Lipschitz continuity of quantum-classical conditional entropies with respect to angular distance and related properties of angular distance Michael Liaofan Liu1 2Florian Kanitschar1 3Amir Arqand1and Ernest Y.-Z. Tan1

2025-05-02 0 0 710.31KB 21 页 10玖币

侵权投诉

Lipschitz continuity of quantum-classical conditional entropies with respect to angular

distance, and related properties of angular distance

Michael Liaofan Liu,1, 2, ∗Florian Kanitschar,1, 3 Amir Arqand,1and Ernest Y.-Z. Tan1

1Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

2Department of Mathematics, Amherst College, Amherst, MA 01002, USA

3Technische Universit¨at Wien, Faculty of Mathematics and Geoinformation, Wiedner Hauptstraße 8, 1040 Vienna, Austria

(Dated: March 24, 2023)

We derive a Lipschitz continuity bound for quantum-classical conditional entropies with respect to

angular distance, with a Lipschitz constant that is independent of the dimension of the conditioning

system. This bound is sharper in some situations than previous continuity bounds, which were either

based on trace distance (where Lipschitz continuity is not possible), or based on angular distance but

did not include a conditioning system. However, we ﬁnd that the bound does not directly generalize

to fully quantum conditional entropies. To investigate possible counterexamples in that setting, we

study the characterization of states which saturate the Fuchs–van de Graaf inequality and thus have

angular distance approximately equal to trace distance. We give an exact characterization of such

states in the invertible case. For the noninvertible case, we show that the situation appears to be

signiﬁcantly more elaborate, and seems to be strongly connected to the question of characterizing

the set of ﬁdelity-preserving measurements.

I. INTRODUCTION

Given two quantum states ρand σon a Hilbert space

H, one of the most natural questions to ask is how similar

ρand σare. Common measures to answer this question

include the trace distance,

T (ρ, σ):=1

2kρ−σk1,(1)

and the (root-)ﬁdelity,

F (ρ, σ):=

√ρ√σ

1.(2)

The trace distance is a metric on the set of density op-

erators D(H), and it has a meaningful interpretation as

the distinguishability of two quantum states. In a quan-

tum hypothesis testing scenario, where Bob randomly

prepares one of two states ρand σ(with equal probabil-

ity) for Alice to distinguish, Alice can correctly identify

the incoming state with probability 1+T(ρ,σ)

2. In contrast,

the ﬁdelity is not a metric, but it can be interpreted as

the probability that a state ρ“passes a test” for being

the same as a pure state σ[1].

Another important task in quantum information the-

ory is to quantify the amount of information present in

a quantum system. The von Neumann entropy

H (ρ):=−tr (ρln ρ)

is one quantity which fulﬁlls this role [2], as it appears

in many fundamental information theoretic tasks such

as Schumacher data compression [3] and randomness ex-

traction [4]. This concept can be extended to condi-

tional entropies H (A|B)ρfor bipartite states ρ:=ρAB ∈

∗mliu24@amherst.edu

D(HA⊗ HB), with one of several equivalent deﬁnitions

being the diﬀerence between the joint entropy and the

marginal entropy,

H (A|B)ρ:= H (ρAB )−H (ρB),(3)

where ρB:= trA(ρ) is the reduced state of ρon HB.

Further details about quantum distance measures and

quantum entropies can be found in e.g. [1,5].

A useful property of the von Neumann entropy is that

it is continuous for ﬁnite-dimensional quantum systems.

This motivates the search for so-called entropic continu-

ity bounds, which capture the notion that two states ρ, σ

close in some metric d, e.g. d(ρ, σ) = δ&0, are expected

to be close in entropy as well, i.e.

|H (ρ)−H (σ)| ≤ f(δ),

where fis some function such that limδ→0f(δ) = 0.

For example, in [6], Audenaert derived the tightest

form of the Fannes-type continuity bound for the von

Neumann entropy in terms of trace distance. Speciﬁcally,

letting d:= dim H ∈ N,ρ, σ ∈ D(H), and T:= T (ρ, σ),

Audenaert showed that

|H (ρ)−H (σ)| ≤ Tln (d−1) + h (T),(4)

where h(x):=−xln x−(1 −x) ln(1 −x) is the binary

entropy function.

