publications

2025

1. Majorization theory for quasiprobabilities

   Twesh Upadhyaya, Zacharie Van Herstraeten, Jack Davis, and 3 more authors

   arXiv [quant-ph], Jul 2025

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   Majorization theory is a powerful mathematical tool to compare the disorder in distributions, with wide-ranging applications in many fields including mathematics, physics, information theory, and economics. While majorization theory typically focuses on probability distributions, quasiprobability distributions provide a pivotal framework for advancing our understanding of quantum mechanics, quantum information, and signal processing. Here, we introduce a notion of majorization for continuous quasiprobability distributions over infinite measure spaces. Generalizing a seminal theorem by Hardy, Littlewood, and Pólya, we prove the equivalence of four definitions for both majorization and relative majorization in this setting. We give several applications of our results in the context of quantum resource theories, obtaining new families of resource monotones and no-goes for quantum state conversions. A prominent example we explore is the Wigner function in quantum optics. More generally, our results provide an extensive majorization framework for assessing the disorder of integrable functions over infinite measure spaces.

2024

1. Exploring the possibility of a complex-valued non-Gaussianity measure for quantum states of light

   Andrew J. Pizzimenti, Prajit Dhara, Zacharie Van Herstraeten, and 2 more authors

   APL Quantum, Sep 2024

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   We consider a quantity that is the differential relative entropy between a generic Wigner function and a Gaussian one. We prove that said quantity is minimized with respect to its Gaussian argument, if both Wigner functions in the argument of the Wigner differential entropy have the same first and second moments, i.e., if the Gaussian argument is the Gaussian associate of the other, generic Wigner function. Therefore, we introduce the differential relative entropy between any Wigner function and its Gaussian associate and we examine its potential as a non-Gaussianity measure. The proposed, phase-space based non-Gaussianity measure is complex-valued, with its imaginary part possessing the physical meaning of the Wigner function’s negative volume. At the same time, the real part of this measure provides an extra layer of information, rendering the complex-valued quantity a measure of non-Gaussianity, instead of a quantity pertaining only to the negativity of the Wigner function. We prove that the measure (both the real and imaginary parts) is faithful, invariant under Gaussian unitary operations, and find a sufficient condition on its monotonic behavior under Gaussian channels. We provide numerical results supporting the aforesaid condition. In addition, we examine the measure’s usefulness to non-Gaussian quantum state engineering with partial measurements.

2. Majorization theoretical approach to entanglement enhancement via local filtration

   Zacharie Van Herstraeten, Nicolas J. Cerf, Saikat Guha, and 1 more author

   Phys. Rev. A, Oct 2024

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   From the perspective of majorization theory, we study how to enhance the entanglement of a two-mode squeezed vacuum (TMSV) state by using local filtration operations. We present several schemes achieving entanglement enhancement with photon addition and subtraction, and then consider filtration as a general probabilistic procedure consisting in acting with local (nonunitary) operators on each mode. From this, we identify a sufficient set of two conditions for these filtration operators to successfully enhance the entanglement of a TMSV state, namely, the operators must be Fock orthogonal (i.e., preserving the orthogonality of Fock states) and Fock amplifying (i.e., giving larger amplitudes to larger Fock states). Our results notably prove that ideal photon addition, subtraction, and any concatenation thereof always enhance the entanglement of a TMSV state in the sense of majorization theory. We further investigate the case of realistic photon addition (subtraction) and are able to upper bound the distance between a realistic photon-added (-subtracted) TMSV state and a nearby state that is provably more entangled than the TMSV, thus extending entanglement enhancement to practical schemes via the use of a notion of approximate majorization. Finally, we consider the state resulting from 𝑘-photon addition (on each of the two modes) on a TMSV state. We prove analytically that the state corresponding to 𝑘=1 majorizes any state corresponding to 2≤𝑘≤8 and we conjecture the validity of the statement for all 𝑘≥9.

3. Classical capacity of quantum non-Gaussian attenuator and amplifier channels

   Zacharie Van Herstraeten, Saikat Guha, and Nicolas J. Cerf

   Int. J. Quantum Inf., Aug 2024

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   We consider a quantum bosonic channel that couples the input mode via a beam splitter or two-mode squeezer to an environmental mode that is prepared in an arbitrary state. We investigate the classical capacity of this channel, which we call a non-Gaussian attenuator or amplifier channel. If the environment state is thermal, we of course recover a Gaussian phase-covariant channel whose classical capacity is well known. Otherwise, we derive both a lower and an upper bound to the classical capacity of the channel, drawing inspiration from the classical treatment of the capacity of non-Gaussian additive-noise channels. We show that the lower bound to the capacity is always achievable and give examples where the non-Gaussianity of the channel can be exploited so that the communication rate beats the capacity of the Gaussian-equivalent channel (i.e. the channel where the environment state is replaced by a Gaussian state with the same covariance matrix). Finally, our upper bound leads us to formulate and investigate conjectures on the input state that minimizes the output entropy of non-Gaussian attenuator or amplifier channels. Solving these conjectures would be a main step toward accessing the capacity of a large class of non-Gaussian bosonic channels.

