Quantum speed limits in arbitrary phase spaces Weiquan Meng1and Zhenyu Xu1 1School of Physical Science and Technology Soochow University Suzhou 215006 China

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Quantum speed limits in arbitrary phase spaces

Weiquan Meng1and Zhenyu Xu1

1School of Physical Science and Technology, Soochow University, Suzhou 215006, China

(Dated: August 25, 2023)

Quantum speed limits (QSLs) provide an upper bound for the speed of evolution of quantum states

in any physical process. Based on the Stratonovich-Weyl correspondence, we derive a universal QSL

bound in arbitrary phase spaces that is applicable for both continuous variable systems and ﬁnite-

dimensional discrete quantum systems. This QSL bound allows the determination of speed limit

bounds in speciﬁc phase spaces that are tighter than those in Wigner phase space or Hilbert space

under the same metric, as illustrated by several typical examples, e.g., a single-mode free ﬁeld and N-

level quantum systems in phase spaces. This QSL bound also provides an experimentally realizable

way to examine the speed limit in phase spaces relevant to applications in quantum information and

quantum optics.

I. INTRODUCTION

Quantum speed limits (QSLs) set the upper bound

on the speed for quantum systems evolving in an arbi-

trary physical process [1–3]. The milestone of describing

the QSL time as the intrinsic time scale of quantum dy-

namics was achieved by Mandelstam and Tamm, who

clariﬁed the longstanding debate on the explanations for

the time-energy uncertainty relationship [4]. The energy

ﬂuctuation restricts the minimum time for a quantum

state to evolve into its orthogonal state. Apart from the

standard deviation of the energy [4–6], alternative QSL

time bounds based on the averaged energy [7–10] and

the interplay among them [11–13] have been developed

in succession. In the last decade, QSLs have even been

extended to the ﬁelds of open systems involving various

metrics [14–20], quantum dynamical speedup [16,21–26],

information geometry [27], time-optimal quantum con-

trol [28–32], quantum and macroscopic stochastic pro-

cesses [33–35], non-Hermitian systems [15,36–38], dy-

namics of many-body quantum systems [39,40], the evo-

lution of observables [41–43], and recently the topological

structure of the dynamics [44]. Indeed, the QSL rep-

resents a powerful tool for evaluating the performance

of quantum computers [45], the accuracy of quantum

metrology [46,47], and the eﬃciency in quantum thermo-

dynamics [48–56], among others. Thus, extensive eﬀort

has been applied toward experimental veriﬁcation to this

end [57–61].

Apart from the well-known Schr¨odinger’s wave func-

tion, Heisenberg’s matrix mechanics, and Feynman’s

path integral, the phase-space formalism of quantum me-

chanics has an advantage in terms of its resemblance to

classical statistical mechanics and supports a profound

understanding of the underlying statistical properties

[62]. The pioneering work regarding the study of QSLs in

phase spaces was presented by Deﬀner, who found that

the QSL in the Wigner (W) phase space is equivalent

to that in the density operator space and much easier to

perform computations involving continuous variable sys-

tems [63]. Later, because the Wphase space retains the

structure of the classical phase space through quantum-

classical correspondence, Shanahan et al. discovered that

the QSL is not a particular quantum phenomenon but a

universal property of the time evolution of continuous

variable systems in the Wphase space [64]. Meantime,

questions regarding the speed limit in the Wphase space

have led to the intensive study of the classical speed limit

over the past few years [64–67].

For continuous variable systems, the Wfunction is the

most widely used phase-space quasiprobability represen-

tation with symmetric ordering of the position and mo-

mentum operators (or equivalently aand a†) [68]. Nev-

ertheless, it is not unique, and a large number of well-

known phase-space quasiprobability distribution func-

tions exist [62]. For instance, the Glauber-Sudarshan P

function [69,70] with a normal ordering of aand a†can

be used to diagonalize the density operator in terms of

coherent states and is fundamental in photoelectric detec-

tion [71]. Additionally, the Husimi Qfunction [72] with

the antinormal ordering of aand a†provides a possible

way to deﬁne a nonnegative quasiprobability distribu-

tion, and the Cahill-Glauber s-parametrized quasiprob-

ability distributions incorporate the above W,P, and Q

functions, with s= 0,1, and −1, respectively [73,74].

