# Emulation of quantum measurements with mixtures of coherent states

A. Mikhalychev  B. I. Stepanov Institute of Physics, NAS of Belarus,
Nezavisimosti ave. 68, 220072 Minsk, Belarus  Y. S. Teo  Department of Physics
and Astronomy, Seoul National University, 08826 Seoul, South Korea  H. Jeong
Department of Physics and Astronomy, Seoul National University, 08826 Seoul,
South Korea  A. Stefanov  Institute of Applied Physics, University of Bern,
Sidlerstrasse 5, CH-3012 Bern, Switzerland  D. Mogilevtsev  B. I. Stepanov
Institute of Physics, NAS of Belarus, Nezavisimosti ave. 68, 220072 Minsk,
Belarus<EMAIL_ADDRESS>
(February 2022)

###### Abstract

We propose a methodology to emulate quantum phenomena arising from any non-
classical quantum state using only a finite set of mixtures of coherent
states. This allows us to successfully reproduce well-known quantum effects
using resources that can be much more feasibly generated in the laboratory. We
present a simple procedure to experimentally carry out quantum-state emulation
with coherent states, illustrate it emulating multi-photon NOON states with
few phase-averaged coherent states, and demonstrate its capabilities in
observing fundamental quantum-mechanical effects, such as the Hong-Ou-Mandel
effect, violating Bell inequalities and witnessing quantum non-classicality.

##  I Introduction

Understanding the extent to which classical elements can be used to reveal
nontrivial quantum effects in experiments can offer a deeper perspective on
the interplay between classical and quantum resources. On the one hand, there
are cornerstone results obtained with essentially non-classical states, such
as antibunching of photons and suppression of the field amplitude noise below
the classical level [1, 2], exceeding the standard quantum limit in
measurement precision [3], violating Bell inequalities [4], and the exhibition
of non-classicality signatures [5, 6, 7]. On the other hand, interesting new
research on how these quantum effects can still be observed using classical
resources have emerged. For example, recently it was shown that Bell
inequalities violation and other quantum-like signatures may be brought to
light by “classical entanglement”, that is, by local classical correlations of
different degrees of freedom [8, 9, 10, 11, 12, 13].

In this work we investigate the potential of emulating experiments over a
quantum state using a set of ”classical” probe states, i.e., those possessing
a non-negative Glauber-Sudarshan P functions [6, 14], examples of which
include mixtures of coherent states. The underlying principle behind emulating
an arbitrary state with the density matrix $\rho$ is the linear-algebraic fact
that $\rho$ is expressible as a linear combination of a set of non-orthogonal
basis states, where some of the coefficients in such a linear combination can
be negative. This approach underlies the so-called “data pattern” method
developed for quantum tomography [15, 16, 17, 18, 19], and allows one to avoid
calibrating the measurement setup by fitting a response from an unknown state
to the responses from other known probe states [20]. We suggest the scheme
that works in somewhat opposite way: we fit the state with “probes” aiming to
produce the same measured response as the true quantum state. In a typical
scenario in quantum mechanics, the observer has access to the complete quantum
state $\rho$, with which the probabilities of all possible experimental
outcomes can be predicted. The essence of our quantum-state emulation is to
employ a classical preparation procedure that samples the basis states we want
in order to reproduce measurement results obtained with this $\rho$: either of
a particular measurement or of an arbitrary one (under certain reasonable
constraints). The respective coefficients that go with the basis states are
incorporated through post-processing. Our emulation scheme may be understood
as an _a priori_ assignment of classical information that specifies the
components of $\rho$ during state preparation, and is otherwise absent in a
non-emulation (usual) scenario. Noticeably, fitting of a particular
measurement does not require high fidelity of the representation with the true
quantum state. On the other hand, high-fidelity representation implies
accurate fitting of the measurement results for any observable with limited
eigenvalues. We show how to achieve such representation of exquisitely quantum
multi-photon NOON states by using just few phase-averaged coherent states.

Figure 1: The measurement of (a) a regular quantum statistical mixture and (b)
a general emulation representation stated in (1). In both kinds of state
preparation: the component $\rho_{i}$ is produced by the source with
probability proportional to $|c_{i}|$. In the case of a statistical mixture
state, estimation of the expectation value $\left<A\right>$ of an observable
$A$ requires no knowledge about how $\rho$ is prepared $(\rho_{i})$. In the
_emulation_ of $\rho$ that requires some negative $c_{i}$’s, classical
information concerning state preparation is necessary: The observer measures
either $A$ or $-A$ depending on the ancillary measurement of
$\rho_{\textsc{c}}$ with $B$.

To demonstrate feasibility of our approach, we show how to emulate fundamental
quantum results such as antibunching, violation of Bell inequalities, and
witnessing non-classicality, and provide estimation of classical resources
required for it. Our scheme has important practical applications. For example,
it allows one to test quantum effects when it is problematic or too expensive
to generate “true” non-classical states, such as the NOON states with several
photons, or, for example, for low wavelengths (such as the microwave spectral
region, where one needs implementing superconducting circuits [21]).

The outline of the paper is as follows. In the Section II we discuss the
ideology of our emulation scheme and show how one includes classical
information on the state preparation in the emulation set-up. In the third
Section we discuss high-fidelity representations of several few-photon states
including Fock and NOON-states. In the Section IV we describe the way of
witnessing non-classicality of emulated states with realistic single-photon
detectors. In ther Sections V, VI and VIII we discuss demonstration of the
Hong-Ou-Mandel effect, phase estimation and Bell testing with our emulated
states.

##  II Emulation scheme

Now let us demonstrate how it is possible to emulate an arbitrary measurement
over an arbitrary quantum state using info about the state preparation. We
assume emulating measurement of the observable $A$. Let us consider a state
described by the density matrix $\rho_{\mathrm{true}}$ and approximate it by
the operator $\rho$ represented through a set of probe states $\\{\rho_{i}\\}$
in the following way:

$\rho=\sum_{j}c_{j}\rho_{j},\quad\sum_{j}c_{j}=1,$ (1)

where the coefficients $c_{j}$ can be negative for non-orthogonal $\rho_{j}$.
The emulation of the measurement results is faithful if for an arbitrary small
$\epsilon>0$ we can find such representation (1) that

$|\mathrm{Tr}\\{A(\rho-\rho_{\mathrm{true}})\\}|<\epsilon.$

For any $A$ with limited eigenvalues, $|\lambda_{i}|\leq M$ ($M=1$ if a
photon(s) detection probability is measured as in the examples below), the
faithfulness condition is satisfied when the representation fidelity is high
enough (see Sec. 1 of the Appendix):

$F(\rho,\rho_{\text{true}})=[\operatorname{Tr}\sqrt{\sqrt{\rho}\rho_{\text{true}}\sqrt{\rho}}]^{2}\geq
1-\epsilon^{2}/(4M^{2})$

It is well-known that one can always build such a representation with a
mixture of coherent states projectors, either by a continuous Glauber-
Sudarshan P representation [6, 14], or its “coarse-grained” discrete version
[22, 23]. Recently in papers discussing the “data pattern” approach it was
shown how to approximate a given state (or its projection on some subspace)
with high fidelity by a finite (and rather small) number of coherent state
projectors on some predefined lattice [15, 16, 18, 19].

To incorporate classical information on the signs of $c_{j}$, we build the
following combined state of our signal and the two-state ancilla labeling the
prepared states:

$\rho_{c}=\frac{\zeta_{+}}{\zeta_{+}+\zeta_{-}}\rho_{+}\otimes|+\rangle\langle+|+\frac{\zeta_{-}}{\zeta_{+}+\zeta_{-}}\rho_{-}\otimes|-\rangle\langle-|,$
(2)

where

$\zeta_{+(-)}=\sum\limits_{c_{j}>0(<0)}|c_{j}|,\quad\rho_{+(-)}=\zeta_{+(-)}^{-1}\sum\limits_{c_{j}>0(<0)}|c_{j}|\rho_{j},$

and the two mutually orthonormal ancilla states $|\pm\rangle$ encode classical
information about the sign of the coefficient $c_{j}$ before the sampled
signal state $\rho_{j}$. Notice that all the weights in the mixture of
combined probe states (2) are positive. To utilize the knowledge about the
state preparation for measuring the observable $A$, we suggest measuring the
combined observable $A\otimes B$, where the ancilla observable allowing to
infer the info about the state preparation is

$B=(\zeta_{+}+\zeta_{-})(|+\rangle\langle+|-|-\rangle\langle-|).$ (3)

Thus, up to the accuracy of the representation,

$\langle A\rangle=\mathrm{Tr}\\{(A\otimes B)\rho_{c}\\}.$

Notice that for the positive-weighted mixtures of coherent-state projectors,
the state (2) becomes trivial, and the measurement procedure is the same as
for a usual, preparation-indifferent measurement. The scheme for measuring
$\rho_{\textsc{c}}$ defined in Eq. (2) for general mixed states and emulated
states are shown in Fig. 1. As follows from Eq.(2), the described measurement
can be realized in the following simple way. One samples the probes $\rho_{j}$
according to the probability distribution

