Better Heisenberg limits coherence bounds and energy-time tradeos via quantum R enyi information Michael J. W. Hall

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Better Heisenberg limits, coherence bounds, and energy-time tradeoﬀs

via quantum R´enyi information

Michael J. W. Hall

Theoretical Physics, Research School of Physics, Australian National University, Canberra ACT 0200, Australia

An uncertainty relation for the R´enyi entropies of conjugate quantum observables is used to obtain

a strong Heisenberg limit of the form RMSE ≥f(α)/(hNi+1

2), bounding the root mean square error

of any estimate of a random optical phase shift in terms of average photon number, where f(α) is

maximised for non-Shannon entropies. Related simple yet strong uncertainty relations linking phase

uncertainty to the photon number distribution, such as ∆Φ ≥maxnpn, are also obtained. These

results are signiﬁcantly strengthened via upper and lower bounds on the R´enyi mutual information

of quantum communication channels, related to asymmetry and convolution, and applied to the

estimation (with prior information) of unitary shift parameters such as rotation angle and time, and

to obtain strong bounds on measures of coherence. Sharper R´enyi entropic uncertainty relations are

also obtained, including time-energy uncertainty relations for Hamiltonians with discrete spectra.

In the latter case almost-periodic R´enyi entropies are introduced for nonperiodic systems.

I. INTRODUCTION

Quantum mechanics places fundamental limits on the information which can be gained in various contexts, ranging

from the accuracy to which the phase shift of an optical probe state can be estimated to the secure key rate that can

be obtained from a cryptographic protocol. Such limits are often formulated via uncertainty relations that restrict,

for example, the degree to which values of two observables can be jointly speciﬁed, or the degree to which both an

intended party and an eavesdropper can access quantum information [1].

Entropic uncertainty relations place particularly strong restrictions, and underlie the main themes of this paper.

One has, for example, the number-phase uncertainty relation [2, 3]

H(N|ρ) + H(Φ|ρ)≥log 2π+H(ρ),(1)

for the number and canonical phase observables of an optical mode, Nand Φ. Here H(A|ρ) = −Pap(a|ρ) log p(a|ρ) is

the Shannon entropy of an observable Awith probability distribution p(a|ρ), for a state described by density operator

ρ, and H(ρ) = −tr[ρlog ρ] denotes the von Neumann entropy of the state. The choice of logarithm base is left open

throughout, corresponding to a choice of units, e.g., to bits for base 2 and nats for base e. It follows that the number

and phase uncertainties of any quantum state, as quantiﬁed by their Shannon entropies, cannot both be arbitrarily

small.

Entropic uncertainty relations have useful counterparts in quantum metrology. For example, if a random phase shift

Θ of an optical probe state ρis estimated via some measurement Θest, then it follows from uncertainty relation (1)

that the error in the estimate, Θest −Θ, is strongly constrained by the tradeoﬀ relation [4]

H(N|ρ) + H(Θest −Θ|ρ)≥log 2π+H(ρ).(2)

This relation applies to arbitrary estimates, rather than to a particular phase observable Φ, and further implies that

the root-mean-square error (RMSE) of the estimate is bounded by [4–6]

RMSE := h(Θest −Θ)2i1/2≥p2π/e3

hNi+ 1 ,(3)

where hNi= tr[ρN] denotes the average photon number of the probe state. This is a strong form of the well-

known Heisenberg limit in quantum metrology, which states that the phase error can asymptotically scale no better

than hNi−1, where such limits cannot be obtained via quantum Fisher information methods without additional

assumptions [7] (see also Section III A). The above bounds are valid for both linear and nonlinear phase shifts, can

be further strengthened to take into account any prior information about the phase shift Θ, and in many cases far

outperform bounds based on quantum Fisher information [4] (see also Section II).