In addition, the ﬂourishing development in quantum

information and quantum technology now requires the

consideration of ﬁnite-dimensional systems, e.g., an en-

semble of spins, to deﬁne corresponding quasiprobability

distribution functions (see a recent review [75]). Com-

monly, there are two methods. The ﬁrst method involves

generating discrete analogues of the quasiprobability dis-

tribution functions [76–78], an approach that has been

used in various applications in quantum-state tomogra-

phy [79], resource theory for quantum computation [80–

82], and studies of the dynamics of many-body quantum

systems [83,84]. The second method involves employ-

ing generalized coherent states [85–90] to provide insights

into the study of quantum foundations, e.g., the recent

construction of a general statistical framework for ﬁnite-

dimensional discrete quantum systems [91] and the inter-

pretation of quantum entanglement with classical trajec-

tories [92].

At this point, the following two questions naturally

arXiv:2210.14278v2 [quant-ph] 24 Aug 2023

arise: (i) Is there a simple QSL bound that can incorpo-

rate continuous variable and ﬁnite-dimensional quantum

systems in arbitrary phase spaces? (ii) What are the

tangible beneﬁts for the QSL in other quasiprobability

distribution phase spaces compared with those in the W

phase space?

To address the above questions, in this paper, we de-

rive a universal QSL bound for quantum systems in ar-

bitrary phase spaces using the Stratonovich-Weyl (SW)

correspondence [93]. The key concept for the SW corre-

spondence is to transform any operator in Hilbert space

into target phase spaces (W,P,Qor others) with an

SW kernel, which is constructible for both continuous

variable and ﬁnite-dimensional quantum systems in s-

parametrized phase spaces [75]. This universal QSL

bound has several tangible beneﬁts. (i) First, our QSL

bound provides a uniﬁed framework for studying the

speed limit for continuous variable and ﬁnite-dimensional

quantum systems in quasiprobability distribution phase

spaces. This uniﬁed QSL bound is not just for aes-

thetic appeal but provides a convenient tool to evaluate

the speed limit directly in various scenarios. (ii) Sec-

ond, our work provides new insight into the search for

tighter QSL bounds that are superior to those obtained

in the Wphase space under the same metric. We use the

traditional Cauchy-Schwartz inequality method for the

ﬁrst scaling and the s-parametrized phase spaces as tools

for secondary scaling, and this latter step is our major

contribution. (iii) Third, in the past, phase spaces with

s̸= 0,±1 were rarely used, as they were usually believed

to be less valuable than traditional spaces in quantum

optics [71]. In our paper, we show that these represen-

tations also have advantages. The tighter QSL bound

achieved with s̸= 0,±1 provides evidence that a speciﬁc

choice of scould also be superior to the commonly used

W,P, and Qphase spaces in studying QSL bounds.

Our paper is organized as follows. In Sec. II, we de-

rive the universal QSL bound in arbitrary phase spaces.

Then, we employ this uniﬁed QSL bound to several typ-

ical quantum systems, including a single-mode free ﬁeld

and an N-level quantum system in s-parametrized phase

spaces in Sec. III. The experimental consideration is an-

alyzed in Sec. IV. Finally, we summarize in Sec. V with

concluding remarks and an outlook of potential applica-

tions.