$p_{j}={|c_{j}|}/(\zeta_{+}+\zeta_{-}),$

labels each probe state by the ancilla state $|+\rangle$ or $|-\rangle$
depending on $\operatorname{sgn}(c_{j})$, and performs the measurement of the
observable $A$ on the signal state and $B$ on the ancilla. If the $k$-th
sample is the probe $\rho_{j_{k}}$, let us denote the particular measurement
result of the observable $A$ as $A_{k}$ and the classical weight (measurement
result for $B$) as $B_{k}$. For $N$ samples, we calculate the following
combination of the measurement results:

$\langle
A\rangle_{N}=\frac{1}{N}\sum_{k}A_{k}B_{k}=\frac{\zeta_{+}+\zeta_{-}}{N}\sum_{k}\operatorname{sgn}(c_{j_{k}})A_{k},$
(4)

with $\left<A\right>=\langle A\rangle_{N}+O(1/\sqrt{N})$ (for details, please,
see Sec. 2 of the Appendix). Eqs. (1)–(4) show that using just coherent
states, it is possible to emulate results of any measurements on the quantum
state, $\rho_{\mathrm{true}}$, with arbitrary precision. However, one needs to
pay for it by the necessity of additional measurements of the ancilla
resulting in positive and negative weights $B_{k}$, and that leads to increase
in statistical errors. Indeed, from Eqs. (1,2,3) and using
$\zeta_{+}-\zeta_{-}=1$, one can get the following expressions for the
variances,

$\displaystyle\Delta_{AB}=(\zeta_{+}+\zeta_{-})(\zeta_{+}\mathrm{Tr}\\{\rho_{+}A^{2}\\}+\zeta_{-}\mathrm{Tr}\\{\rho_{-}A^{2}\\})-\left<A\right>^{2},$
$\displaystyle\Delta_{A}=(\zeta_{+}-\zeta_{-})(\zeta_{+}\mathrm{Tr}\\{\rho_{+}A^{2}\\}-\zeta_{-}\mathrm{Tr}\\{\rho_{-}A^{2}\\})-\left<A\right>^{2},$

where the variance $\Delta_{AB}$ is of $A\otimes B$ evaluated with the mixed
state $\rho_{c}$, and the variance $\Delta_{A}$ is of the observable $A$
evaluated with $\rho$. Thus, one has for the difference

$\begin{gathered}\Delta_{AB}-\Delta_{A}=2\zeta_{+}\zeta_{-}\left(\mathrm{Tr}\\{\rho_{+}A^{2}\\}+\mathrm{Tr}\\{\rho_{-}A^{2}\\}\right)\geq
0.\end{gathered}$ (5)

The price for the ability to model quantum states by mixtures of non-negative-
P-function states is a larger number of the measurement runs for getting the
same statistical error ( some consideration on the sampling accuracy in non-
classical quantum-state emulation are provided in Sec. 2 of the Appendix).
However, as we shall see below, the emulation procedure might be quite
economical in terms of used resources.

##  III Emulation feasibility

It is already established that for few-photon and few-mode non-classical
states one can achieve higher than $0.99$ fidelity of the approximation (1)
and reproduction of the experiment results with just few tens of the probe
coherent states [15, 16, 19]. For few-photon Fock states it is possible to
develop quite economical representations in terms of phase-averaged coherent
states. Let us show here how it is possible to represent even entangled states
using few phase-averaged coherent states and a set of simple optical devices
such as beam-splitters and phase-shifters.

Figure 2: Representation of the single-photon state in terms of phase-averaged
coherent states: decomposition coefficients (a), diagonal elements of the
optimal linear combination of the coherent states, shown in different scales
in the main plot and the inset (b). Figure 3: Representation of the two-photon
state in terms of phase-averaged coherent states: decomposition coefficients
(a) and diagonal elements of the optimal linear combination of the coherent
states, shown in different scales in the main plot and the inset (b).

###  III.1 Single-photon and two-photon states emulation

First of all, let us show examples of representations for single-photon and
two-photon states. In general, given a fixed set of $\rho_{j}$’s, the
coefficients $c_{j}$, which approximate $\rho_{\text{true}}$ in the best way,
can be determined by solving the following numerical problem:

$\displaystyle\,\max_{\\{c_{j}\\}}\,F(\rho_{\text{true}},\rho)$
$\displaystyle\,\text{subject to: }\sum_{j}c_{j}=1,\,\,\rho\geq 0\,,$ (6)

where the fidelity

$F(\rho_{\text{true}},\rho)=(\mathrm{Tr}\left\\{\sqrt{\sqrt{\rho_{\text{true}}}\,\rho\sqrt{\rho_{\text{true}}}}\right\\})^{2}$

is maximized over the coefficients $c_{j}$ conditioned on the positive
semidefiniteness of $\rho$ [19]. The solution to this problem can be found
using semidefinite programming.

When $\rho_{\text{true}}=|{1}\rangle\langle{1}|$ is the single-photon state,
the set of probes can be chosen in the form of 5 phase-averaged coherent
states with amplitudes $\alpha_{i}=0$, 0.25, 0.5, 0.75, 1 (Fig. 2a):

$\rho_{j}=\overline{|\alpha_{j}\rangle\langle\alpha_{j}|}\equiv\frac{1}{2\pi}\int\limits_{0}^{2\pi}d\varphi|\alpha_{j}e^{i\varphi}\rangle\langle\alpha_{i}e^{i\varphi}|\\\
{}=\sum_{n=0}^{\infty}\frac{|\alpha_{j}|^{2n}}{n!}e^{-|\alpha_{j}|^{2}}|n\rangle\langle
n|.$ (7)

The semidefinite program in (6) produces the resulting optimal coefficients
$c_{j}=-21.8,25.6,-3.1,0.33,-0.0028$. The fidelity of the constructed
representation for the single photon state exceeds 0.9996 (Fig. 2b).

Similarly, as it is shown in Fig. 3, just seven phase-averaged coherent states
allow to represent two-photon Fock states with the representation fidelity
exceeding 0.998.

###  III.2 NOON state emulation

Now let us demonstrate a potential of our approach by emulating entangled
bipartite states, namely, the NOON states,

$|\Psi_{N}\rangle_{ab}=\frac{1}{\sqrt{2}}\left(|N\rangle_{a}|0\rangle_{b}-|0\rangle_{a}|N\rangle_{b}\right),$
(8)

with only the phase-averaged coherent states, and with practically the same
effort as the $(N-1)$-photon Fock states (for $N\geq 2$). For $N=1,2$ it is
trivially accomplished by 50/50 beam-splitting and single-photon/vacuum or two
single-photon inputs. Let us show that beam-splitting and phase-shifting
allows easy producing the NOON states with an arbitrary N.

We have already shown above that Fock states can be decomposed in terms of
phase-averaged coherent states and require affordable resources. Linear
optical transformation of the Fock states correspond to trivial arithmetic
operations with the amplitudes of the coherent states, used for their
representation. Therefore, finding a way to represent a NOON state as a result
of some linear operations applied to Fock states will be sufficient for
construction of its efficient decomposition.

First, let us consider the linear optical transformation

$a^{\dagger}\rightarrow(a^{\dagger}+e^{i\theta}b^{\dagger})/\sqrt{2},\quad
b^{\dagger}\rightarrow(a^{\dagger}-e^{i\theta}b^{\dagger})/\sqrt{2}$ (9)

applied to the 2-mode Fock state $|n\rangle_{a}|m\rangle_{b}$ with $n+m=N$.
The density matrix of the resulting state is

$\rho(n,m,\theta)=\sum_{j=0}^{N}\sum_{l=0}^{N}R_{nm}^{(j)}R_{nm}^{(l)}e^{i\theta(l-j)}\\\
{}\times|j\rangle_{a}\langle l|\otimes|N-j\rangle_{b}\langle N-l|,$ (10)

where

$R_{nm}^{(j)}=\sum_{k=\max(0,j-m)}^{\min(n,j)}\frac{\sqrt{n!m!j!(N-j)!}(-1)^{m-j+k}}{k!(n-k)!(j-k)!(m-j+k)!}2^{-N/2}.$
(11)

In the symmetric case $n=m=N/2$, only even indices $j=2s$ yield non-zero
coefficients (it can be considered as a generalization of Hong-Ou-Mandel
effect):

$R_{nn}^{(2s)}=\frac{\sqrt{(2s)!(N-2s)!}(-1)^{n-s}}{s!(n-s)!}2^{-N/2}.$ (12)

To generate the target NOON state, we need to remove all the terms from Eq.
(10), except for those with $j$ and $l$ equal to 0 or $N$. Here, we can use
the equality

$\frac{1}{N}\sum_{k=0}^{N-1}e^{i\frac{2\pi
jk}{N}}=\left[\begin{array}[]{cc}1,&k=0,\pm N,\pm 2N,\ldots\\\
0,&\text{otherwise}.\end{array}\right.$ (13)

Therefore,

$\frac{1}{N}\sum_{k=0}^{N-1}\rho(n,m,\theta_{0}+2\pi k/N)\\\
{}=\frac{N!}{n!m!}2^{-N+1}|\Psi_{N}(\theta_{0},m)\rangle_{ab}\langle\Psi_{N}(\theta_{0},m)|\\\
{}+\sum_{j=1}^{N-1}\left(R_{nm}^{(j)}\right)^{2}|j\rangle_{a}\langle
j|\otimes|N-j\rangle_{b}\langle N-j|,$ (14)

where

$|\Psi_{N}(\theta_{0},m)\rangle_{ab}=\frac{1}{\sqrt{2}}\left(|N\rangle_{a}|0\rangle_{b}+(-1)^{m}e^{iN\theta_{0}}|0\rangle_{a}|N\rangle_{b}\right).$
(15)

By the choice $\theta_{0}=\pi/N$ for even $m$ and $\theta_{0}=0$ for odd $m$,
one can ensure that the first term of Eq. (14) corresponds to the target state
(8): $|\Psi_{N}(\theta_{0},m)\rangle_{ab}=|\Psi_{N}\rangle_{ab}$.