Similar continuity bounds also exist for conditional en-

tropies. As shown by Winter [7], letting dA:= dim HA∈

N,dB:= dim HB∈N,ρ, σ ∈ DHA⊗ HB, and

T:= T (ρ, σ), the following holds:

H (A|B)ρ−H (A|B)σ

≤2Tln dA+ (1 + T) h T

1 + T.

(5)

Such continuity bounds have been applied in vari-

ous contexts. For example, in [8], Upadhyaya et al.

arXiv:2210.04874v2 [quant-ph] 22 Mar 2023

constructed a ﬁnite-dimensional cutoﬀ formulation for a

class of inﬁnite-dimensional entropy optimization prob-

lems. Qualitatively, that work argues that if an inﬁnite-

dimensional state is “close” (under some metric) to

a ﬁnite-dimensional state, then an entropic continuity

bound allows us to replace the former with the latter

and compensate for the resulting change in entropy by

applying a correction based on the continuity bound.

This so-called dimension-reduction method plays an im-

portant role in quantum key distribution (QKD) secu-

rity proofs [9]. However, it relies heavily on the continu-

ity bound in Eq. (5) to compute the required correction

term. An improved continuity bound would lead to a

smaller correction term in this method and hence a larger

secret key rate.

Another application of entropic continuity bounds

arises in unstructured entropy optimization problems, as

studied in e.g. [10]. In that work, the approach is that in

order to minimize the entropy over some set of states, one

simply computes the entropy on a suﬃciently ﬁne discrete

“grid” of states in the set, then uses the continuity bound

to ensure that the true minimum does not lie more than

f(δ) away from the minimum over the grid. Again, an

improved continuity bound would result in tighter results

from such an approach.

In the above contexts, two desirable properties of the

continuity bound f(δ) (for conditional entropies) are as

follows.

Condition 1. f(δ)should be independent of dB, the di-

mension of the conditioning system HB.

Condition 2. f(δ)should have ﬁnite (and ideally small)

derivative at δ= 0.

The ﬁrst property is useful (or in some cases required) for

the applications mentioned above, since in those contexts

the conditioning system may have large or unbounded di-

mension. The second property is desirable for obtaining

better scaling at small δ, since then we would not require

extremely small values of δin order to force the entropy

diﬀerence to be small.

While the Winter bound (Eq. (5)) satisﬁes condition 1,

it does not satisfy condition 2due to the binary entropy

term h, which has unbounded derivative as δ→0. In

fact, such scaling of the conditional entropy with respect

to trace distance is in some sense unavoidable, since there

is an explicit family of states that saturates the Aude-

naert bound (Eq. (4)), which has the binary entropy term

as well. To work around this issue and obtain a bound

that satisﬁes both conditions 1and 2, one approach is

to consider an alternative distance measure such as the

angular distance, deﬁned as

A (ρ, σ):= arccos F (ρ, σ).

We remark that this is not simply an arbitrary change

of distance measure: in the context of the applications

mentioned above, the quantity that arises “naturally” in

the analysis is the ﬁdelity rather than the trace distance,

hence working with the bound in Eq. (5) is somewhat

suboptimal.

This approach is promising in light of the following re-

sult. In [10], Sekatski et al. proved Lipschitz continuity

of the von Neumann entropy with respect to angular dis-

tance. That is, for d:= dim H ∈ N,ρ, σ ∈ D(H), and

x0:= exp W0−2

e≈4.922, where W0is the principal

branch of the Lambert-W function, it was shown that

|H (ρ)−H (σ)| ≤ u(d) A (ρ, σ),(6)

where the Lipschitz constant u(d) is

u(d):=(q8ln x0

x0√d−1 1 ≤d≤4

2 ln d d ≥5.(7)

Now, a naive application of Eq. (6) to conditional en-

tropies, using Eq. (3) and the triangle inequality, would

yield

H (A|B)ρ−H (A|B)σ

≤u(dAdB) A (ρ, σ) + u(dB) A (ρB, σB)

≤(u(dAdB) + u(dB)) A (ρ, σ),

(8)

where in the last line we used the monotonicity of the

angular distance under quantum channels. While this

bound satisﬁes condition 2, it violates condition 1. How-

ever, the estimates to obtain Eq. (8) from Eq. (6) are

crude and leave room for reﬁnement. Thus, we ask

whether it is possible to obtain Lipschitz continuity of

the conditional entropy with respect to angular distance,

while avoiding dependence on dBin the ﬁnal bound.