While the above results arise via properties of standard Shannon and von Neumann entropies, it is known that

various elements of quantum information theory can be generalised to the family of R´enyi entropies, Hα(A|ρ) and

Hα(ρ), and associated R´enyi relative entropies Dα(ρkσ) [1]. These quantities are labelled by a real index, α≥0,

and reduce to the standard entropies and relative entropy for α= 1. One has, for example, the R´enyi uncertainty

arXiv:2210.14613v2 [quant-ph] 17 Nov 2022

relation [8, 9]

Hα(N|ρ) + Hβ(Φ|ρ)≥log 2π, 1

α+1

β= 2,(4)

analogous to Equation (1). Several questions then immediately arise. Are such generalisations to R´enyi entropies

advantageous? Why are the uncertainties of Nand Φ characterised by two diﬀerent R´enyi entropies, Hαand Hβ,

in Equation (4)? And why is there no term depending on the degree of purity of the state, analogous to H(ρ) in

Equation (1)?

Several positive answers to the ﬁrst question above are known, in contexts such as mutually unbiased bases [8],

quantum cryptography [10], and quantum steering [11]. An aim of this paper is to demonstrate further unambiguous

advantages of R´enyi entropic uncertainty relation (4), in the context of quantum metrology. For example, it will be

shown in Section II to lead to a generalised Heisenberg limit of the form

RMSE ≥f(α)

hNi+1

(5)

for random phase shifts, where the function f(α) is maximised for the choice α≈0.7471. This choice not only

improves on the denominator in Equation (3) (corresponding to α= 1), but also improves on the numerator, by

around 4%, with the result being independent of R´enyi entropies and any interpretation thereof. Further entropic

bounds on the RMSE are obtained in Section II, as well as related simple yet strong uncertainty relations for number

and canonical phase observables, such as

∆Φ ≥max

np(n|ρ).(6)

A second aim of the paper is to further strengthen uncertainty relations and metrology bounds such as Equations (2)–

(6), achieved in Section III via ﬁnding upper and lower bounds for the classical R´enyi mutual information of quantum

communication channels [12–15], which also shed light on the second and third questions above. The upper bounds

are based on the notion of R´enyi asymmetry [16], recently applied to energy-time uncertainty relations for conditional

R´enyi entropies by Coles et al. [17]. The lower bounds relate to the convolution of the prior and error distributions.

For example, the number-phase uncertainty relation

α(ρ) + Hα(Φ|ρ)≥log 2π, α ≥1

2,(7)

is obtained in Section III, which generalises Equation (1) for Shannon entropies, and strengthens Equation (4) for

Renyi entropies to take the degree of purity of the state into account. Here AN

α(ρ) denotes the associated R´enyi

asymmetry, which may be interpreted as quantifying the intrinsically ‘quantum’ uncertainty of N, and satisﬁes a

duality property for pure states that underpins the relationship between the indexes αand βin Equation (4).

The results in Section III hold for the general case of unitary displacements generated by an operator with a discrete

spectrum (such as N). Applications to strong upper and lower bounds for several measures of coherence [18, 19], the

estimation of rotation angles, and energy-time metrology and uncertainty relations, are brieﬂy discussed in Section IV.

In the latter case, almost-periodic R´enyi entropies are introduced for the time uncertainties of non-periodic systems,

analogously to the case of standard entropies [20]. Conclusions are given in Section V, and proof technicalities are

largely deferred to appendices.

II. METROLOGY BOUNDS, HEISENBERG LIMIT AND UNCERTAINTY RELATIONS VIA R´

ENYI

ENTROPIES

In this section an analogue of metrology relation (2) is derived for R´enyi entropies, via uncertainty relation (4).

The improved Heisenberg limit (5) follows as a consequence, as well as several simple uncertainty relations for number

and phase, including Equation (6). Stronger versions of these results will be obtained in Section III.

A. Deﬁnition of R´enyi entropies and R´enyi lengths

To proceed, several deﬁnitions are necessary. First, the photon number of an optical mode is described by a

Hermitian operator Nhaving eigenstates {|ni},n= 0,1,2, . . . , with associated probability distribution p(n|ρ) =

hn|ρ|nifor a state described by density operator ρ. A phase shift θof the ﬁeld is correspondingly described by

ρθ=e−iNθρeiNθ.