Finally, the target state can be expressed from Eq. (14):

$|\Psi_{N}\rangle_{ab}\langle\Psi_{N}|=\frac{n!m!2^{N-1}}{N!}\Biggl{[}\frac{1}{N}\sum_{k=0}^{N-1}\rho\left(n,m,\theta_{0}+\frac{2\pi
k}{N}\right)\\\
{}-\sum_{j=1}^{N-1}\left(R_{nm}^{(j)}\right)^{2}|j\rangle_{a}\langle
j|\otimes|N-j\rangle_{b}\langle N-j|\Biggr{]}.$ (16)

As discussed above, the derived expression implies that emulation of the NOON-
state is not much more complex than emulation of the Fock state with $N-1$
photons.

When $N$ is even, the procedure can be simplified. Eq. (12) implies that only
even multipliers $(l-j)$ of the parameter $\theta$ are present in Eq. (10) if
$n=m=N/2$. Therefore, $\rho(n,n,\theta+\pi)=\rho(n,n,\theta)$, and the
summation over $j$ can be limited by $N/2-1$ instead of $N-1$:

$|\Psi_{N}\rangle_{ab}\langle\Psi_{N}|\\\
=\frac{(n!)^{2}2^{N-1}}{N!}\Biggl{[}\frac{2}{N}\sum_{k=0}^{N/2-1}\rho\left(n,m,\theta_{0}+\frac{2\pi
k}{N}\right)\\\
{}-\sum_{k=1}^{N/2-1}\left(R_{nm}^{(2k)}\right)^{2}|2k\rangle_{a}\langle
2k|\otimes|N-2k\rangle_{b}\langle N-2k|\Biggr{]}.$ (17)

In comparison with Eq. (16), the derived expression contains almost twice
smaller number of terms and requires emulation of Fock states with up to
$\max(N/2,N-2)$ photons only.

The expressions (16) and (17) have exactly the same form as required by Eq.
(1). Therefore, the emulation of the target NOON-state can be performed as
discussed above, but applied in two steps. First, one randomly chooses one of
the states from the right-hand side of Eq. (16) or (17) with their
probabilities being proportional to the decomposition coefficients. Then, the
selected state is emulated ”classically” according to its decomposition in
terms of coherent states. Let us consider the emulation of the states from the
right-hand side of Eqs. (16) and (17) in more details.

Suppose that the Fock states $|n\rangle$ can be approximated by linear
combinations of phase-averaged coherent states in the following way:

$|n\rangle\langle
n|\approx\sum_{i}c_{ni}\overline{|\alpha_{ni}\rangle\langle\alpha_{ni}|}.$
(18)

Therefore, 2-mode Fock states can be emulated as

$|n\rangle_{a}\langle n|\otimes|m\rangle_{b}\langle
m|\approx\sum_{i,j}c_{ni}c_{mj}\overline{|\alpha_{ni}\rangle_{a}\langle\alpha_{ni}|}\otimes\overline{|\alpha_{mj}\rangle_{b}\langle\alpha_{mj}|}.$
(19)

Finally, the states $\rho(n,m,\theta)$ can be decomposed as

$\rho(n,m,\theta)=\sum_{i,j}c_{ni}c_{mj}\rho_{ij}(n,m,\theta),$ (20)

where

$\rho_{ij}(n,m,\theta)=\frac{1}{2\pi}\int_{0}^{2\pi}d\varphi_{1}\int_{0}^{2\pi}d\varphi_{2}\\\
{}\times\left|\frac{|\alpha_{ni}|e^{i\varphi_{1}}+|\alpha_{mj}|e^{i\varphi_{2}}}{\sqrt{2}}\right\rangle_{a}\left\langle\frac{|\alpha_{ni}|e^{i\varphi_{1}}+|\alpha_{mj}|e^{i\varphi_{2}}}{\sqrt{2}}\right|\otimes{}\\\
\left|e^{i\theta}\frac{|\alpha_{ni}|e^{i\varphi_{1}}-|\alpha_{mj}|e^{i\varphi_{2}}}{\sqrt{2}}\right\rangle_{b}\left\langle
e^{i\theta}\frac{|\alpha_{ni}|e^{i\varphi_{1}}-|\alpha_{mj}|e^{i\varphi_{2}}}{\sqrt{2}}\right|.$
(21)

Technically, generation of a phase-averaged coherent state corresponds to
generation of a coherent state with the given amplitude and addition of a
random uniformly distributed phase shift. I.e. to emulate the state
$\rho(n,m,\theta)$, one chooses the pair of indices $(i,j)$ with the
probabilities proportional to $c_{ni}c_{mj}$, then choose two random phases
$\varphi_{1},\varphi_{2}\in[0,2\pi)$, and finally generates the 2-mode
coherent state according to the integrand of Eq. (21).

For $N=1$, $n=1$, and $m=0$, Eq. (16) yields

$|\Psi_{-}\rangle_{ab}\langle\Psi_{-}|=\rho(1,0,\pi),$ (22)

which completely agrees with the previously obtained results.

For $N=2$, $n=1$, and $m=1$, one can use Eq. (17) to obtain the representation

$|\Psi_{2}\rangle_{ab}\langle\Psi_{2}|=\rho(1,1,0),$ (23)

known from Hong-Ou-Mandel effect.

For $N=3$, $n=2$, $m=1$ and $N=4$, $n=2$, $m=2$ the results are

$|\Psi_{3}\rangle_{ab}\langle\Psi_{3}|=\frac{4}{9}\left[\rho(2,1,0)+\rho(2,1,2\pi/3)+\rho(2,1,4\pi/3)\right]\\\
{}-\frac{1}{6}\left(|1\rangle_{a}\langle 1|\otimes|2\rangle_{b}\langle
2|+|2\rangle_{a}\langle 2|\otimes|1\rangle_{b}\langle 1|\right)$ (24)

and

$|\Psi_{4}\rangle_{ab}\langle\Psi_{4}|=\frac{2}{3}\left[\rho(2,2,\pi/4)+\rho(2,2,3\pi/4)\right]\\\
{}-\frac{1}{3}|2\rangle_{a}\langle 2|\otimes|2\rangle_{b}\langle 2|$ (25)

respectively.

So, we can see that for emulation of NOON-states with $N\leq 4$ it is
sufficient to implement representation of just a single and two-photon states
considered in the previous Subsection. The final fidelity of NOON-states
decomposition with the considered representations are listed in Table 1.

| $N=1$ | $N=2$ | $N=3$ | $N=4$
---|---|---|---|---
Fidelity | 0.9996 | 0.9992 | 0.99 | 0.982
$\zeta_{+}+\zeta_{-}$ | 51 | $2.6\times 10^{3}$ | $2.8\times 10^{4}$ | $1.8\times 10^{5}$
Table 1: Results for ”classical” emulation of NOON-states

As mentioned above, the derived expression implies that emulation of the NOON-
state is not much more complex than emulation of the Fock state with $N-1$
photons.

Below it is shown how to perform phase estimation with emulated NOON states.
By experimenting with such NOON states, one can confirm the expected quantum
effects without the troubles related to the generation and preservation of
complex non-classical quantum states. Here it is also useful to mention that
decoherence very quickly deteriorates metrological advantage expected from the
true NOON states [24].

One should emphasize that the measurement-oriented representation might be
more economical that the one discussed above. Indeed, for faithful emulation
of measuring the observable $A$ diagonal in the Fock-state basis, it is
sufficient to emulate Fock-state mixtures instead of superpositions. Also, the
complexity of representation is not connected with the state energy, but
rather with the number of required basis states. For example, to represent the
“cat-state” $|\alpha\rangle+|\alpha+\delta\rangle$ with $|\delta|\ll|\alpha|$
and an arbitrary $\alpha$, one might need only a few coherent projectors with
amplitudes close to $\alpha$.