In this work, we answer this question in the aﬃr-

mative when ρand σare quantum-classical states on

HA⊗ HB(i.e. when there exists an orthonormal ba-

sis {|gki}kfor HBsuch that both ρand σare of the

form Pkγkτk⊗|gkihgk|for some density operators τk∈

D(HA) and probabilities γk∈[0,1]). We present this re-

sult in Sec. II. However, we ﬁnd that our bound does not

hold in general for fully quantum states. To further in-

vestigate counterexamples in this setting, we study char-

acterizations of states saturating the Fuchs–van de Graaf

inequalities. In particular, the states saturating the up-

per bound in the inequality have T (ρ, σ)≈A (ρ, σ) when

A (ρ, σ) is small, so these states could pose an obstruc-

tion to deriving continuity bounds in terms of A (ρ, σ)

that scale better than those in terms of T (ρ, σ). While

it is well-known that any pair of pure states saturate

the upper Fuchs–van de Graaf inequality, we show that

these are not the only such states. In Sec. III, we pro-

vide a characterization of all such pairs (ρ, σ) in the case

where both of them are invertible. This result may be of

independent interest in other applications such as com-

puting QKD keyrates (we discuss this further in the ap-

pendices). However, we ﬁnd that such a characterization

in the general case where (ρ, σ) are noninvertible appears

signiﬁcantly more challenging, and we discuss how it re-

lates to identifying the set of measurements that preserve

the ﬁdelity between states. Finally, we provide some con-

cluding remarks in Sec. IV.

II. CONTINUITY BOUND

We now state and prove the main result of our

manuscript, a continuity bound for the conditional en-

tropy of quantum-classical states with respect to angular

distance. Subsequently, we discuss the tightness of this

bound, and we highlight some challenges for generalizing

our result to classical-quantum or fully quantum states.

A. Main theorem and proof

Theorem 1. Let HAand HBbe Hilbert spaces of ﬁ-

nite dimension dAand dB, respectively. Let ρ, σ ∈

DHA⊗ HB. Let u(·)and x0be deﬁned as in Eq. (7)

and the preceding text. Suppose in addition that ρand σ

are both quantum-classical states with respect to HAand

HB. Then

H (A|B)ρ−H (A|B)σ≤u(dA) A (ρ, σ).(9)

Proof. Since ρand σare quantum-classical states, we can

write

ρ=

k=1

αkρk⊗ |fkihfk|

σ=

k=1

βkσk⊗ |fkihfk|

for some density operators ρk, σk∈ D(HA), probabilities

αk, βk∈[0,1] which satisfy PdB

k=1 αk= 1 = PdB

k=1 βk,

and orthonormal basis {|fki}kfor HB. For each k, con-

sider a spectral decomposition of ρkand σk,

ρk=

j=1

pjk |ejkihejk|

σk=

j=1

qjk |˜ejkih˜ejk|,

where the eigenvalues pjk, qjk ≥0 satisfy PdA

j=1 pjk =

1 = PdA

j=1 qjk, and the eigenvectors form orthonormal

bases {|ejki}j,{|˜ejki}jfor HA. Deﬁning ρjk :=pjkαk

and σjk :=qjkβkfor all jand k,ρand σcan be written

ρ=

k=1

j=1

ρjk |ejkihejk|⊗|fkihfk|

σ=

k=1

j=1

σjk |˜ejkih˜ejk|⊗|fkihfk|,

and their partial traces can be written as

ρB:= trA(ρ) =

k=1 



j=1

ρjk

|fkihfk|

σB:= trA(σ) =

k=1 



j=1

σjk

|fkihfk|.