4. Wigner entropy conjecture and the interference formula in quantum phase space

   Zacharie Van Herstraeten and Nicolas J. Cerf

   arXiv [quant-ph], Nov 2024

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   Wigner-positive quantum states have the peculiarity to admit a Wigner function that is a genuine probability distribution over phase space. The Shannon differential entropy of the Wigner function of such states - called Wigner entropy for brevity - emerges as a fundamental information-theoretic measure in phase space and is subject to a conjectured lower bound, reflecting the uncertainty principle. In this work, we prove that this Wigner entropy conjecture holds true for a broad class of Wigner-positive states known as beam-splitter states, which are obtained by evolving a separable state through a balanced beam splitter and then discarding one mode. Our proof relies on known bounds on the p-norms of cross-Wigner functions and on the interference formula, which relates the convolution of Wigner functions to the squared modulus of a cross-Wigner function. Originally discussed in the context of signal analysis, the interference formula is not commonly used in quantum optics although it unveils a strong symmetry exhibited by Wigner functions of pure states. We provide here a simple proof of the formula and highlight some of its implications. Finally, we prove an extended conjecture on the Wigner-Rényi entropy of beam-splitter states, albeit in a restricted range for the Rényi parameter α≥1/2.

2023

1. Complex-valued Wigner entropy of a quantum state

   Nicolas J. Cerf, Anaelle Hertz, and Zacharie Van Herstraeten

   Quantum Stud. Math. Found., Oct 2023

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   It is common knowledge that the Wigner function of a quantum state may admit negative values, so that it cannot be viewed as a genuine probability density. Here, we examine the difficulty in finding an entropy-like functional in phase space that extends to negative Wigner functions and then advocate the merits of defining a complex-valued entropy associated with any Wigner function. This quantity, which we call the complex Wigner entropy, is defined via the analytic continuation of Shannon’s differential entropy of the Wigner function in the complex plane. We show that the complex Wigner entropy enjoys interesting properties, especially its real and imaginary parts are both invariant under Gaussian unitaries (displacements, rotations, and squeezing in phase space). Its real part is physically relevant when considering the evolution of the Wigner function under a Gaussian convolution, while its imaginary part is simply proportional to the negative volume of the Wigner function. Finally, we define the complex-valued Fisher information of any Wigner function, which is linked (via an extended de Bruijn’s identity) to the time derivative of the complex Wigner entropy when the state undergoes Gaussian additive noise. Overall, it is anticipated that the complex plane yields a proper framework for analyzing the entropic properties of quasiprobability distributions in phase space.

2. Majorization ladder in bosonic Gaussian channels

   Zacharie Van Herstraeten, Michael G. Jabbour, and Nicolas J. Cerf

   AVS Quantum Sci., Mar 2023

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   We show the existence of a majorization ladder in bosonic Gaussian channels, that is, we prove that the channel output resulting from the nth energy eigenstate (Fock state) majorizes the channel output resulting from the (n+1)th energy eigenstate (Fock state). This reflects a remarkable link between the energy at the input of the channel and a disorder relation at its output as captured by majorization theory. This result was previously known in the special cases of a pure-loss channel and quantum-limited amplifier, and we achieve here its non-trivial generalization to any single-mode phase-covariant (or -contravariant) bosonic Gaussian channel. The key to our proof is the explicit construction of a column-stochastic matrix that relates the outputs of the channel for any two subsequent Fock states at its input. This is made possible by exploiting a recently found recurrence relation on multiphoton transition probabilities for Gaussian unitaries [Jabbour and Cerf, Phys. Rev. Res. 3, 043065 (2021)]. Possible generalizations and implications of these results are then discussed.