Also, the choice of state mixtures for emulation is not limited to coherent
states or their phase-averaged version. One might guess that the
representations akin to ones discussed above can be developed with other
states, and tailored for a particular measurement. For example, thermal states
were used to represent Fock states in Ref.[16]. Some considerations on
choosing the best mixtures for a particular measurements are given in Sec. 4
of the Appendix.

Figure 4: The values of non-classicality witness operator $W$ (Eq. (28)) for
the coherent states of the single-photon state representation depicted in Fig.
2a. The dashed horizontal line shows the expectation value of the non-
classicality witness operator for the single-photon state.

##  IV Non-classicality witness

Now let us demonstrate how can one certify non-classicality of the emulation
results.

To prove non-classicality of a given state (or a class of states), one can
construct a witness operator $W$, find the classical limit

$W_{0}=\max_{\rho^{\prime}{\in\mathcal{C}}}\mathrm{Tr}\\{W\rho^{\prime}\\},$
(26)

where $\mathcal{C}$ is the set of non-negative-P-function states, and check
that the investigated state $\rho$ yields

$\mathrm{Tr}\\{W\rho\\}>W_{0}.$ (27)

For example, to prove non-classicality of the single-photon state, one can
build the following witness operator:

$W=2|1\rangle\langle 1|-|0\rangle\langle 0|-|2\rangle\langle 2|.$ (28)

Using the the developed representation for $\rho^{\prime}$ and taking into
account diagonality of the operator $W$ in Fock state basis, one can show that

$\begin{gathered}W_{0}=\max_{\rho^{\prime}{\in\mathcal{C}}}\mathrm{Tr}\\{W\rho^{\prime}\\}\\\
{}=\max_{P:P(\alpha)\geq 0}\int d^{2}\alpha
P(\alpha)W(|\alpha|),\end{gathered}$ (29)

where

$W(|\alpha|)=\langle\alpha|W|\alpha\rangle=\left(2|\alpha|^{2}-1-\frac{|\alpha|^{4}}{2}\right)e^{-|\alpha|^{2}},$
(30)

and the normalization condition holds:

$\int d^{2}\alpha P(\alpha)=1.$ (31)

A non-negativity of $P(\alpha)$ implies that the maximal classical value
$W_{0}$ corresponds to the maximum of the function $W(|\alpha|)$, which equals

$W_{0}=\max_{\alpha}W(|\alpha|)=0.206$ (32)

and is reached for the coherent state with the amplitude
$|\alpha^{(0)}|=1.134$.

For the single-photon state, the expectation value of the witness operator
equals $\langle 1|W|1\rangle=2>W_{0}$.

The mean value $\langle W\rangle$ for the emulated state (7) equals 1.9992,
which clearly exceeds the classical limit $W_{0}=0.206$. On the other hand,
Fig. 4 shows that the values $\mathrm{Tr}\\{W\rho_{j}\\}$ fit into the
classical region $[-1,W_{0}]$ for all $j$. The two reasons for the final
result exceeding the classical limit are:

  * •

minus sign for certain $\rho_{j}$: classical maximum of $-W$ is $1>0.206$ (but
still less than 2);

  * •

the measurement results are multiplied by the factor
$\zeta_{+}+\zeta_{-}=50.8$.

The excess variance of a single-trial measurement (given by Eq. (5) of the
main text) is $2.0\times 10^{3}$. Therefore, to demonstrate the non-
classicality reliably, one needs of about $10^{5}$ copies of the state.

Figure 5: Approximation of the single-photon non-classicality witness by the
four-detector measurement setup: detection scheme (a), Fock-basis
decomposition coefficients of the POVM elements $\Pi_{m}$ (b), and comparison
of the ideal witness $W$ (gray bars) and the constructed witness $W_{4}$ (blue
line) - plot (c). The inset in plot (c) shows the found decomposition
coefficients $z_{m}$ in Eq. (35). The detection efficiency $\eta=0.8$ and the
dark count rate $\varepsilon=0.001$ were used for the calculations.

###  IV.1 Non-classicality witnessing under realistic measurement conditions

The non-classicality witness $W$, described by Eq. (28), requires a photon
number resolving measurement. To stay more realistic, it is worth constructing
a witness, which can be measured with usual single-photon detectors,
possessing final efficiency and dark count rate.

Let us consider the measurement setup, shown in Fig. 5a and consisting of 4
single-photon detectors with the detection efficiency $\eta$ and the dark
count rate $\varepsilon$. The five possible outcomes of the measurement
correspond to detection of $m=0$, 1, 2, 3, and 4 counts respectively and can
be described by the positive operator-valued measure (POVM)
$\\{\Pi_{0},\ldots,\Pi_{4}\\}$. The POVM elements have the following Fock-
state basis representation:

$\Pi_{m}=\sum_{n=0}^{\infty}|n\rangle\langle n|p(m|n),$ (33)

where

$p(m|n)=\sum_{k=0}^{m}\frac{4!(-1)^{m-k}(1-\varepsilon)^{4-k}}{(4-m)!k!(m-k)!}\\\
{}\times\left(1-\frac{4-k}{4}\eta\right)^{n}$ (34)

is the probability of detecting $m$ counts if the input state of the
measurements scheme in Fig. 5a is the Fock state $n$ (Fig. 5b).

Following the ideas from Eq. (1), one can try to approximate the witness
operator $W$, introduced by Eq. (28), in terms of the available POVM elements:

$W\approx W_{4}\equiv\sum_{m=0}^{4}z_{m}\Pi_{m},$ (35)

where the coefficients $z_{m}$ (see the inset in Fig. 5c) can be found, for
example, by minimization of the quadratic distance between $W$ and $W_{4}$
(the fidelity $F$ cannot be used here because neither $W$ nor $W_{4}$ are
positive semi-definite operators):

$\min_{\\{z_{m}\\}}\sum_{n=0}^{\infty}\left(\langle n|W|n\rangle-\langle
n|W_{4}|n\rangle\right)^{2}.$ (36)

Fig. 5c shows the resulting witness operator $W_{4}$. While being different
from the ideal witness $W$ because of detectors’ non-ideality, it is still
suitable for detection of non-classicality. During the performed numerical
calculations, we assumed that the detection efficiency equals $\eta=0.8$ and
the dark count probability is $\varepsilon=0.001$.

Similarly to Eq. (32), the maximal classical value $W_{40}$ of the constructed
witness corresponds to the coherent state $|\alpha\rangle$ with
$\alpha=1.176$, maximizing
$W_{4}(|\alpha|)=\langle\alpha|W_{4}|\alpha\rangle$:

$W_{40}=\max_{\alpha}W_{4}(|\alpha|)=0.248.$ (37)

The witness value, reached for the single-photon state, equals $\langle
1|W_{4}|1\rangle=1.538>W_{40}$. Unlikely the ideal witness $W$ yielding zero
variance for the state $|1\rangle$, the variance of the observable $W_{4}$ for
the single-photon state is $1.59$. Therefore, one needs to perform at least
several repetitions of the measurement to be sure that the results are
incompatible with the assumption of a non-negative-P-function input state if
the state $|1\rangle$ is supplied.

The ”classically” emulated single-photon state, discussed in the previous
sections, yields the mean value $\langle W_{4}\rangle=1.537$, which still
noticeably exceeds the classical limit. The excess variance of the observable
$W_{4}$ is $1.8\times 10^{3}$. Therefore, the number of the measurement
repetitions, required for reliable proof of the single-photon state non-
classicality, remains approximately the same as for the measurement of the
ideal witness $W$.

Figure 6: Scheme of observing Hong-Ou-Mandel effect with single photons (a)
and the setup for its ”classical emulation” (b). Gray dashed lines show how
the phase-averaged coherent states can be generated by variable splitting a
reference coherent state with subsequent application of a random phase shift
$\varphi$ in one of the arms.

##  V Hong-Ou-Mandel effect

Let us illustrate a measurement emulation scheme by an archetypal quantumness
demonstrator: the Hong-Ou-Mandel single-photon interference. If one has a
single photon per each entry port of the 50/50 BS (Fig. 6a), in case of the
ideal interference of both input fields $a$ and $b$, the probability $p_{12}$
of having the detectors ${D_{1}}$ and $D_{2}$ clicking simultaneously is zero.
If the interference isn’t ideal (for example, due to imperfect overlapping of
the pulses or misaligned polarization), $p_{12}\neq 0$ and increasing with
worsening of interference [25, 26]. If the detectors have the efficiency
$\eta$ and do not distinguish modes in the impinging fields, a registration of
a click on $j$-th detector is described by the expression [26]

$\Pi_{j}=1-{:\exp\left\\{-\frac{\eta}{2}(a^{\dagger}a+b^{\dagger}b\pm
fa^{\dagger}b\pm f^{*}b^{\dagger}a)\right\\}:},$

where the signs “+” and “$-$” correspond to the $j=1$ and 2 respectively, the
operators $x^{\dagger}$, $x$ are creation and annihilation operators for
$x$-th mode, $x=a,b$; $::$ denotes the normal ordering, and the parameter $f$
describes the degree of the overlap. Upon considering, for simplicity, a real
and positive $f$, the probability of _both_ detectors clicking is

$p_{12}=\langle
1_{a},1_{b}|\Pi_{1}\Pi_{2}|1_{a},1_{b}\rangle=(1-f^{2})\eta^{2}/2,$

where the $|1_{a},1_{b}\rangle$ describes the single-photon Fock states in the
modes $a$ and $b$.