Now, observe that the eigenvalues ρjk and σjk of ρand

σcompletely determine the eigenvalues of their partial

traces ρBand σB, respectively. This allows us to “map”

the problem to Rd, where d:=dAdB, as follows. For

each k∈ {1, . . . , dB}, let us choose the ordering of the

eigenvalues pjk (and corresponding eigenvectors |ejki) to

be such that p1k≥p2k≥... ≥pdAk; similarly, choose

the ordering of the eigenvalues qjk to be such that q1k≥

q2k≥... ≥qdAk. Now, consider the vectors

r:=√ρjk k,j

s:=√σjk k,j

(10)

in Rd, where the entries of rand sare ordered with

kas the outer index and jas the inner index. We

observe that the angular distance A (ρ, σ) between ρ

and σis always lower bounded by the angular distance

θ0:= arccos (r·s)∈[0,π

2] between rand s. To see

this, we decompose the ﬁdelity as a sum over kusing

the quantum-classical structure, then apply a variational

characterization of the trace norm [1] and the von Neu-

mann trace inequality [11], which yields



√ρ√σ

1=

k=1 pαkβkk√ρk√σkk1

k=1 pαkβk|tr (√ρk√σkUk)|

≤

k=1 pαkβk

j=1

√pjk√qjk

k=1

j=1

√ρjk√σjk

=r·s,

where the Ukare some unitaries on HA. Thus, we see

that

θ0= arccos (r·s)≤arccos 

√ρ√σ

1= A (ρ, σ),(11)

as needed.

Next, since the eigenvalues of ρand σcompletely deter-

mine the eigenvalues of their partial traces, it is possible

to compute the conditional entropy of ρand σgiven only

the vectors rand s. To see this, consider the following

function

Hc(v):=−

k=1

j=1

jk ln v2

k=1 



j=1

jk

ln dA

l=1

lk!,

where v= (vjk)k,j can be any vector in Rd. Then

Hc(r) = −

k=1

j=1

ρjk ln ρjk +

k=1 



j=1

ρjk

ln dA

l=1

ρlk!

= H (A|B)ρ

Hc(s) = −

k=1

j=1

σjk ln σjk +

k=1 



j=1

σjk

ln dA

l=1

σlk!

= H (A|B)σ,

(12)

so the vectors rand sare suﬃcient to determine the

conditional entropies H (A|B)ρand H (A|B)σ.

The idea of our proof is now to integrate from rto sin

Rd, tracking the inﬁnitesimal changes in the conditional

entropy and angular distance. To see this formally, ﬁrst

note that rand sare unit vectors (with respect to the

standard inner product on Rd), since r·r= tr (ρ) =

1 = tr (σ) = s·s. Moreover, we have r, s ≥0 by deﬁ-

nition (10). Now, note that if r·s= 1, then r=s, so

we have Hc(r)=Hc(s) i.e. H (A|B)ρ= H (A|B)σ. Since

u(dA)≥0 and A (ρ, σ)≥0, Eq. (9) holds trivially in this

case. Now consider the remaining case r·s∈[0,1). Let

˜sbe the normalized projection of sonto the orthogonal

complement of Span {r},

˜s:=s−(s·r)r

|s−(s·r)r|.

Using ˜s, we deﬁne the path

v(θ):= cos(θ)r+ sin(θ)˜s

from rto s, where θ∈[0, θ0]. Note that v(0) = r,v(θ0) =

s, and v(θ) traverses the great circle along the (d−1)-

sphere from rto s. In addition, note that |v(θ)|= 1 for

all θ∈[0, θ0]. Now, the tangent to the path v(θ) is

w(θ):=v0(θ) = −sin(θ)r+ cos(θ)˜s,

which satisﬁes |w(θ)|= 1 and v(θ)·w(θ) = 0 for all

θ∈[0, θ0].

For notational simplicity, we now deﬁne Hc(θ):=

Hc(v(θ)), so

Hc(θ) = −

k=1

j=1

v(θ)2

jk ln(v(θ)2

jk)

k=1 



j=1

v(θ)2

jk

ln dA

l=1

v(θ)2

lk!.