Second, the canonically conjugate phase observable Φ is described by the positive-operator-valued measure (POVM)

{|φihφ|}, with φranging over the unit circle and

|φi:= 1

√2π

∞

n=0

e−inφ|ni,(8)

and associated canonical phase probability density p(φ|ρ) = hφ|ρ|φi[21, 22]. It is straightforward to check that this

density is translated under phase shifts, i.e., p(φ|ρθ) = p(φ−θ|ρ).

Third, the classical R´enyi entropies of Nand Φ are deﬁned by [1]

Hα(N|ρ) := 1

1−αlog ∞

n=0

p(n|ρ)α, Hα(Φ|ρ) := 1

1−αlog Idφ p(φ|ρ)α,(9)

for α∈[0,∞). These reduce to the standard Shannon entropies in the limit α→1 (using, e.g., limα→1[g(α)−

g(1)]/[α−1] = g0(1) for g(α) = log Pnp(n|ρ)α). They provide measures of uncertainty that are small for highly

peaked distributions and large for spread-out distributions. In particular, Hα(N) = 0 and Hα(Φ|ρ) = log 2πfor any

number state ρ=|nihn|. Direct measures of uncertainty are given by the associated R´enyi lengths

Lα(N|ρ) := "∞

n=0

p(n|ρ)α#1

1−α

, Lα(Φ|ρ) := Idφ p(φ|ρ)α1

1−α

,(10)

which quantify the eﬀective spreads of Nand Φ over the nonnegative integers and the unit circle, respectively [23].

Note that uncertainty relation (4) can be rewritten in the form

Lα(N|ρ)Lβ(Φ|ρ)≥2π, 1

α+1

β= 2 (11)

for these spreads, akin to the usual Heisenberg uncertainty relation.

B. Entropic tradeoﬀ relation for phase estimation

If some estimate θest is made of a phase shift θapplied to a probe state, then the estimation error, θerr =θest −θ,

will have a highly-peaked probability density for a good estimate, and a spread-out probability density for a poor

estimate. Hence, the quality of the estimate can be quantiﬁed in terms of the R´enyi entropy of p(θerr). The following

theorem imposes a tradeoﬀ between the quality of any estimate of a completely unknown phase shift and the number

entropy of the probe state.

Theorem 1. For any estimate Θest of a uniformly random phase shift Θapplied to a probe state ρ, the estimation

error Θest −Θsatisﬁes the tradeoﬀ relation

Hα(Θest −Θ|ρ) + Hβ(N|ρ)≥log 2π, 1

α+1

β= 2.(12)

Note that the condition on αand βimplies α, β ≥1

2. For the case of Shannon entropies, i.e., α=β= 1, this

result has been previously obtained via entropic uncertainty relation (1) [5]. A similar method is used in Appendix A

to prove the general result of the theorem via entropic uncertainty relation (4). It is worth emphasising that, unlike

uncertainty relation (4), Theorem 1 applies to any estimate of the random phase shift, including the canonical phase

measurement Φ as a special case (for this case Equation (4) is recovered).

Theorem 1 implies that no phase-shift information can be gained via a probe state |nihn|, as expected since number

eigenstates are insensitive to phase shifts. In particular, the number entropy Hβ(N|ρ) vanishes for any index βand

so the error entropy in Equation (12) must reach its maximum value of log 2π, which is only possible if the error has

a uniform probability density, i.e., p(θerr)=1/(2π).

Conversely, Theorem 1 connects informative estimates with probe states that have a high number entropy. For

example, if a canonical phase measurement is used to estimate a random phase shift of the pure probe state |ψi=

(nmax + 1)−1/2(|0i+|1i+. . . |nmaxi), the error distribution may be calculated, using Equation (A2) of Appendix A

for Θest ≡Φ, as

p(θerr) = hθerr|ρ|θerri=|hθerr|ψi|2=1

2π(nmax + 1) 

nmax

n=0

einθerr 

,(13)

and the Shannon entropy of the error then follows via Equation (69) of Reference [24] as

H(Θest −Θ|ρ) = log 2π+ log(nmax + 1) + 2[1 −1−1−2−1− ··· − (nmax + 1)−1] log e

≤log 2π−log(nmax + 1) + 2(1 −γ) log e, (14)

where γ≈0.5772 is Euler’s constant. Hence such a probe state leads to an arbitrarily low uncertainty for the

error as nmax increases. Moreover, Theorem 1 implies that this estimate is near-optimal, under the constraint of at

most nmax photons, in the sense that the error entropy is within 2(1 −γ) log e≈1.2 bits of the minimum possible,

log 2π−log(nmax + 1), allowed by Equation (12) under this constraint.

