The Complementary Information Principle of Quantum Mechanics

Yunlong Xiao,1, ∗ Kun Fang,2, † and Gilad Gour1, ‡

1Department of Mathematics and Statistics and Institute for Quantum Science and Technology,University of Calgary, Calgary, Alberta T2N 1N4, Canada

2Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, UK

The uncertainty principle bounds the uncertainties about incompatible measurements, clearly setting quantumtheory apart from the classical world. Its mathematical formulation via uncertainty relations, plays an irreplace-able role in quantum technologies. However, neither the uncertainty principle nor uncertainty relations can fullydescribe the complementarity between quantum measurements. As an attempt to advance the efforts of com-plementarity in quantum theories, we formally propose a complementary information principle, significantlyextending the one introduced by Heisenberg. First, we build a framework of black box testing consisting ofpre- and post-testing with two incompatible measurements, introducing a rigorous mathematical expression ofcomplementarity with definite information causality. Second, we provide majorization lower and upper boundsfor the complementary information by utilizing the tool of semidefinite programming. In particular, we provethat our bounds are optimal under majorization due to the completeness of the majorization lattice. Finally, asapplications to our framework, we present a general method to outer-approximating all uncertainty regions andalso establish fundamental limits for all qualified joint uncertainties.

Introduction.— Quantum mechanics has revolutionizedour understanding of the physical world, leaving us a keymessage that our world is inherently unpredictable. The well-known Heisenberg’s uncertainty principle [1] states that it isimpossible to prepare a state such that its outcome probabilitydistributions from two incompatible measurements are bothsharp. This heuristic idea was first formulated by Kennard [2](see also the work of Weyl [3]) into a mathematically rig-orous inequality for position and momentum measurements,where the uncertainties were quantified through the standarddeviation [4–10]. Later on, an alternative quantitative for-mulation based on entropic measures has been introduced byBiałynicki-Birula and Mycielski [11]. Recently, physicistshave also employed majorization to study uncertainty rela-tions [12–17]. All of these are known as preparation uncer-tainty with detailed reviews given in [18–20].

The uncertainty principle manifests another deeper conceptin quantum theory, namely complementarity [21]—a givenphysical attribute can only be revealed at the price of anothercomplementary attribute being suppressed. As a more gen-eral concept, complementarity can also be exhibited through“duality paradoxes”, such as wave-particle duality [22]. Thearise of complementarity differentiates quantum theory fromits classical counterpart, leading to a plethora of applicationssuch as entanglement detection [23–26], Einstein-Podolsky-Rosen (EPR) steering detection [27–32] as well as quantumkey distribution [33–36].

In recent years, uncertainty relations have been used as astandard tool to explore the complementarity nature of quan-tum measurements. A series of efforts have been devoted toseeking the optimal bounds on uncertainty relations for givenspecific uncertainty measures, such as Shannon or Renyi en-tropies [37–61]. However, most of this enormous body ofwork focuses on a single entropy measure, whereas the com-plete information about the physical attributes of incompatiblemeasurements is fully contained in the outcome probabilityvectors themselves [62]. Hence, inevitable losses of informa-

tion occurs whenever a high dimensional probability vector isprojected into its one-dimensional entropic value. To have afull-scale understanding of complementarity between incom-patible measurements and to be able to find more practical ap-plications, it is therefore essential to consider a more generalframework, in which the full probability vectors are consid-ered.

In this paper, we introduce a new principle which we callthe complementary information principle (CIP), characteriz-ing the information trade-off between incompatible measure-ments, as characterized with respect to their full outcomeprobability vectors. This principle is different than the un-certainty principle as it does not rely on an uncertainty quan-tifier such as entropy or standard deviation. Since the CIP in-volves pairs of probability vectors (corresponding to the twoincompatible measurements), it introduces a new type of par-tial order between pairs of probability vectors that we callmarginal majorization. Based on this order and with the helpof semidefinite programming (SDP), we investigate qualita-tive and quantitative aspects of our CIP, providing insightsinto the fundamental limits of quantum measurements. As anapplication to our CIP, we provide tight outer approximationof uncertainty regions with any given uncertainty measuresin general [63], improving and generalizing the bounds givenin [64] which are only restricted to specific uncertainty mea-sures, and in [65–68] which are limited to weak measures.The generality and efficiency of this approach makes it suit-able as a benchmark for the forthcoming research on uncer-tainty regions. Finally, we discuss another application of ourframework in bounding general forms of joint uncertainties.Indeed, our results outperform the universal uncertainty re-lations (UUR) [13] and direct-sum majorization uncertaintyrelations (DSMUR) [15, 16] since we make an efficient useof the information gain from the pre-testing. We also show-case that our uncertainty regions are more informative than thebest known Maassen and Uffink’s (MU) entropic uncertaintyrelations [40], such as breaking the limitation of the harmonic

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condition. We stress that our principle is universal and cap-tures the essence of complementarity in quantum mechanicsas it does not rely on any specific form of uncertainty mea-sures.