The following scheme reproduces the Hong-Ou-Mandel effect (Fig. 6b): the
randomly chosen phase-averaged coherent states are produced by appropriate
splitting of an input coherent state, while the additional random phase shift
$\varphi$ introduces the effect of the phase averaging. In this manner the
representation of $\rho^{\prime}$ for the input single-photon states can be
built in terms of the phase-averaged coherent states,

$\rho^{\prime}=\sum_{k,l}c_{k}c_{l}\,\rho^{a}_{k}\rho^{b}_{l}.$ (38)

For the probe state $\rho^{a}_{k}\rho^{b}_{l}$ with a non-ideal overlap, the
two-detector click probability now reads

$p_{12}^{kl}=\frac{1}{2\pi}\int
d\varphi\left[1-p_{+}^{kl}(\varphi)\right]\left[1-p_{-}^{kl}(\varphi)\right]\,,$
(39)

where

$p_{\pm}^{kl}(\varphi)=\exp\left[-\frac{\eta}{2}\left(|\alpha_{k}|^{2}+|\alpha_{l}|^{2}\pm
2f|\alpha_{k}||\alpha_{l}|\cos\varphi\right)\right].$

Let us now estimate to which extent the representation (38) is more expensive
in terms of the necessary number of state copies. From Eq. (38), for
$\eta=0.8$ and the overlap $f^{2}=0.95$, one gets $p_{12}=0.017$. For the
input states being true single photons, a single-trial variance is less than
unity. However, the single-trial variance, estimated according to Eq. (5), is
$\operatorname{Var}(p_{12})=1.5\times 10^{4}$. So, one needs the number of
measurement runs (samples of probe states), $N$, of about $10^{6}$ for
reliable demonstration of Hong-Ou-Mandel effect. However, it is worth
reminding here that to generate just a single photon, for example, by the
spontaneous down-conversion, one needs about $10^{5}-10^{12}$ pumping photons
[27].

Figure 7: Scheme of interferometer for phase estimation using 2-photon NOON-
state.
Figure 8: Dependence of normalized coincidence rates $g_{2}(\theta)$ on the
measured phase shift $\theta$. Solid lines indicate the dependence for
interference of two single-photon states. Points and error bars show the
values and the standard deviations for ”classically” emulated single-photon
states for $N=10^{8}$ repetitions.

##  VI Phase estimation with 2-photon NOON-state

Here we show how one can emulate the phase estimation with NOON states using
our scheme.

The state obtained after the interference of 2 photons at a beamsplitter, is
the 2-photon NOON-state
$(|2\rangle_{a}|0\rangle_{b}-|0\rangle_{a}|2\rangle_{2})/\sqrt{2}$ and,
therefore, can be used for sensitivity enhancement in phase estimation. For
the scheme, shown in Fig. 7, the probability of encountering both photons in
one arm (coupled to either $D_{1}$ or $D_{2}$) equals

$p_{2}^{(0)}(\theta)=\frac{1+f}{4}\sin^{2}\theta,$ (40)

while the probability of having one photon in each arm is

$p_{11}^{(0)}(\theta)=\frac{1-f}{2}+\frac{1+f}{2}\cos^{2}\theta,$ (41)

where $\theta$ is the phase shift to be measured.

Similarly to the previous section, one can calculate the probability of
coincidence count $p_{11}$ and the unconditional probabilities of single-
photon detection $p_{1\cdot}$ and $p_{\cdot 1}$ and introduce the normalized
coincidence rate:

$g_{2}(\theta)=\frac{p_{11}}{p_{1\cdot}p_{\cdot 1}}\\\
{}=\frac{16\left(\eta(1-\epsilon)(z+\eta(3-f-4\epsilon)+8\epsilon)+4\epsilon^{2}\right)}{\left(\eta(1-\epsilon)z+\eta(1-\epsilon)(8-f\eta-\eta)+8\epsilon\right)^{2}}$
(42)

where $z=(1+f)\eta\cos 2\theta$. The solid line in Fig. 8a shows the
dependence of $g_{2}(\theta)$ on the phase shift $\theta$.

For the 2-mode probe state $\rho_{i}\otimes\rho_{j}$, the probabilities of
clicks on both detectors are

$p_{11}(i,j;\theta)=\frac{1}{2\pi}\int
d\varphi\left[1-p_{-}(|\alpha_{i}|,|\alpha_{j}|,\theta,\varphi)\right]\\\
{}\times\left[1-p_{+}(|\alpha_{i}|,|\alpha_{j}|,\theta,\varphi)\right],$ (43)
$p_{1\cdot}(i,j)=\frac{1}{2\pi}\int
d\varphi\left[1-p_{-}(|\alpha_{i}|,|\alpha_{j}|,\theta,\varphi)\right],$ (44)

and

$p_{\cdot 1}(i,j)=\frac{1}{2\pi}\int
d\varphi\left[1-p_{+}(|\alpha_{i}|,|\alpha_{j}|,\theta,\varphi)\right],$ (45)

where

$p_{\pm}(x,y,\theta,\varphi)=(1-\varepsilon)\exp\Bigl{[}-\frac{\eta}{2}\Bigl{(}x^{2}(1\pm\cos\theta)\\\
{}+y^{2}(1\mp\cos\theta)\pm 2\sqrt{f}xy\sin\theta\sin\varphi\Bigr{)}\Bigr{]}.$
(46)

The results of the calculation of the normalized second-order functions for
the true and emulated states are shown in Fig. 8. The values $\eta=0.8$,
$\varepsilon=0.001$, $f=0.95$ and $N=10^{8}$ repetitions were assumed.

Figure 9: Detection scheme for classical emulation of 2-mode entanglement and
Bell-type measurements with phase-averaged probe coherent states. Alice and
Bob prepare the same amplitudes and opposite phases of the coherent states.

##  VII Bell inequalities violation

Another famous manifestation of quantumness is Bell-type inequalities
violation for a distinguishable (for example, spatially separated) quantum
systems. Let us show here how it is possible to emulate the state of two
entangled modes $a$ and $b$ sharing a single photon state
$|\Psi_{ab}\rangle\langle\Psi_{ab}|$, where
$|\Psi_{ab}\rangle=\left(|1\rangle_{a}|0\rangle_{b}-|0\rangle_{a}|1\rangle_{b}\right)/\sqrt{2}$,
and to demonstrate violation of the Clauser-Horn inequality [28] using a
modification of the scheme discussed in Ref. [29]. In Ref. [29] the coherently
displaced signals are measured with simple on-off detectors and the following
inequality is considered:

$-1\leq
j_{0}=q(\alpha,\gamma)-q(\alpha,\delta)+q(\beta,\gamma)+q(\beta,\delta)\\\
{}-q_{a}(\beta)-q_{b}(\gamma)\leq 0,$ (47)

where the single-detector no-click probabilities and coinciding two-detector
no-click probabilities to be measured for the coherently displaced input state
are defined as

$q_{x}(\mu)=\langle Q_{x}(\mu)\rangle=\left\langle
D_{x}(\mu)\Pi_{\text{off}}^{(x)}D_{x}^{\dagger}(\mu)\right\rangle,$

and

$q(\mu,\nu)=\langle Q_{a}(\mu)Q_{b}(\nu)\rangle,$

where the operator $D_{x}(\mu)=\exp\\{\mu x^{\dagger}-\mu^{*}x\\}$ describes
coherent displacement of $x$-th mode by the amplitude $\mu$, $x=a,b$,
implemented by mixing the mode with a local oscillator (LO) field at a
beamsplitter (Fig. 9). The operator $\Pi_{\text{off}}^{(x)}$ describes
appearance of count absence on the detector measuring the $x$-th mode. For the
detectors with the efficiency $\eta$ this operator can be expressed as

$\Pi_{\text{off}}^{(x)}=(1-\eta)^{n_{x}}$

in terms of the number operator $n_{x}=x^{\dagger}x$. The value of $j_{0}$ can
be minimized for all possible shifts $\alpha$, $\beta$, $\gamma$, and
$\delta$. To have the minimal $j_{0}$, one needs setting
$\alpha=-\delta=\mu_{1}$ and $\gamma=-\beta=\mu_{2}$. Particular values of
$\mu_{1,2}$ depend on the efficiency. For example, the minimal value of
$j_{0}$ and the optimal amplitudes $\mu_{1}$ and $\mu_{2}$ for ideal detectors
($\eta=1$) are equal to ($-1.172$, 0.563, 0.165); for $\eta=0.95$ the optimal
values are ($-1.118$, 0.587, 0.177) , for $\eta=0.90$ they are ($-1.066$,
0.615, 0.191). Taking into account the relations between the optimal
amplitudes and inequality (47), one can introduce the observable