II. QUANTUM SPEED LIMITS IN ARBITRARY

PHASE SPACES

Consider an operator Ain Hilbert space; its SW sym-

bol in an arbitrary phase space is [75]

A(η) := Tr[A∆s(η)],(1)

where ∆s(η) denotes the SW kernel, and ﬁve criteria (lin-

earity, reality, standardization, covariance, and tracing

properties) ensure such a legitimate SW correspondence

(see Appendix A). ηis a point in a phase space that de-

termines a state |η⟩(η→ |η⟩) in Hilbert space and the

index slabels a family of phase spaces, e.g., F0

A(η) is the

well-known Wigner function [75]. Since A→Fs

A(η) is a

one-to-one linear map, it is natural to deﬁne the inverse

of the SW symbol as

A=Zdµ(η)Fs

A(η)∆−s(η),(2)

where dµ(η) is the invariant integration measure. Then,

the trace of the product of the two operators is immedi-

ately obtained (see Appendix A)

Tr(AB) = Zdµ(η)Fs

A(η)F−s

B(η).(3)

Consider a quantum system evolving under the Hamil-

tonian H, the overlap between the initial state ρ0and

the ﬁnal state ρtis captured by the relative purity

Pt(ρ0, ρt) := Tr(ρ0ρt) [12,15,64], which is then trans-

formed by the above SW correspondence

Pt(ρ0, ρt) = Zdµ(η)F−s

ρ0(η)Fs

ρt(η).(4)

As the relative purity is a metric of the state evolution,

its time derivative can be regarded as the speed of the

state evolution. The primary procedure for obtaining

the changing rate of the relative purity is to calculate

∂tFs

ρt(η), which is achieved by transforming the von Neu-

mann equation ∂tρt=1

iℏ[H, ρt] to the phase space

∂F s

ρt(η)

∂t =Fs

H, F s

ρt(η),(5)

where we employ a generalized Moyal bracket

{{Fs

A, F s

B}} := 1

iℏ(Fs

A⋆ F s

B−Fs

B⋆ F s

A),(6)

and a generalized star product

(Fs

A⋆ F s

B)(η):= Zdµ(η′)Zdµ(η′′)Fs′

A(η′)

×Fs′′

B(η′′)Tr[∆−s′(η′)

×∆−s′′ (η′′)∆s(η)].(7)

Equation (5) is also called the generalized Liouville equa-

tion [75] and governs the dynamics of a phase space func-

tion.

To this end, the changing rate of the relative purity is

given by

Pt(ρ0, ρt) = Zdµ(η)F−s

ρ0(η)Fs

H, F s

ρt(η)

=Zdµ(η)F−s

ρt(η)Fs

ρ0, F s

H(η).

(8)

Note that the second line is more convenient for

use in experimental veriﬁcation (see Sec. IV) and

that the derivation can be found in Appendix B. By

means of the Cauchy-Schwarz inequality in L2, i.e.,

Rf(x)g(x)∗dx

2≤R|f(x)|2dx R|g(x)|2dx, the norm of

the changing rate for the purity is bounded by



Pt(ρ0, ρt)≤Vs

QSL(t),(9)

with

QSL(t) := min χ−s

tvs

QSL(0) , χ−s

0vs

QSL(t),(10)

where the two terms in min{·,·} are equal if Vs

QSL is

time-independent. The two components χ−sand vs

QSL

of Vs

QSL(t) are in dual phase spaces with opposite s[94].

Here

χs

t:= Zdµ(η)Fs

ρt(η)2

,(11)

and

QSL(t) := Zdµ(η)Fs

ρt, F s

H(η)

2

,(12)

where the reality postulate of the SW correspondence is

employed in the deﬁnition of Eq. (11). Speciﬁcally, χ−s

and vs

QSL in Eq. (10) are in a self-dual phase space when

s= 0. Upon integration of Eq.(9) with respect to time

from 0 to τ, we obtain the usual deﬁnition of QSL time

τ≥τQSL := 1−Pτ

DVs

QSL(t)Eτ

,(13)

where ⟨f(t)⟩τ:= 1

τRτ

0f(t)dt.