The main results of our study are presented as Theorems,with all the technical details of the proofs delegated to theSupplemental Material [69].

(a) PUR: J(p,q) > c.

Γρ

Γρ

M

N

p

q

J(p,q)

H

H

Γρ

Γρ

H

H

Pre-testing

Post-testing

M

N

p

q

(b) CIR: r ≺ q ≺ t.

FIG. 1: (color online) Scenarios of preparational uncertainty rela-tion (PUR) (a) and complementary information relation (CIR) (b).PUR gives a state-independent bound, namely c, and conditioned oninformation gain p, CIR provides the optimal bounds for q undermajorization “≺”; that is r ≺ q ≺ t.

Framework of Black Box Testing.— The basic task ofblack box testing is shown in Fig. 1. An unknown blackbox prepares two independent and identically distributed re-sources ρ, which are being tested by incompatible measure-ments M and N chronologically. Without loss of generality,we assume that the test with M (a.k.a. pre-testing) is per-formed first. After that, we do the test with N (a.k.a. post-testing). Each test is performed repeatedly, returning us anoutcome probability distribution. Prior to the test, the knowl-edge associated with the outcomes is associated with the “un-certainty” of the quantum measurement. Once the test is com-pleted and its outcome is physically observed, this uncertaintyturns into an “information gain”. In this paper we focus on thecomplementarity between the information gain from the pre-testing and the uncertainty associated with post-testing (be-fore the quantum measurement is performed).

Formally, consider a preparation channel Γρ inside theblack box generating quantum state ρ [100]. Then the out-come probability distribution of the pre-testing with (basis)measurement M = {|u j〉〈u j|}

nj=1 is given by the Born’s rule

p := (〈u j| ρ |u j〉)nj=1. Since the pre-testing is repeated with the

same measurement, M, rather than a complete tomographyprocess, there exist many density matrices leading to the sameprobability distribution of the measurement outcomes. If thepre-testing outcome occurs according to the probability distri-bution p = (c j)n

j=1, we collect all candidates of ρ compatiblewith this outcome as

S (M,p) := { ρ > 0 | 〈u j| ρ |u j〉 = c j,∀ j = 1, · · · , n }. (1)

This set of quantum states forms the information gain fromthe pre-testing, manifesting the physical attribute of M via itsclassical outcome p, and narrowing down the possible range

of ρ being tested. Since the post-testing will be performedover the same quantum state ρ, its outcome is necessarily con-fined by the information gain. For a fixed post-testing mea-surement N = {|v`〉〈v` |}n`=1, we denote the set of all possibleprobability vectors q := (〈v` | ρ |v`〉)n

`=1 by

Q(N |M,p) :={(〈v` | ρ |v`〉

)n`=1 | ρ ∈ S (M,p)

}. (2)

We will also abbreviate Q(N |M,p) as Q for simplicity. Sincethe tuple {p,Q} captures fully the complementarity betweenthe measurements M and N, we will work with this set di-rectly, and quantify the uncertainty of Q based on the infor-mation gain p.

Complementary Information Principle.— Following theobservation given in [13], the uncertainty associated with aprobability vector cannot be larger than the uncertainty asso-ciated with its randomly relabelled version. That is, a prob-ability vector x ∈ Rn is more uncertain than y ∈ Rn if andonly if x is majorized by y, x ≺ y, i.e.

∑kj=1 x↓j 6

∑kj=1 y↓j for

all 1 6 k 6 n − 1 [101]. Here the down-arrow indicates thatthe components of the corresponding vectors are arranged ina non-increasing order. Since we compare probability vectorsin Q with the same information gain p, we define the right-majorization as (p1,q1) ≺R (p2,q2) if p1 = p2 and q1 ≺ q2,emphasizing that two sets of the black box testing admitthe same pre-testing distribution, while the first post-testingoutcome is more uncertain than the second one. The left-majorization can be similarly defined as (p1,q1) ≺L (p2,q2)if q1 = q2 and p1 ≺ p2. Both ≺R and ≺L are called heremarginal majorizations, which is different from the relativemajorization discussed in [102].

Based on the notion of marginal majorizations, we are nowin a position to state our main results. Due to the fact that aprobability simplex with majorization forms a complete lattice[103–105], then any possible probability vector q ∈ Q mustbe confined within two unique probability vectors r, t. Thefollowing theorems establish an explicit construction of theoptimal choices of r and t.