$\displaystyle
J_{0}=Q_{a}(-\mu_{2})Q_{b}(\mu_{2})-Q_{a}(\mu_{1})Q_{b}(-\mu_{1})-$
$\displaystyle
Q_{a}(-\mu_{2})[1-Q_{b}(-\mu_{1})]-[1-Q_{a}(\mu_{1})]Q_{b}(\mu_{2}),$

corresponding to the quantity $j_{0}=\langle J_{0}\rangle$. To emulate
$|\Psi_{ab}\rangle\langle\Psi_{ab}|$, it is convenient to represent this state
as a result of beam-splitting of a single-mode one-photon state. Emulating
this single-photon state with a mixture of phase-averaged coherent states,
$|\Psi_{ab}\rangle\langle\Psi_{ab}|$ can be rewritten in the form of Eq. (1)
with

$\displaystyle\rho_{j}=\frac{1}{2\pi}\int_{0}^{2\pi}d\varphi|f_{j}(\varphi)\rangle\langle
f_{j}(\varphi)|,$

where the states $|f_{j}(\varphi)\rangle$ are products of the coherent states
of the modes $a$ and $b$:

$\displaystyle|f_{j}(\varphi)\rangle={|{\alpha_{j}}e^{i\varphi}/{\sqrt{2}}\rangle_{a}}{|-{\alpha_{j}}e^{i\varphi}/{\sqrt{2}}\rangle_{b}}.$

Factorization of the probe states implies that, in contrast to the emulated
splitting of a single-photon state, they can be generated separately by Alice
and Bob.

A scheme, suitable for implementation of the discussed emulation of the state
$|\Psi_{ab}\rangle\langle\Psi_{ab}|$ is shown in Fig. 9. For each trial, the
two parties choose the same random index $j$ of the probe state and the same
random phase shift $\varphi$ and prepare the coherent states
$|\alpha_{j}e^{i\varphi}/\sqrt{2}\rangle_{a}$ and
$|-\alpha_{j}e^{i\varphi}/\sqrt{2}\rangle_{b}$ of the modes $a$ and $b$
respectively. Input BS’s in Fig. 9 allows one to realize required coherent
shifts. The expectation values of $J_{0}$ over such probe states can be found
using the rules

$\operatorname{Tr}\left\\{\rho_{j}Q_{a}(\mu)Q_{b}(\nu)\right\\}=\,\frac{1}{2\pi}\int
d\varphi q_{j}(\mu,\varphi)q_{j}(-\nu,\varphi),$

and

$\operatorname{Tr}\left\\{\rho_{j}Q_{a(b)}(\mu)\right\\}=\,\frac{1}{2\pi}\int
d\varphi q_{j}(\pm\mu,\varphi),$

where
$q_{j}(\nu,\varphi)=\exp(-\eta|{\alpha_{j}e^{i\varphi}}/{\sqrt{2}}-\nu|^{2})$.
Since the Alice’s and Bob’s parts of the probe states interfere with their LO
fields only, the phase stability between Alice’s and Bob’s coherent-state
sources is not required.

As it is to be expected, the mean value of the measured observable remains
approximately the same as for the true state, $\langle
J_{0}\rangle=-1.118,-1.077$ for $\eta=0.95,0.9$, while the single-trial
variances are expectedly large: $\operatorname{Var}(J_{0})=5.1\times 10^{3}$
and $5.0\times 10^{3}$ respectively. The numbers of the measurement
repetitions, required for reliable non-classicality demonstration, are
$N\gtrsim 1.5\times 10^{6}$ for $\eta=0.95$ and $N\gtrsim 3.4\times 10^{6}$
for $\eta=0.90$.

##  VIII Conclusions

We have shown that results of quantum measurements can be emulated using only
quantum states with non-negative P functions. For that purpose, one just needs
to know how the quantum state can be prepared with a set of non-negative-P-
function states by binary labeling each probe and using that classical
information during the measurement. The price for the ability to use such
“classical” light sources is the necessity to tailor our method for each
particular state, and larger number of measurement runs required for getting
reliable results. Generally, this number might become so large as to render
our emulation procedure unfeasible. However, we have demonstrated that for
some important and interesting states and measurement schemes our approach is
feasible and can be even more efficient in terms of “classical” resources
necessary to generate and measure the state. The proposed approach is likely
to be a handy toolbox for proof-of-principle experiments for testing quantum
effects; for verification of proper functioning of measurement setups for
fundamental quantum experiments before the required non-classical states
become available; and for design of affordable demonstration breadboards for
education and science dissemination. The presented procedure provides a
systematic tool for “quantum-inspired” metrology by translating quantum
measurement protocols into their classical counterparts, where the non-
negative-P-function-state signals are combined in the optimal way in order to
enhance the sensitivity.

###### Acknowledgements.

A. M. and D. M. gratefully acknowledge support from The Belarusian Republican
Foundation for Fundamental Research (F21KOR-002,F21IKR-003), the EU project
PhoG 820365 and the NATO project NATO SPS - G5860. Y. S. T. and H. J.
acknowledge support by the National Research Foundation of Korea (NRF) (Grant
Nos. NRF-2019R1A6A1A10073437, NRF2019M3E4A1080074, NRF-2020R1A2C1008609 and
2020K2A9A1A06102946) via the Institute of Applied Physics at Seoul National
University and by Institute of Information & communications Technology
Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (Nos.
2021-0-01059 and 2021-2020-0-01606).

## Appendix: Developing classical representations

###  .1 Representation fidelity and systematic error of emulation

Let us consider emulation of measuring an observable $A$ in a quantum state
$\rho_{\text{true}}$ by using the representation

$\rho_{\text{true}}\approx\rho=\sum_{i}c_{i}\rho_{i}$

in terms of semi-classical probe states $\\{\rho_{i}\\}$. The systematic error
of the emulation equals

$\delta_{\text{sys.}}=|\langle A\rangle_{\rho}-\langle
A\rangle_{\text{true}}|=|\mathrm{Tr}\\{A(\rho-\rho_{\mathrm{true}})\\}|.$ (48)

If the exact operator $A$, describing the measured observable, is known when
the emulation protocol is designed, Eq. (48) can be used directly for
assessment of the particular representation quality, since it defines the
distance between $\rho$ and $\rho_{\text{true}}$ appropriate for the
particular measurement. Otherwise, it is instructive to connect the upper
bound of the systematic error for an arbitrary observable $A$, satisfying
certain reasonable constraints, with the representation fidelity

$F(\rho,\rho_{\text{true}})=\Bigl{[}\operatorname{Tr}\sqrt{\sqrt{\rho}\rho_{\text{true}}\sqrt{\rho}}\Bigr{]}^{2}.$
(49)

Measurement of an observable $A$ is characterized by a set of projective
operators $E_{i}$, forming a positive operator valued measure (POVM)
$\\{E_{i}\\}$ with $i$-th outcome being mapped to the eigenvalue $\lambda_{i}$
of the operator $A$:

$\langle A\rangle_{\rho}=\sum_{i}\lambda_{i}p_{i},\quad
p_{i}=\operatorname{Tr}(E_{i}\rho),$ (50) $\langle
A\rangle_{\text{true}}=\sum_{i}\lambda_{i}q_{i},\quad
q_{i}=\operatorname{Tr}(E_{i}\rho_{\text{true}}),$ (51)

and

$\delta_{\text{sys.}}=\Bigl{|}\sum_{i}\lambda_{i}(p_{i}-q_{i})\Bigr{|}.$ (52)

The fidelity of the representation is connected to distinguishability of the
states $\rho$ and $\rho_{\text{true}}$ by the optimal POVM measurement [30],
which is at least as sensitive as the particular measurement characterized by
the POVM $\\{E_{i}\\}$ and associated with the observable $A$:

$F(\rho,\rho_{\text{true}})=\min\limits_{\text{POVM
}\\{E^{\prime}_{i}\\}}F_{\text{B}}^{2}(\\{\operatorname{Tr}(E^{\prime}_{i}\rho)\\},\\{\operatorname{Tr}(E^{\prime}_{i}\rho_{\text{true}})\\})\\\
\leq F_{\text{B}}^{2}(p,q),$ (53)

where $F_{\text{B}}(u,v)=\sum_{i}\sqrt{u_{i}v_{i}}$ is the Bhattacharyya
coefficient of classical probability distributions $u$ and $v$;
$p=\\{p_{i}\\}$ and $q=\\{q_{i}\\}$ are the probability distributions for the
measurement of $A$ for $\rho$ and $\rho_{\text{true}}$.

