ORIGINAL RESEARCH article
Sec. Quantum Engineering and Technology
Volume 10 - 2022 | https://doi.org/10.3389/fphy.2022.955786
On Non-Convexity of the Nonclassicality Measure via Operator Ordering Sensitivity
- 1School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, China
- 2Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China
- 3School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China
- 4State Key Laboratory of Mesoscopic Physics, School of Physics, Frontiers Science Center for Nano-optoelectronics, Peking University, Beijing, China
- 5Beijing 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.
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. . This measure is well motivated and has operational significance stemmed from operator ordering sensitivity , which is also known as squared quadrature coherence scale in measuring quadrature coherence , and proved to be closely related to the entanglement . 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. .
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 . 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. . Consider a single-mode bosonic field with annihilation operator a and creation operator a† satisfying the commutation relation
on the phase space
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. , 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 . Here
is precisely the operator ordering sensitivity.
The purpose of this work is to demonstrate that the nonclassicality measure
The structure of the remainder of the paper is as follows. In Section 2, we demonstrate that the nonclassicality measure
2 Non-Convexity of the Nonclassicality Measure
In this section, we show that
Now we give a family of counterexamples to show that
of the vacuum state ρ1 = |0⟩⟨0| (which is classical) and the Fock state ρ2 = |n⟩⟨n| with n > 1, then by direct calculation, we have
To evaluate So(ρ), noting that
we have, by direct calculation, that
from which we obtain
It follows from the inequality chain (4) that
Since when n > 24, the following inequality holds
it follows that
This implies that
3 Relating the Operator Ordering Sensitivity So(ρ) to the Wigner-Yanase Skew Information
As a side issue, in this section, we show that although the operator ordering sensitivity So(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 So(ρ) through the following counterexamples. Take
where |n⟩ are the Fock (number) states with
Now direct evaluation yields
Substituting the above into Eq. 2, we obtain
This implies that So(ρ) is not convex.
In the above counterexamples showing non-convexity of So(ρ), both the constituent states ρ1 and ρ2 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 τ2 = |1⟩⟨1|, and their mixture
then by direct calculation, we have
To evaluate So(τ), noting that from Eq. 2, we have
Now direct calculation leads to
from which we obtain
By continuity, this implies that So(⋅) is not convex for λ close to 1. More explicitly, for λ = 0.9, we have
which explicitly shows that So(⋅) is not convex.
The non-convex quantity So(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 
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 
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
Due to the convexity of the Wigner-Yanase skew information ,
It is amusing to note the analogy between the passing from classical probability distributions to quantum mechanical amplitudes and that from So(ρ) to
By the way, we present an alternative and simple proof of the interesting fact that 
for any classical state ρ, which implies that So(⋅) is convex when the component states are restricted to coherent states (noting that So(|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
from which we obtain
In particular, if ρ is a classical state, then P(α) ≥ 0, and this implies that So(ρ) ≤ 1 for any classical state ρ. In contrast, the fact that
for any classical state follows readily from the convexity of
4 A Convex Measure of Nonclassicality
Motivated by the Wigner-Yanase skew information, we propose a measure of nonclassicality defined as
It is clear that
Considering the convex combination of quantum states ρ1 and ρ2 with probabilities p1 = p and p2 = 1 − p respectively, the mixed state is denoted by
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 σ is a classical state, and the last inequality can be directly obtained from inequality (8). So we have
In other words,
We have demonstrated that
By the way, we have also demonstrated that although the important operator ordering sensitivity So(⋅) 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.
All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.
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.
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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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, China
Reviewed by:Shao-Ming Fei, Capital Normal University, China
Zhaoqi Wu, Nanchang University, China
Copyright © 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, firstname.lastname@example.org