This upper bound of the evolution speed Vs

QSL(t) is

applicable to quantum systems in arbitrary phase spaces

as long as the SW kernel is given. The universality of

the derived QSL bound lies in the fact that the gen-

eralized Moyal bracket and star product are convenient

for use with diﬀerent kinds of systems in various phase

spaces, such as continuous and discrete phase spaces, in

the same framework. In the following, we examine the de-

rived QSL bound of both continuous variable and ﬁnite-

dimensional quantum systems in several typical phase

spaces, e.g., a single-mode of the quantized radiation

ﬁeld in Cahill-Glauber s-parametrized quasiprobability

distribution phase spaces, N-level quantum systems in

s-parametrized phase spaces with continuous degrees of

freedom, and qubit systems in a toroidal lattice phase

space with discrete degrees of freedom.

III. EXAMPLES

A. A single-mode free ﬁeld or a one-dimensional

harmonic oscillator in phase spaces

For a single-mode free ﬁeld or a one-dimensional har-

monic oscillator, the s-parametrized SW kernel is given

by [73]

∆s(η) = Zdµ(ζ)D(ζ)eηζ∗−η∗ζ+s|ζ|2/2,(14)

deﬁned in terms of the displacement operator D(ζ) :=

eζa†−ζ∗a, and the integration measure dµ(ζ) = (1/π)d2ζ.

This s-parametrized SW kernel has been widely studied

in the ﬁeld of quantum optics [95,96]. Considering the

system initially prepared in a coherent state |α0⟩, the

evolved state under the control of H=ℏωa†a+1

2

is |αt⟩=e−iωt/2α0e−iωt=e−iωt/2|α⟩[95], where

we use α:= α0e−iωt as an abbreviation. Then, the

SW symbol of the coherent state is given by Fs

ρt(η) =

1−sexp 2

s−1|α−η|2with −1⩽s < 1 [74]. In terms of

Eqs.(11) and (12), we are ready to obtain the two com-

ponents of Vs

QSL (see Appendix C):

χs=1

√1−s(15)

and

QSL =√2ω|α0|

1−s,(16)

which are time-independent (thus, we omit the subscript

tfor notation simplicity) and lead to the ﬁnal result

QSL =√2ω|α0|

(1 −s)√1 + s.(17)

It is convenient to obtain the QSL bound in Hilbert

space VQSL =pTr(|˙ρt|2) = √2

ℏ∆E=√2ω|α0|, which is

equivalent to the case of s= 0 in Eq. (17). Thus, Eq.

(17) can be rewritten as Vs

QSL =√2

ℏ(1−s)√1+s∆Ein terms

of the standard deviation of H. This is a QSL bound

of the Mandelstam-Tamm type for the s-parametrized

phase space, which is in agreement with the QSL bound

derived in Ref. [64] in Wigner phase space with s= 0.

The tightest bound is immediately obtained as V−1

QSL =

3√3

4ℏ∆E.

In Fig. 1,Vs

QSL versus parameter sis depicted given

ω|α0|= 1. One can ﬁnd that the self-dual case, i.e.,

s= 0, is not the tightest one (neither are the other well-

known representations, e.g., s=−1). Obviously, there

is a representation with s̸= 0 that has the lowest upper

bound. Indeed, by choosing an appropriate parameter of

the phase space, i.e., s=−1/3as analyzed above, we

can ﬁnd the corresponding tightest dual phase spaces, as

marked by the red dot in Fig. 1.

The above analysis implies that the s-parametrized

phase spaces provide an approach to search for tighter

QSL bounds than the one given initially in Wphase space

or Hilbert space. Such a phase space is unlike the usual

W,P, and Qrepresentations. Indeed, the rarely used

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QuantumspeedlimitsinarbitraryphasespacesWeiquanMeng1andZhenyuXu11SchoolofPhysicalScienceandTechnology,SoochowUniversity,Suzhou215006,China(Dated:August25,2023)Quantumspeedlimits(QSLs)provideanupperboundforthespeedofevolutionofquantumstatesinanyphysicalprocess.BasedontheStratonovich-Weylcorrespondenc...

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Quantum speed limits in arbitrary phase spaces Weiquan Meng1and Zhenyu Xu1 1School of Physical Science and Technology Soochow University Suzhou 215006 China

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