The right-hand side of Eq.(53) depends on the particular choice of the
measured observable $A$. To get a universal bound, one can maximize
$F_{\text{B}}(p,q)$ over appropriate measurements:

$F(\rho,\rho_{\text{true}})\leq\max\limits_{p,q,\lambda}F_{\text{B}}^{2}(p,q),$
(54)

conditioned by

$\sum_{i}p_{i}=1,\quad\sum_{i}q_{i}=1,\quad\Bigl{|}\sum_{i}\lambda_{i}(p_{i}-q_{i})\Bigr{|}=\delta_{\text{sys.}}$
(55)

Let us assume that any considered observable $A$ is known a priori to have
limited eigenvalues: $|\lambda_{i}|\leq M$. For example, if the probability of
detecting a photon or a coincidence count is measured, we have $M=1$.

Optimization of Eq. (54) over $\lambda$ immediately shows that the choice

$\lambda_{i}=M\mathrm{sgn}(p_{i}-q_{i})$

is optimal for maximization of $F_{\text{B}}(p,q)$. The last constraint in Eq.
(55) is transformed into

$\sum_{i}|p_{i}-q_{i}|=\delta_{\text{sys.}}/M.$

Pairwise variations of $(p_{i},q_{i})$ and $(p_{j},q_{j})$ for all such $i$
and $j$ that $\mathrm{sgn}(p_{i}-q_{i})=\mathrm{sgn}(p_{j}-q_{j})$ lead to the
optimality condition $p_{i}:q_{i}=p_{j}:q_{j}$. Finally, one arrives at the
condition

$\sum_{i:\;p_{i}>q_{i}}(p_{i}+q_{i})=\sum_{i:\;p_{i}<q_{i}}(p_{i}+q_{i})=1$
(56)

yielding the maximal value

$\max\limits_{p,q,\lambda}F_{\text{B}}^{2}(p,q)=1-\delta_{\text{sys.}}^{2}/(2M)^{2}.$
(57)

Combining Eqs. (54) and (57) and solving for $\delta_{\text{sys.}}$, we arrive
at the following upper bound for the systematic error of emulation:

$\delta_{\text{sys.}}\leq 2M\sqrt{1-F(\rho,\rho_{\text{true}})}.$ (58)

###  .2 Rate of convergence of $\left<A\right>_{N}$

Suppose that the set of ”classical” component probe states $\\{\rho_{j}\\}$
are used to emulate the target true state
$\rho_{\text{true}}\approx\sum_{j}c_{j}\rho_{j}$. In the perspective of
statistical inference, one can rewrite $\left<A\right>_{N}$ defined in Eq. (3)
of the main text as

$\left<A\right>_{N}=\dfrac{\zeta}{N}\sum_{j}\sum_{l}\mathrm{sgn}(c_{j})a_{l}n_{jl}\,,$
(59)

where

$\zeta=\sum_{j}|c_{j}|=\zeta_{+}+\zeta_{-},$

and now $\left<A\right>_{N}$ is a double sum over all probe states employed
and eigenvalues $a_{l}$ of the observable $A$ measured. The frequencies
$n_{jl}$ of having $l$-th result of the measurement with $j$-th probe are
summed to the total number of trials, $N$: $\sum_{j,l}n_{jl}=N$. Thus the
relative frequencies $\nu_{jl}=n_{jl}/N$ inherently follow a multinomial
distribution with respect to the indices $j$ and $l$, with the statistical
average

$\overline{\nu_{jl}\nu_{j^{\prime}l^{\prime}}}=\begin{cases}[\delta_{l,l^{\prime}}\widetilde{p}_{jl}+(N-1)\widetilde{p}_{jl}\widetilde{p}_{jl^{\prime}}]/N&\text{when
}j=j^{\prime}\,,\\\
(N-1)\widetilde{p}_{jl}\widetilde{p}_{j^{\prime}l^{\prime}}/N&\text{otherwise}\,.\\\
\end{cases}$ (60)

determined by the observation
$\overline{\nu_{jl}}=\widetilde{p}_{jl}=p_{j}p_{jl}$, $p_{j}=|c_{j}|/\zeta$
and $p_{jl}=\langle{a_{l}}|{\rho_{j}}|{a_{l}}\rangle$. These immediately give

$\displaystyle\,\overline{\left(\left<A\right>_{N}-\left<A\right>\right)^{2}}$
$\displaystyle=$
$\displaystyle\,\sum_{j,j^{\prime},l,l^{\prime}}\mathrm{sgn}(c_{j})\mathrm{sgn}(c_{j^{\prime}})a_{l}a_{l^{\prime}}\overline{(\nu_{jl}\nu_{j^{\prime}l^{\prime}}-\widetilde{p}_{jl}\widetilde{p}_{j^{\prime}l^{\prime}})}$
$\displaystyle=$
$\displaystyle\,\dfrac{\zeta^{2}}{N}\left[\sum_{j}p_{j}\mathrm{Tr}\\{\rho_{j}A^{2}\\}-\left(\sum_{j}p_{j}\mathrm{sgn}(c_{j})\mathrm{Tr}\\{\rho_{j}A\\}\right)^{2}\right]$
$\displaystyle=$
$\displaystyle\,\dfrac{1}{N}\left[\zeta\,\mathrm{Tr}\\{(\zeta_{+}\rho_{+}+\zeta_{-}\rho_{-})A^{2}\\}-\left<A\right>^{2}\right]=\dfrac{\Delta_{AB}}{N}\,.$
(61)

Alternatively, one arrives at this result by considering the extended model
[$\left<A\right>=\mathrm{Tr}\\{(A\otimes B)\rho_{\textsc{c}}\\}$] discussed in
the text and recognizing the fact that each _independently_ sampled eigenvalue
incurs a quantum variance of $\Delta_{AB}$, such that scaling $\Delta_{AB}$
with $N$ gives the right answer. Hence, in the limit of large $N$, we indeed
expect that

$\left<A\right>-\left<A\right>_{N}=O(1/\sqrt{N}).$

On the other hand, for the same number of copies $N$, if one can generate
$\rho_{\text{true}}$ directly, then the naive linear estimator
$\left<A\right>_{\textsc{lin}}=\sum_{l}a_{l}\nu_{l}$ for $\left<A\right>$,
where $\nu_{l}=n_{l}/N\rightarrow
p_{l}=\langle{a_{l}}|{\rho_{\text{true}}}|{a_{l}}\rangle$, leads to

$\overline{\left(\left<A\right>_{\textsc{lin}}-\left<A\right>\right)^{2}}=\dfrac{\Delta_{A}}{N}$
(62)

owing to the basic multinomial rule

$\overline{(\nu_{j}-p_{j})(\nu_{k}-p_{k})}=\frac{1}{N}(p_{j}\delta_{j,k}-p_{j}p_{k}).$

As argued in the text, the fact that $\Delta_{AB}>\Delta_{A}$ simply
reiterates that non-classical state emulation using ”classical” component
states reduces the complexity of quantum-state generation at the expense of a
larger $N$ to estimate $\left<A\right>$ up to some fixed accuracy.

###  .3 Sampling accuracy in non-classical quantum-state emulation

Let us show how well sampling of the probe states from the set with the help
of $\rho_{\textsc{c}}$ (equivalent to using a classical random-number
generator) can approximate the state $\rho$ from Eq. (1) of the main text.

After a multinomial sampling of $N_{\text{s}}$ copies of $\rho$ (not to be
confused with $N$, the total number of copies used to estimate
$\left<A\right>_{N}$), one obtains the estimator
$\widehat{\rho}=\zeta\sum_{j}\nu_{j}\mathrm{sgn}(c_{j})\rho_{j}$, where the
relative frequencies $\nu_{j}$ of the probe states $\rho_{j}$ tend to the
probabilities $p_{j}=|c_{j}|/\zeta$ for $N_{\text{s}}\gg 1$. We may consider
the mean squared-error (MSE),
$\mathrm{MSE}=\overline{\mathrm{Tr}\\{(\widehat{\rho}-\rho)^{2}\\}}$ with
$\overline{\,\vphantom{\operatorname{Tr}{(\widehat{\rho}-\rho)^{2}}}\cdot\,}$
denoting the statistical mean (expectation value), as the figure of merit for
determining the accuracy of such a sampling with a given value of
$N_{\text{s}}$ with respect to the actual state $\rho=\sum_{j}c_{j}\rho_{j}$
being classically emulated. For a multinomial distribution, as
$\overline{\nu_{j}}=p_{j}$ and

$\overline{(\nu_{j}-p_{j})(\nu_{k}-p_{k})}=\frac{1}{N_{\text{s}}}(p_{j}\delta_{j,k}-p_{j}p_{k}),$

the MSE can be easily computed to be

$\mathrm{MSE}=\frac{1}{N_{\text{s}}}\sum_{j,j^{\prime}}\\!\left(|c_{j}||c_{j^{\prime}}|\mathrm{Tr}\\{\rho_{j}^{2}\\}-c_{j}c_{j^{\prime}}\mathrm{Tr}\\{\rho_{j}\rho_{j^{\prime}}\\}\right).$
(63)

An important special case corresponds to pure probe states
$(\mathrm{Tr}\\{\rho_{j}^{2}\\}=1)$. They allow to see more clearly into the
essence of statistical noise introduced by ”classical” emulation of quantum
states. For mixed probe states $\rho_{j}$, their intrinsic classical noise is
masked by the assumption about their noise-less sampling. For pure probes, one
can represent the MSE (63) as

$\mathrm{MSE}=\frac{1}{N_{\text{s}}}\left\\{\left(1-\mathrm{Tr}\\{\rho^{2}\\}\right)+\left(\zeta^{2}-1\right)\right\\}.$
(64)

The first term corresponds to the internal classical noise of the mixed state
$\rho$, while the latter one describes the additional sampling noise
introduced by classical representation of a non-classical state. If the state
$\rho$ is ”classical”, one can find its representation with positive weights
$c_{j}>0$. Therefore, according to standard normalization of the density
matrices $\rho$ and $\rho_{j}$, we have
$\zeta=\sum_{j}|c_{j}|=\sum_{j}c_{j}=1$ and the second term in Eq. (64)
vanishes. Notice that this term also vanishes when one samples the actual
physical state (2) of the main text used for reproducing measurement results
instead of $\rho$ having
$\mathrm{MSE}=\left(1-\mathrm{Tr}\\{\rho_{c}^{2}\\}\right)/{N_{\text{s}}}$.