This last result strongly contrasts with Fisher information methods, which suggest that the best possible single-mode

probe state, under the constraint of at most nmax photons, is the simple superposition state 2−1/2(|0i+|nmaxi) [25].

However, it follows from Theorem 1 that this probe state cannot be optimal for the estimation of a random phase

shift. In particular, noting that Hβ(N|ρ) = log 2 for this case, Equations (10) and (12) give

log Lα(Θest −Θ|ρ) = Hα(Θest −Θ|ρ)≥log π(15)

for any value of nmax, in stark contrast to Equation (14). Indeed, choosing α= 2 gives

Idθerr p(θerr)−1

2π2

=L2(Θest −Θ|ρ)−1−1

π+1

4π2≤1

4π2,(16)

implying that p(θerr) cannot be too diﬀerent from a uniform distribution. Hence the simple superposition state has

a poor performance in comparison to the probe state in Equation (14), for the case of uniformly random estimates.

A more direct comparison with Fisher information bounds is made in in the following subsection, and the diﬀerence

explained in Section III A.

C. Lower bounds for RMSE, a strong Heisenberg limit, and number-phase uncertainty relations

Equation (10) and Theorem 1 imply that the R´enyi length of the error for any estimate of a random phase shift

has the lower bound Lα(Θest −Θ|ρ)≥2π/Lβ(N|ρ). However, a more familiar length measure for characterising

the performance of an estimation scheme is the root-mean-square error (RMSE) of the estimate, given by RMSE =

h(Θest −Θ)2i1/2. Note that, in contrast to the case of entropies and R´enyi lengths, a well-known ambiguity arises:

θ2

err = (θest −θ)2is not a periodic function, and hence evaluation of the RMSE depends on the choice of a phase

reference interval for the error θerr. Fortunately this is easily resolved: a perfect estimate corresponds to a zero error,

and hence the reference interval centred on zero, i.e., θerr ∈[−π, π), will be used.

The following theorem gives three strong lower bounds for the RMSE, where the third has the form of a generalised

Heisenberg limit as discussed in Section I. A corollary to this theorem, further below, gives corresponding preparation

uncertainty relations for number and phase.

Theorem 2. For any estimate Θest of a uniformly random phase shift Θapplied to a probe state ρ, the root-mean-

square error RMSE=h(Θest −Θ)2i1/2has the lower bounds

RMSE ≥π

√3L1/2(N|ρ),RMSE ≥max

np(n|ρ),RMSE ≥fmax

hNi+1

,(17)

where L1/2(N|ρ) = Pnpp(n|ρ)2is a R´enyi length as deﬁned in Equation (10), and fmax ≈0.5823 denotes the

maximum value of the function

f(α) := 









2α−1π

3α−11

23α−1

21

1−α(1−α)1

2Γ( 1

1−α)

Γ( 1

1−α−1

2),1

2≤α≤1,

2α−1π

3α−11

23α−1

21

1−α(α−1) 1

2Γ( α

α−1+1

Γ( α

α−1), α ≥1,

(18)

which is achieved for the choice α≈0.7471.

In Equation (18), Γ(x) denotes the Gamma function and the value of f(1) is deﬁned by taking the limit α→1

in either expression and using limx→0(1 −3x/2)1/x =e−3/2and limx→∞ x1/2Γ(x)/Γ(x+1

2) = 1, to obtain f(1) =

p2π/e3≈0.5593. The scaling function f(α) is plotted in Figure 1. The proof of the theorem relies on Theorem 1

and upper bounds on R´enyi entropies under various constraints, and is given in Appendix B.