Theorem 1. Let M = {|u j〉〈u j|}nj=1 and N = {|v`〉〈v` |}n`=1

be the measurements of pre- and post-testing respectively.If the outcome probability distribution from M is given byp = (c j)n

j=1, then any outcome probability q from N is boundedas (p, r) ≺R (p,q) ≺R (p, s) with

r = (R1,R2, . . . ,Rn) := (r1, r2 − r1, . . . , rn − rn−1) , (3)s = (S 1, S 2, . . . , S n) := (s1, s2 − s1, . . . , sn − sn−1) . (4)

Each elements rk, sk can be efficiently computed via the fol-lowing SDPs respectively,

rk = min{x∣∣∣ x > Tr

(NIk ρ

), ∀Ik ⊂ [n], ρ ∈ S (M,p)

}(5)

sk = max{Tr

(NIk ρ

) ∣∣∣ ∀Ik ⊂ [n], ρ ∈ S (M,p)}, (6)

where Ik denotes a subset of [n] ≡ {1, · · · , n} with cardinalityk and NIk :=

∑`∈Ik|v`〉〈v` | is a partial sum with index Ik.

3

An intuitive understanding of this result can be illustratedin terms of Lorenz curves [101]. For any probability vectorx = (xi)n

i=1 in an non-increasing order, its associated Lorenzcurve L(x) is defined as the linear interpolation of the points{(

k,∑k

i=1 xi)}n

k=0 with the convention (0, 0) for k = 0. Then themajorization relation x ≺ y can be geometrically interpretedas L(x) laying everywhere below L(y). Accordingly, to findthe optimal r and t such that r ≺ q ≺ t for all q ∈ Q is equiv-alent to find the tightest Lorenz curves bounding L(q) frombelow and above for all q ∈ Q, respectively. For this pur-pose, the curves {(k, rk)}nk=0 and {(k, sk)}nk=0 with r0 = s0 = 0from Theorem 1 are taken as the minimum and maximum ofall the Lorenz curves generating from the set Q. Note thatany Lorenz curve is concave by definition, and the minimumof a set of concave functions remains concave. Therefore,{(k, rk)}nk=0 gives us a well-defined Lorenz curve, indicatingthe optimality of the lower bound r. However, this is not thecase for {(k, sk)}nk=0 since it is not necessarily concave in gen-eral. An explicit example is given in the Supplemental Mate-rial [69].

To obtain the optimal upper bound t, it suffices to per-form an additional flatness process [106], constructing thetightest Lorenz curve from {(k, sk)}nk=0. Specifically, let j bethe smallest integer in {2, . . . , n} such that S j > S j−1, andi be the greatest integer in {1, . . . , j − 1} such that S i−1 >(∑ j

k=i S k)/( j − i + 1) := a. Define the vector t as

t := (T1, . . . ,Tn) with Tk =

a for k = i, . . . , jS k otherwise.

(7)

Finally, the bounds r in (3) and t in (7) can be ensured asoptimal for the set Q under the order of majorization.

Theorem 2. Based on the same settings as in Theorem 1, forany probability vectors x and y such that x ≺ q ≺ y for allq ∈ Q, it holds x ≺ r ≺ q ≺ t ≺ y.

Since majorization only forms a partial order, the optimalbounds r and t not necessarily belong to Q in general. Thatis, these bounds may not be achieved by any quantum stateρ ∈ S (M,p) except for the qubit case where majorization isa total order. Moreover, the flatness process is required ingeneral except for the qubit and qutrit cases where the linearinterpolation of {(k, sk)}nk=0 is already concave. In terms of thecomputational complexity of our results, we need to solve oneSDP of size n × n with n = dim(ρ) for every lower boundelement rk and solve

(nk

)SDPs of size n × n for every upper

bound element sk. Since each SDPs are independent for dif-ferent index Ik, the element sk can still be solved efficientlyvia parallel computations even for large dimension n. More-over, we would like to mention that by applying the flatnessprocess directly, we could tighter the bound of UUR and ob-tain the optimal bound of DSMUR.

An explicit example of the above theorems is given inFig. 2 where we choose the pre-testing with measurementM = {|0〉〈0|, |1〉〈1|, |2〉〈2|} with outcome probability distribu-tion p = (1/6, 1/6, 2/3) and the post-testing with measure-ment N given by |v0〉 = (1/2, 0,

√3/2), |v1〉 = (0, 1, 0) and

0

0.2

0.4

0.6

0.8

1

1 2 3

r1

r2t1

t2

r

q ∈ Q

(1/3, 1/3, 1/3)

(1, 0, 0)

t

FIG. 2: (color online) An example of Theorem 1 and 2. The dottedline are the Lorenz curves of trivial bounds (1/3, 1/3, 1/3), (1, 0, 0).The thick black lines are Lorenz curves of the optimal lower andupper bounds given by r and t, respectively. Each solid color line isthe Lorenz curve of a probability vector q from Q.

|v2〉 = (−√

3/2, 0, 1/2). Each solid color line is the Lorenzcurve of a probability vector q ∈ Q. The optimal lower boundr and the optimal upper bound t are depicted as thick blacklines. These two boundaries fully characterize the uncertaintyof the setQ based on the information gain from the pre-testing.