It is also worth noting that if $\rho_{\text{true}}$ is also pure, the MSE in
(64) is defined by accuracy and purity of the representation, and goes to zero
with the fidelity going to unity.

###  .4 Optimizing the representation

Our task is to simulate the result of quantum-state measurement. So, the
optimization task for developing the representation would consist of choosing
the minimal possible number of ”classical” probes providing for the least
error is estimating a specified observable. Also, these probes themselves
should be specified. The choice of the probes for the quantum state
reconstruction was discussed in a number of works [16, 17, 19]. However,
measurement-oriented optimization as discussed in the current contribution,
was not carried on. We leave this discussion for the future works. Here we
consider a particular case of optimization which allows shedding some light on
the best choice of the probe states for the experiments.

First, let us search for the optimal decomposition of the state
$\rho_{\text{true}}$, which simultaneously minimizes the additional sampling
noise $(\zeta^{2}-1)/N_{\text{s}}$ (see Eq. (64)) and the decomposition error
$\mathrm{Tr}\\{(\rho-\rho_{\text{true}})^{2}\\}$:

$\min\limits_{\\{\rho_{j}\\},\\{c_{j}\\}}D,\quad
D=\left(\frac{\zeta^{2}-1}{N_{\text{s}}}+\operatorname{Tr}\\{(\rho-\rho_{\text{true}})^{2}\\}\right).$
(65)

A variation in $\rho_{j}$ and $c_{j}$ gives

$\delta
D=2\sum_{j}\left(\frac{\zeta\mathrm{sgn}(c_{j})}{N_{\text{s}}}+\operatorname{Tr}\\{\rho_{j}(\rho-\rho_{\text{true}})\\}\right)\delta
c_{j}\\\
+2\sum_{j}c_{j}\operatorname{Tr}\\{\delta\rho_{j}(\rho-\rho_{\text{true}})\\}.$
(66)

The normalization of density operators imposes the constraints $\sum_{j}\delta
c_{j}=0$ and $\mathrm{Tr}\\{\delta\rho_{j}\\}=0$ on the variations. Strictly
speaking, one should also impose certain constraints ensuring semi-positivity
of $\rho$ during the considered variation. However, if the state $\rho$ is
mixed with all its eigenvalues being strictly positive, the semi-positivity
condition is not violated for any infinitely small variation.

First, let us assume that we do not require the states $\rho_{j}$ to be
classical, i.e. we do not impose any additional constraints on
$\delta\rho_{j}$. In that case, the variation in $\rho_{j}$ (the lower line in
Eq. (66)) implies that the optimal representation should be accurate:
$\rho=\rho_{\text{true}}$. Then, variation in $c_{j}$ leads to the equation
$\sum_{j}\mathrm{sgn}(c_{j})\,\delta c_{j}=0$ and, finally, to positivity of
all the weights: $c_{j}>0$ for all $j$. Therefore, if we are not limited in
terms of the structure of $\rho_{j}$, the optimal choice is either
$\rho_{\text{true}}$ itself or any of its representations in the form of a
positive-weight mixture (if $\rho_{\text{true}}$ is mixed).

The latter conclusion about the positivity of optimal $c_{j}$ as soon as the
representation is accurate ($\rho=\rho_{\text{true}}$) stems from the
variation in $c_{j}$ and remains valid regardless of any constraints imposed
on $\delta\rho_{j}$. If, for the available set of probe states $\rho_{j}$, an
accurate decomposition of $\rho_{\text{true}}$ requires negative weights, such
a decomposition is not optimal. For usage of a simpler representation with a
smaller number of components $\rho_{j}$, the advantage of having smaller
sampling noise will exceedingly compensate for the loss in representation
accuracy.

Let us now assume that the set of probe states is limited by coherent states
only: $\rho_{j}=|\alpha_{j}\rangle\langle\alpha_{j}|$. Their variations can be
written as

$\delta\rho_{j}=(a^{\dagger}-\alpha_{j}^{\ast})|\alpha_{j}\rangle\langle\alpha_{j}|\delta\alpha_{j}+|\alpha_{j}\rangle\langle\alpha_{j}|(a-\alpha_{j})\delta\alpha_{j}^{\ast}.$
(67)

Independence of the variations of the coherent states amplitudes implies that
the following condition should be satisfied for the optimal representation:

$\langle\alpha_{j}|[a,\rho-\rho_{\text{true}}]|\alpha_{j}\rangle=0.$ (68)

From this equation it follows, for example, that for the set of coherent
probes not limited to just a single state, difference between the optimally
represented $\rho$ and $\rho_{\text{true}}$ cannot be proportional to a
coherent or a number state. Generally, the operator
$[a,\rho-\rho_{\text{true}}]$ should be expressible as a sum of linearly
independent operators. The number of these operators should be at least that
of the probe states.

Now let us consider a measurement-oriented version of the procedure described
above. We look for the optimal component set that minimizes sampling noise
(63) for a given $N_{\text{s}}$ and under the condition that a certain
measurement result is to be obtained. As the properties of the optimal
components $\rho_{j}$ are of interest here, we shall focus only on their
variation. Furthermore, we suppose that there is some desirable physical
property about the decomposition $\rho$ that needs to be fixed in the form of
an expectation-value constraint, namely $\mathrm{Tr}\\{{\rho}\,O\\}=\mu$ for
some observable $O$. The corresponding distance to be minimized,
$D^{\prime}=N_{\text{s}}\mathrm{MSE}-\lambda(\mathrm{Tr}\\{{\rho}\,O\\}-\mu)$.
This distance is parametrized by a Lagrange scalar $\lambda$. A variation in
$\rho_{j}$ therefore gives

$\displaystyle\delta D^{\prime}=$
$\displaystyle\,2\zeta\sum_{j}|c_{j}|\mathrm{Tr}\\{\rho_{j}\delta\rho_{j}\\}-2\sum_{j}c_{j}\mathrm{Tr}\\{\rho\delta\rho_{j}\\}$
$\displaystyle\,-\lambda\sum_{j}c_{j}\mathrm{Tr}\\{O\delta\rho_{j}\\}.$ (69)

Since the $\rho_{j}$’s are quantum states, they take the form
$\rho_{j}=W_{j}^{\dagger}W_{j}/\mathrm{Tr}\\{W_{j}^{\dagger}W_{j}\\}$. This
implies the variation

$\delta\rho_{j}=\dfrac{\delta W^{\dagger}_{j}W_{j}+W^{\dagger}_{j}\delta
W_{j}}{\mathrm{Tr}\\{W_{j}^{\dagger}W_{j}\\}}-\rho_{j}\,\dfrac{\mathrm{Tr}\\{\delta
W^{\dagger}_{j}W_{j}+W^{\dagger}_{j}\delta
W_{j}\\}}{\mathrm{Tr}\\{W_{j}^{\dagger}W_{j}\\}}$ (70)

that leads to the extremal equation

$\displaystyle c_{j}\rho_{j}M=$
$\displaystyle\,2\zeta|c_{j}|(\rho_{j}^{2}-\rho_{j}\mathrm{Tr}\\{\rho_{j}^{2}\\})+c_{j}\rho_{j}\mathrm{Tr}\\{\rho_{j}M\\}\,,$
$\displaystyle M=$ $\displaystyle\,2\rho+\lambda\,O,$ (71)

when $\delta D^{\prime}$ is set to zero.

In order to satisfy (71), we now need $\rho_{j}$ to commute with
$2\rho+\lambda\,O$. This practically means that $\rho_{j}$ shares common
eigenstates with $\rho$ and $O$. A specific situation is when $O=a^{\dagger}a$
and $\mu$ is the mean photon number of the system. Then, an extremal set of
$\rho_{j}$’s is some set of Fock-state mixture ($\rho_{j}=\sum_{n}|n\rangle
w_{jn}\langle n|$ with $\sum_{n}w_{jn}=1$), which is compatible with a $\rho$
that is also a Fock state.

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