0.5

1.0

1.5

2.0

2.5

0.35

0.40

0.45

0.50

0.55

0.60

fHΑL

FIG. 1. The scaling function f(α) for the Heisenberg limit in Theorem 2. Particular values of interest are f(1/2) = 1/2,

f(1) = p2π/e3≈0.5593, and the maximum value fmax ≈f(0.7471) ≈0.5823.

The lower bounds in Theorem 2 are relatively strong, and indeed the ﬁrst inequality in Equation (17) is tight,

being saturated for number states. In particular, the error distribution is always uniform for this case, as noted below

Theorem 1, yielding RMSE = 1

2πRπ

−πdφerr (φerr)2=π/√3, as per the ﬁrst lower bound.

Moreover, the third bound in Equation (17) of Theorem 2 is stronger than the Heisenberg limit in Equation (3),

both in the numerator and the denominator. In particular, the scaling factor f(1) ≈0.5593 in Equation (3) is

outperformed by ≈4% when compared to fmax ≈0.5823 in Equation (17). Note that while the derivation of this

bound relies on properties of R´enyi entropies (see Appendix B), no R´enyi entropies appear in the bound itself. The

bound thus demonstrates an unambiguous advantage of using R´enyi entropies in quantum metrology that is completely

independent of their interpretation.

Theorem 2 can also be directly compared to the Fisher information bound

RMSEθ:= h(Θest −θ)2i1/2

ρθ=Z∞

−∞

dθest (θest −θ)2p(θest|ρθ)1/2

≥1

2∆N,(19)

for the root mean square error of any locally unbiased estimate Θest of a given phase-shift θof probe state ρ[25]. Here

phase shifts are ‘unwrapped’ from the unit circle to the real line, ∆Nis the root mean square deviation of the number

operator for the probe state, and local unbiasedness is the requirement that hΘestiρχ=R∞

−∞ dθest θestp(θest|ρχ) = χ

for all phase shifts χin some neighbourhood of θ. Note the bound implies that RMSEθbecomes inﬁnite for number

states. Under the constraint of a maximum photon number nmax, the probe state minimising the Fisher bound is

the simple superposition 2−1/2(|0i+|nmaxi), considered in Section II, yielding RMSEθ≥1/nmax, which approaches

zero as nmax increases [25]. In contrast, for any estimate of a uniformly random phase shift, the ﬁrst two bounds in

Equation (17) of Theorem 2 give the much stronger lower bounds RMSE > π/(2√3) ≈0.9069 and RMSE ≥1

2for this

probe state, irrespective of the value of nmax. Thus the optimal single-mode probe state for the Fisher information

bound is not optimal for estimating uniformly random phase shifts. The underlying reason for this diﬀerence is related

to the degree of prior information available about the phase shift [4, 26], as discussed further in Section III A.

Finally, it is of interest to note that each of the bounds in Theorem 2 can be used to obtain a corresponding

preparation uncertainty relation for the canonical phase and photon number observables. In particular, deﬁning the

standard deviation ∆χΦ of the canonical phase observable Φ with respect to reference angle χvia [24]

(∆χΦ)2:= Zχ+π

χ−π

dφ (φ−χ)2p(φ|ρ),(20)

one has the following corollary of Theorem 2, also proved in Appendix B.

Corollary 1. The canonical phase and photon number of an optical mode satisfy the family of uncertainty relations

Lβ(N|ρ)∆χΦ≥αα

α−1f(α),1

α+1

β= 2,(21)

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BetterHeisenberglimits,coherencebounds,andenergy-timetradeosviaquantumRenyiinformationMichaelJ.W.HallTheoreticalPhysics,ResearchSchoolofPhysics,AustralianNationalUniversity,CanberraACT0200,AustraliaAnuncertaintyrelationfortheRenyientropiesofconjugatequantumobservablesisusedtoobtainastrongHeisenbe...

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Better Heisenberg limits coherence bounds and energy-time tradeos via quantum R enyi information Michael J. W. Hall

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