Observe that Hc(θ) is continuous on [0, θ0] (under the

standard convention for entropy deﬁnitions that 0 ln 0 ≡

0). Thus, if we show that Hc(θ) is diﬀerentiable on (0, θ0)

and its derivative satisﬁes H0

c(θ)≤u(dA) on that inter-

val, then the desired result follows immediately, since

H (A|B)σ−H (A|B)ρ=|Hc(θ0)−Hc(0)|

=Zθ0

c(θ)dθ

≤Zθ0

0H0

c(θ)dθ

≤u(dA)θ0

≤u(dA) A (ρ, σ),

where the ﬁrst line follows from Eq. (12), and the last

line follows from Eq. (11). Thus, all that remains is to

bound H0

c(θ)by u(dA).

To do this, we ﬁrst handle a technicality regarding

zero eigenvalues. For each k∈ {1, . . . , dB}, let Sk

Abe

the set of j∈ {1, . . . , dA}such that at least one of

rjk, sjk is nonzero. Furthermore, let SBbe the set of

k∈ {1, . . . , dB}such that Sk

Ais nonempty. Then for

any (k, j) with k∈SBand j∈Sk

A, at least one of

rjk, sjk is nonzero, which implies that v(θ)2

jk >0 for

all θ∈(0, θ0). Moreover, for all other (k, j), we have

that rjk =sjk = 0, so v(θ)2

jk = 0 for all θ∈(0, θ0),

which implies that the value of Hc(θ) would not change

upon removing the term v(θ)2

jk. Thus, in the remainder

of the argument, summations of the form Pk,j should be

understood to mean Pk∈SBPj∈Sk

A(and analogously, Pl

means Pl∈Sk

A), which ensures that all terms appearing in

the summations satisfy v(θ)2

jk >0 and Pl∈Sk

Av(θ)2

lk >0

for all θ∈(0, θ0). With this, we see that Hc(θ) is indeed

diﬀerentiable on (0, θ0), and

H0

c(θ)=−X

k,j

2v(θ)jkw(θ)jk ln(v(θ)2

jk)

−X

k,j

v(θ)2

2v(θ)jkw(θ)jk

v(θ)2

k,j

2v(θ)jkw(θ)jk ln X

v(θ)2

lk!

k,j

v(θ)2

jk Pl2v(θ)lkw(θ)lk

Plv(θ)2

lk 

Using v(θ)·w(θ) = 0, this simpliﬁes to

H0

c(θ)= 2 X

k,j

v(θ)jkw(θ)jk ln Plv(θ)2

v(θ)2

jk !

≤2v

k,j

v(θ)2

jk ln2 Plv(θ)2

v(θ)2

jk !,

(13)

where in the last line we used the Cauchy–Schwarz in-

equality with |w(θ)|= 1.

Now, recall that the v(θ)2

jk form a valid probability

distribution (i.e. they are non-negative values summing

to 1), since |v|= 1. Also, note that the argument of ln2

in the ﬁnal line above, i.e. Plv(θ)2

v(θ)2

, lies in the interval

[1,∞). Thus, we now construct an increasing concave

upper bound f(x) for ln2xon x∈[1,∞), as this would

allow us to “move the summation” over k, j (weighted by

the probabilities v(θ)2

jk) into the argument of the func-

tion. To begin, note that

dx ln2x= 2ln x

dx2ln2x= 21−ln x

x2,

so ln2xis convex for all x∈[1, e] and concave for all

x∈[e, ∞). Then to produce f(x), we seek a line y(x) =

m(x−a) such that y(1) = ln2(1) = 0, and such that

there exists x0∈[1,∞) with x0≥e,y(x0) = ln2(x0) and

y0(x0) = d

dx ln2xx0= 2ln x0

x0. Then we must solve the

system

ln2x0=y(x0)=2ln x0

(x0−1),

for x0, which has solutions x0= 1 and x0=

exp W0−2

e ≈4.922. We discard the solution x0=

1< e and keep the other solution x0≈4.922 ≥e. Thus,

our increasing concave upper bound for ln2xis

f(x):=(2ln x0

x0(x−1) 1 ≤x≤x0

ln2x x ≥x0

With f(x), we can write

H0

c(θ)≤2v

k,j

v(θ)2

jk ln2 Plv(θ)2

v(θ)2

jk !