3. Continuous majorization in quantum phase space

   Zacharie Van Herstraeten, Michael G. Jabbour, and Nicolas J. Cerf

   Quantum, Mar 2023

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   We explore the role of majorization theory in quantum phase space. To this purpose, we restrict ourselves to quantum states with positive Wigner functions and show that the continuous version of majorization theory provides an elegant and very natural approach to exploring the information-theoretic properties of Wigner functions in phase space. After identifying all Gaussian pure states as equivalent in the precise sense of continuous majorization, which can be understood in light of Hudson’s theorem, we conjecture a fundamental majorization relation: any positive Wigner function is majorized by the Wigner function of a Gaussian pure state (especially, the bosonic vacuum state or ground state of the harmonic oscillator). As a consequence, any Schur-concave function of the Wigner function is lower bounded by the value it takes for the vacuum state. This implies in turn that the Wigner entropy is lower bounded by its value for the vacuum state, while the converse is notably not true. Our main result is then to prove this fundamental majorization relation for a relevant subset of Wigner-positive quantum states which are mixtures of the three lowest eigenstates of the harmonic oscillator. Beyond that, the conjecture is also supported by numerical evidence. We conclude by discussing some implications of this conjecture in the context of entropic uncertainty relations in phase space.

2021

1. Quantum Wigner entropy

   Zacharie Van Herstraeten and Nicolas J. Cerf

   Phys. Rev. A, Oct 2021

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   We define the Wigner entropy of a quantum state as the differential Shannon entropy of the Wigner function of the state. This quantity is properly defined only for states that possess a positive Wigner function, which we name Wigner-positive states, but we argue that it is a proper measure of quantum uncertainty in phase space. It is invariant under symplectic transformations (displacements, rotations, and squeezing) and we conjecture that it is lower bounded by ln⁡𝜋+1 within the convex set of Wigner-positive states. It reaches this lower bound for Gaussian pure states, which are natural minimum-uncertainty states. This conjecture bears a resemblance with the Wehrl-Lieb conjecture, and we prove it over the subset of passive states of the harmonic oscillator which are of particular relevance in quantum thermodynamics. Along the way, we present a simple technique to build a broad class of Wigner-positive states exploiting an optical beam splitter and reveal an unexpectedly simple convex decomposition of extremal passive states. The Wigner entropy is anticipated to be a significant physical quantity, for example, in quantum optics where it allows us to establish a Wigner entropy-power inequality. It also opens a way towards stronger entropic uncertainty relations. Finally, we define the Wigner-Rényi entropy of Wigner-positive states and conjecture an extended lower bound that is reached for Gaussian pure states.

