# On Non-Convexity of the Nonclassicality Measure *via* Operator Ordering Sensitivity

^{1}School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, China^{2}Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China^{3}School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China^{4}State Key Laboratory of Mesoscopic Physics, School of Physics, Frontiers Science Center for Nano-optoelectronics, Peking University, Beijing, China^{5}Beijing Academy of Quantum Information Sciences, Beijing, China

In quantum optics, nonclassicality in the sense of Glauber-Sudarshan is a valuable resource related to the quantum aspect of photons. A desirable and intuitive requirement for a consistent measure of nonclassicality is convexity: Classical mixing should not increase nonclassicality. We show that the recently introduced nonclassicality measure [Phys. Rev. Lett. **122**, 080402 (2019)] is not convex. This nonclassicality measure is defined via operator ordering sensitivity, which is an interesting and significant probe (witness) of nonclassicality without convexity but can be intrinsically connected to the convex Wigner-Yanase skew information [Proc. Nat. Acad. Sci. United States **49**, 910 (1963)] *via* the square root operation on quantum states. Motivated by the Wigner-Yanase skew information, we also propose a faithful measure of nonclassicality, although it cannot be readily computed, it is convex.

## 1 Introduction

In the conventional scheme of Glauber-Sudarshan, nonclassicality of light refers to quantum optical states that cannot be expressed as classical (probabilistic) mixtures of Glauber coherent states [1–7]. Its detection and quantification are of both theoretical and experimental importance in quantum optics [8–17]. Recently, a remarkable and interesting nonclassicality measure is introduced in Ref. [18]. This measure is well motivated and has operational significance stemmed from operator ordering sensitivity [18], which is also known as squared quadrature coherence scale in measuring quadrature coherence [19], and proved to be closely related to the entanglement [20]. Here we demonstrate that this nonclassicality measure, as well as the operator ordering sensitivity, are not convex. This means that classical (probabilistic) mixing of states can increase nonclassicality, as quantified by this nonclassicality measure via the operator ordering sensitivity. Our result complements the key contribution in Ref. [18].

By the way, we show that the operator ordering sensitivity, though not convex, can be connected to a convex quantity via the very simple and straightforward operation of square root. The modified quantity has both physical and information-theoretic significance, and is actually rooted in an amazing quantity of Wigner and Yanase, introduced in 1963 [21]. Motivated by the Wigner-Yanase skew information, we also propose a faithful measure of nonclassicality which is convex.

To be precise, let us first recall the basic idea and the key quantities in Ref. [18]. Consider a single-mode bosonic field with annihilation operator *a* and creation operator *a*^{†} satisfying the commutation relation

Let *α*⟩ = *D*(*α*)|0⟩ are the coherent states [1–3]. For a bosonic field state *ρ*, consider the parameterized phase space distributions [18]

on the phase space *s* ∈ [−1, 1], *d*^{2}*α* = *dxdy* with *s* = 1, 0, −1, the corresponding phase space distributions are the Glauber-Sudarshan *P* functions, the Wigner functions, and the Husimi functions, respectively.

Motivated by operator ordering due to noncommutativity and in terms of the Hilbert-Schmidt norm, the quantity

is introduced as a probe of nonclassicality of *ρ* in Ref. [18], and is called operator ordering sensitivity. Here

It turns out that.

where [*X*, *Y*] = *XY* − *YX* denotes operator commutator, and

are the conjugate quadrature operators. Simple manipulation shows that

Moreover, the following nonclassicality measure

is introduced as a key result [18]. Here

In particular,

is precisely the operator ordering sensitivity.

The purpose of this work is to demonstrate that the nonclassicality measure *S*_{o}(⋅) defined by Eq. 2 is not convex either.

The structure of the remainder of the paper is as follows. In Section 2, we demonstrate that the nonclassicality measure *S*_{o}(⋅) is not convex, it can be directly connected to a convex quantity related to the celebrated Wigner-Yanase skew information. By the way, we also present a simple proof of the fact that *S*_{o}(*ρ*) ≤ 1 for any classical state. In Section 4, we bring up a convex measure of nonclassicality based on the Wigner-Yanase skew information. Finally, a summary is presented in Section 5.

## 2 Non-Convexity of the Nonclassicality Measure $\mathcal{N}(\rho )$

In this section, we show that

with

Now we give a family of counterexamples to show that *ρ*. Considering the mixture

of the vacuum state *ρ*_{1} = |0⟩⟨0| (which is classical) and the Fock state *ρ*_{2} = |*n*⟩⟨*n*| with *n* > 1, then by direct calculation, we have

To evaluate *S*_{o}(*ρ*), noting that

we have, by direct calculation, that

from which we obtain

It follows from the inequality chain (4) that

while

Consequently,

Since when *n* > 24, the following inequality holds

it follows that

This implies that

## 3 Relating the Operator Ordering Sensitivity *S*_{o}(*ρ*) to the Wigner-Yanase Skew Information

As a side issue, in this section, we show that although the operator ordering sensitivity *S*_{o}(*ρ*) is not convex either with respect to *ρ*, it can be intrinsically related to the celebrated Wigner-Yanase skew information, which is convex.