≤2v

k,j

v(θ)2

jkf Plv(θ)2

v(θ)2

jk !

≤2v

tf

X

k,j

v(θ)2

jk Plv(θ)2

v(θ)2

jk 



≤2v

tf

dAX

k,l

v(θ)2

lk



= 2pf(dA)

=(q8ln x0

x0√dA−1 1 ≤dA≤4

2 ln dAdA≥5

=u(dA).

(14)

In the above, the ﬁrst line is Eq. (13). The second line

follows since fis an upper bound on ln2xfor x≥1. The

third line follows since fis concave and |v(θ)|= 1. The

fourth line follows since fis increasing and Sk

A≤dAfor

all k. The ﬁfth line follows since |v(θ)|= 1. The sixth

and seventh lines follow from the deﬁnitions and the fact

that dA∈N.

In comparison to the proof in [10] for unconditioned

entropies, the main diﬀerence in our proof here is es-

sentially that there are additional contributions to the

derivative H0

c(θ) arising from the H (ρB) term in the con-

ditional entropy. Informally, these contributions act in

the “opposite direction” from those of the H (ρAB ) term,

reducing the magnitude of the derivative and yielding

a ﬁnal bound that is independent of dB, in contrast to

what we would have obtained had we only considered

the derivative of the H (ρAB ) term alone — see Eq. (8).

Another small diﬀerence is that we have constructed the

concave upper bound in a slightly diﬀerent and arguably

simpler way.

B. Potential improvements

How tight is the bound in Eq. (9)? To address this

question empirically, we began by randomly sampling

100,000 pairs of quantum-classical states according to

the procedure in Appendix D 1, and for each pair (ρ, σ),

we computed their angular distance A (ρ, σ) and con-

ditional entropy diﬀerence H (A|B)ρ−H (A|B)σ. We

then plotted the conditional entropy diﬀerences against

the angular distances in Fig. 1. This provides an empir-

ical estimate for the tightness of Eq. (9) at all feasible

angular distances A (ρ, σ)∈0,π

2.

Next, we explore the tightness of Eq. (9) at small

angular distances (since in most applications we are

mainly interested in this case). For each angular distance

A (ρ, σ)∈0.1×10−5,0.2×10−5,...,1.0×10−5, we

randomly sampled 10,000 pairs of classical states (ρ, σ)

with that angular distance, as described in Appendix D 2.

Then, we computed the conditional entropy diﬀerences

H (A|B)ρ−H (A|B)σand plotted them in Fig. 2. The

results suggest that at small angular distances, our conti-

nuity bound is close to the “true” tight expression when

dAis small, but there may be room for improvement

when dAis larger (note that random sampling typically

yields less representative results in high dimensions, so

the latter claim should not be taken as conclusive).

Note that in order to saturate Eq. (9), a pair of states

(ρ, σ) must saturate both inequalities (13) (Cauchy–

Schwarz) and (14) (which roughly speaking is due to the

concavity of f). However, it seems that these inequalities

cannot be simultaneously saturated, which is consistent

with the above empirical evidence that there is room for

sharpening the bound.

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Lipschitzcontinuityofquantum-classicalconditionalentropieswithrespecttoangulardistance,andrelatedpropertiesofangulardistanceMichaelLiaofanLiu,1,2,FlorianKanitschar,1,3AmirArqand,1andErnestY.-Z.Tan11InstituteforQuantumComputing,UniversityofWaterloo,Waterloo,Ontario,CanadaN2L3G12DepartmentofMathemati...

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Lipschitz continuity of quantum-classical conditional entropies with respect to angular distance and related properties of angular distance Michael Liaofan Liu1 2Florian Kanitschar1 3Amir Arqand1and Ernest Y.-Z. Tan1.pdf

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Lipschitz continuity of quantum-classical conditional entropies with respect to angular distance and related properties of angular distance Michael Liaofan Liu1 2Florian Kanitschar1 3Amir Arqand1and Ernest Y.-Z. Tan1

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