2. Majorization theoretical approach to quantum uncertainty – From Wigner entropy to Gaussian bosonic channels

   Zacharie Van Herstraeten

   Université libre de Bruxelles, Oct 2021

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   This thesis is centered on a novel approach to quantum uncertainty based on applying the theory of continuous majorization to quantum phase-space distributions. Majorization theory is a powerful mathematical framework that is aimed at comparing distributions with respect to intrinsic disorder. It is particularly significant in the sense that establishing a majorization relation between two distributions amounts to proving that every (Shur-concave) measure of disorder will categorize one distribution as more ordered than the other. Although this is less known, the distributions here do not need to be normalized nor positive for majorization theory to apply, so the latter even extends beyond probability distributions. Further, a majorization relation can rigorously be defined for both discrete and continuous distributions over a finite-size domain, as well as for (discrete and continuous) distributions that are positive over an infinite-size domain.The central thrust of this thesis is to characterize quantum uncertainty in phase space by applying the tools of majorization theory to the Wigner function, which is the most common (quasi)distribution that embodies a quantum state in phase space. Wigner functions are in general positive and negative, putting them beyond the reach of most information-theoretical measures but perfect candidates for the theory of majorization. We start our manuscript with a succinct overview of the basics of quantum optics in phase space, which are a prerequisite for the characterization of disorder in phase space. This gives us the occasion to present a secondary achievement of the thesis consisting in establishing a resource theory for local Gaussian work extraction, which exploits the symplectic formalism within quantum thermodynamics. In this context, work can be defined as the difference between the trace and symplectic trace of the covariance matrix of the state, and it displays a number of interesting properties. Back to our primary interest, our first contribution is to construct an extended formulation of majorization with the applicability to Wigner functions as our main objective. It must be stressed that when relaxing the positivity condition, majorization relations have not been addressed in the literature for (discrete or continuous) distributions defined over an infinite-size domain. Here, we write extended majorization relations that equally apply to discrete or continuous, positive or negative distributions defined over a finite-size or infinite-size domain. Then, applying these to Wigner functions (which can be negative and have an infinite-size domain), our first finding is that all pure states are either incomparable or equivalent as regards majorization. Hence, any pure state cannot be objectively deemed more disordered than any other one in phase space, so that majorization appears to be better suited to compare mixed states.A large part of the thesis is then concerned with the convex set of quantum states that have a positive Wigner function, seeking a better understanding of its structure. As a consequence of Hudson theorem, this set only contains mixed states, with the notable exception of Gaussian pure states. We highlight a large subset of Wigner-positive states that can be prepared using a balanced beam-splitter and show that these states play a key role in the geometry of the Wigner-positive set as they are extremal states. Restricting ourselves to Wigner-positive states, we formulate the conjecture that the Gaussian pure states (most notably, the vacuum state) are the states of least disorder, which is expressed via a fundamental majorization relation. This conjecture is supported by numerical simulations and is analytically proven over the subset of phase-invariant states containing up to two photons. It can be viewed as a precursor to all uncertainty relations in phase space.Our next contribution pertains to the usage of information theory in quantum phase space. The conjectured fundamental majorization relation for Wigner-positive states implies in turn an infinite number of inequalities relating Schur-concave functionals, most notably the Rényi entropy and the Shannon differential entropy of the Wigner function. We define the latter as the Wigner entropy of a state, and conjecture that it is lower bounded by lnpi+1, namely the value it takes for Gaussian pure states. We then prove that this bound holds over the thermodynamically-relevant set of passive states in arbitrary dimension. The Wigner entropy is itself a lower bound on the sum of the entropies of the marginal distributions in phase space, which makes a clear connection with the entropic uncertainty relations of Białynicki-Birula and Mycielski (which are implied by our conjecture). Interestingly, the Wehrl-Lieb conjecture is also a consequence of our conjecture (while it does not imply it). Turning back to uncertainty measures that could also be applicable to Wigner-negative states, we also investigate which Schur-concave functionals can be considered as relevant uncertainty measures in phase space.A last contribution of the thesis concerns (discrete) majorization in state space, which is the usual way of applying majorization theory to quantum physics as described in the literature. Here, we focus on Gaussian phase-invariant bosonic channels and demonstrate the existence of a majorization ladder for Fock states at the input of the channel. This extends a prior analysis that was restricted to special (quantum-limited) channels. We also find a simple relation to determine whether such channels produce a Wigner-positive state for any input state, echoing the condition for entanglement-breaking channels. We hope that these considerations may help resolving some of the pending questions regarding entropies in bosonic systems and channels.

2019

1. Quantum thermodynamics in a multipartite setting: A resource theory of local Gaussian work extraction for multimode bosonic systems

   Uttam Singh, Michael G. Jabbour, Zacharie Van Herstraeten, and 1 more author

   Phys. Rev. A, Oct 2019

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   Quantum thermodynamics can be cast as a resource theory by considering free access to a heat bath, thereby viewing the Gibbs state at a fixed temperature as a free state and hence any other state as a resource. Here, we consider a multipartite scenario where several parties attempt at extracting work locally, each having access to a local heat bath (possibly with a different temperature), assisted with an energy-preserving global unitary. As a specific model, we analyze a collection of harmonic oscillators or a multimode bosonic system. Focusing on the Gaussian paradigm, we construct a reasonable resource theory of local activity for a multimode bosonic system, where we identify as free any state that is obtained from a product of thermal states (possibly at different temperatures) acted upon by any linear-optics (passive Gaussian) transformation. The associated free operations are then all linear-optics transformations supplemented with tensoring and partial tracing. We show that the local Gaussian extractable work (if each party applies a Gaussian unitary, assisted with linear optics) is zero if and only if the covariance matrix of the system is that of a free state. Further, we develop a resource theory of local Gaussian extractable work, defined as the difference between the trace and symplectic trace of the covariance matrix of the system. We prove that it is a resource monotone that cannot increase under free operations. We also provide examples illustrating the distillation of local activity and local Gaussian extractable work.

2017

1. Differential entropies in phase space for quantum photonic systems

   Zacharie Van Herstraeten

   Université libre de Bruxelles, Oct 2017

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   This report applies the theory of continuous majorization to quantum phase distributions. To this purpose, we restrict our study to nonnegative Wigner distributions. The physicality conditions in phase space formalism are addressed. Wigner entropy is introduced as the differential entropy of nonnegative Wigner distributions. The initial objective of this report is to prove the conjecture of Hertz, Jabbour and Cerf showing that vacuum is the state of least Wigner entropy. That aim is not reached here, but a scheme of demonstration is proposed. The demonstration involves the properties of pure loss channels and decreasing rearrangements. A majorization criterion is derived using Lindblad equation applied to pure loss channels. That criterion ensures that the instantaneous output of a pure loss channel majorizes its input, under some conditions. Finally, numerical simulations are presented to illustrate the validity of the conjecture.