First, we illustrate non-convexity of *S*_{o}(*ρ*) through the following counterexamples. Take

where |*n*⟩ are the Fock (number) states with

and

Now direct evaluation yields

and

Substituting the above into Eq. 2, we obtain

and

This implies that *S*_{o}(*ρ*) is not convex.

In the above counterexamples showing non-convexity of *S*_{o}(*ρ*), both the constituent states *ρ*_{1} and *ρ*_{2} are nonclassical in the sense that they cannot be represented as probabilistic mixtures of coherent states [1–3]. The following counterexamples illustrates that even the mixture of a classical thermal state and a nonclassical state can demonstrate non-convexity. Considering the thermal state

which is classical and the Fock state *τ*_{2} = |1⟩⟨1|, and their mixture

then by direct calculation, we have

To evaluate *S*_{o}(*τ*), noting that from Eq. 2, we have

Now direct calculation leads to

from which we obtain

Clearly

By continuity, this implies that *S*_{o}(⋅) is not convex for *λ* close to 1. More explicitly, for *λ* = 0.9, we have

which explicitly shows that *S*_{o}(⋅) is not convex.

The non-convex quantity *S*_{o}(*ρ*) can be modified to a convex one if we formally replace *ρ* by the square root

which is precisely the sum of the Wigner-Yanase skew information [21]

Remarkably,

which is essentially (up to a constant factor 1/2) an extension of the Wigner-Yanase skew information, as can be readily seen if we recast the original Wigner-Yanase skew information [21]

of the quantum state *ρ* with respect to (skew to) the observable (Hermitian operator) *K* as

and formally replace the Hermitian operator *K* by the non-Hermitian annihilation operator *a*. An apparent interpretation of *Q*, *P*) in the state *ρ* [22–24].

Due to the convexity of the Wigner-Yanase skew information [21], *ρ*, in sharp contrast to *S*_{o}(*ρ*). Moreover,

It is amusing to note the analogy between the passing from classical probability distributions to quantum mechanical amplitudes and that from *S*_{o}(*ρ*) to

By the way, we present an alternative and simple proof of the interesting fact that [18]

for any classical state *ρ*, which implies that *S*_{o}(⋅) is convex when the component states are restricted to coherent states (noting that *S*_{o}(|*α*⟩⟨*α*|) = 1 for any coherent state |*α*⟩), though it is not convex in the whole state space. To this end, let the Glauber-Sudarhsan *P* representation of *ρ* be

then

from which we obtain

In particular, if *ρ* is a classical state, then *P*(*α*) ≥ 0, and this implies that *S*_{o}(*ρ*) ≤ 1 for any classical state *ρ*. In contrast, the fact that

for any classical state follows readily from the convexity of *α*⟩.

## 4 A Convex Measure of Nonclassicality

Motivated by the Wigner-Yanase skew information, we propose a measure of nonclassicality defined as

Here *A*^{†}*A*, and

It is clear that

Considering the convex combination of quantum states *ρ*_{1} and *ρ*_{2} with probabilities *p*_{1} = *p* and *p*_{2} = 1 − *p* respectively, the mixed state is denoted by

Supposing that

due to the fact that the convex combination of classical states is also a classical state, we have

Here the second inequality holds due to

which can be obtained from the fact that |*A*+ *λB*|^{2} ≥ 0 for all real *λ*. While the third inequality follows from the convexity of the celebrated Wigner-Yanase skew information, the convexity of the measure

Similarly from inequality (10) and the convexity of the Wigner-Yanase skew information, we have

where

where *σ* is a classical state, and the last inequality can be directly obtained from inequality (8). So we have

In other words,

## 5. Conclusion

We have demonstrated that

By the way, we have also demonstrated that although the important operator ordering sensitivity *S*_{o}(⋅) is not convex either, it can be simply connected to the convex Wigner-Yanase skew information via the square root operation on quantum states, which is reminiscent of the passing from probabilities to amplitudes via square roots, so fundamental in going from classical to quantum.

Due to the remarkable properties and information-theoretic significance of the Wigner-Yanase skew information, it is desirable to employ this quantity to study nonclassicality of light in particular, and nonclassicality of arbitrary quantum states in general.

## Data Availability Statement

The original contributions presented in the study are included in the article/supplementary material further inquiries can be directed to the corresponding author.

## Author Contributions

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

## Funding

This work was supported by the Fundamental Research Funds for the Central Universities (Grant No. FRF-TP-19-012A3), the National Natural Science Foundation of China (Grant No. 11875317), the China Postdoctoral Science Foundation (Grant No. 2021M690414), the Beijing Postdoctoral Research Foundation (Grant No. 2021ZZ091), and the National Key R&D Program of China (Grant No. 2020YFA0712700).

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Keywords: coherent states, nonclassicality, operator ordering sensitivity, convexity, Wigner-Yanase skew information

Citation: Fu S, Luo S and Zhang Y (2022) On Non-Convexity of the Nonclassicality Measure *via* Operator Ordering Sensitivity. *Front. Phys.* 10:955786. doi: 10.3389/fphy.2022.955786

Received: 29 May 2022; Accepted: 13 June 2022;

Published: 08 July 2022.

Edited by:

Dong Wang, Anhui University, ChinaCopyright © 2022 Fu, Luo and Zhang. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Yue Zhang, zhangyue@baqis.ac.cn