Gaussian ψ(x) given by Eq. (4.10), G can also be computed analytically (with the help of the Symbolic Math Toolbox on Matlab (ver. 2020b, Mathworks)): To prevent the error "The solution of this Lyapunov equation does not exist or is not unique" in using the lyap function on Matlab, a tiny positive number δ is artificially introduced toΓ 33 andΓ 44 . To check that δ does not affect the results significantly, the numerical analysis is repeated with various δ, and it is found that the plots do not show any perceptible change with δ ranging from 10 −13 to 10 −5 . Figure 1 uses δ = 10 −13 .
The Matlab code is reproduced below. The lbmap routine to generate a color map for colorblind viewers can be downloaded from Ref. [49].