Our results inspire a new form of “information causal-ity” [107]: the information that an observer can gain from astate in the past confines the uncertainty associated with thesame state in the future. In other words, the physical attributeof the uncertainty associated with N can only be exhibited atthe expense of the information gained from M. One extremecase is that without any information gain from the pre-testing,we can only infer that S (M,p) is the set of all quantum states,and hence obtaining the trivial bounds for the post-testing,i.e. (1/n, . . . , 1/n) ≺ q ≺ (1, 0, . . . , 0). Another extremecase is the Heisenberg’s uncertainty principle, which corre-sponds to the situation where we obtain an outcome from pre-testing with certainty. That is, the outcome probability vectorp has one entry equal to 1 and 0 otherwise. This implies thatthe information gain S (M,p) = {|u j〉〈u j|}

nj=1. Direct calcula-

tions gives the first element T1 in upper bound t equal to themaximal overlap between measurements M and N, namelymax jk |〈u j|vk〉|

2. Hence, we have q ≺ t , (1, 0, . . . , 0), indicat-ing a non-zero uncertainty of the measurement N, wheneverthe M and N do not have any common eigenvectors.

Universal Uncertainty Regions.— Since the pioneeringwork of Deutsch [37], much has been done in the direction oflower-bounding the joint uncertainties f (p(ρ)) + g (q(ρ)) > b,where p(ρ) and q(ρ) are outcome probability distributions ofthe pre- and post-testing on state ρ respectively, and f , g arevalid uncertainty measures [38–61], i.e., non-negative Schur-concave functions, including Shannon entropy, Renyi entropy,elementary symmetric functions and so forth. In the case of

4

uncertainty region

R( f , g) :=⋃ρ

{( f (p(ρ)) , g (q(ρ)))

}, (8)

the relation f (p(ρ)) + g (q(ρ)) is nothing but a straight linewith slope −1 in the coordinate plane of ( f (p), g(q)), and itsoptimal lower bound b = minρ { f (p(ρ)) + g (q(ρ))} is thenachieved at the tangent line to the bottom left of R( f , g) asshown in Fig. 3. Namely the description of uncertainty re-gions is much more informative than uncertainty relations.However, no efficient method is known to characterize the re-gion R ( f , g) in general. As an application of the majorizationbounds in Theorem 1 and 2, we can provide a general ap-proach for tight outer-approximations.

Consider the statistics set R :=⋃ρ

{(p(ρ),q(ρ))

}in Rn ×

Rn by collecting all compatible pairs of pre- and post-testingoutcomes. Note that the set of all quantum states ρ can bedivided into equivalent classes {S (M,p) : p ∈ Sn} based onthe outcome probability distribution p from the pre-testing,where Sn := { x ∈ Rn :

∑i xi = 1, xi ≥ 0,∀i } is the probability

simplex of dimension n. Then R can be fine-grained as

R =⋃

p∈Sn

Rp, with Rp :={(p,q) : q ∈ Q(M,N,p)

}. (9)

For any fixed p, the majorization bounds r, t set a boundaryfor the set Q(M,N,p). Thus we have the relaxation

Rp ⊆ Rp, with Rp :={(p,q) : r ≺ q ≺ t

}. (10)

As a consequence, by taking the union with respect to p, wehave the relaxation of the whole region,

R ⊆ R with R :=⋃

p∈Sn

Rp. (11)

Finally, any uncertainty regionR( f , g) can be retrieved by pro-jectingR from Rn×Rn to R×R via the uncertainty measures f ,g, and the projection of R will give us an outer-approximationof R( f , g) accordingly. Since R can be used to generate an ap-proximation of uncertainty region with any measures, we thusname it a universal uncertainty region. We emphasize that thisuniversal region R as well as its projections can be explictlydepicted by running p over the probability simplex, which issignificantly more tractable than characterizing R by taking ρover the set of all quantum states [108].

Theorem 3. Let M = {|u j〉〈u j|}nj=1 and N = {|v`〉〈v` |}n`=1 be

the measurements of pre- and post-testing respectively. Forany uncertainty measures f for M and g for N, their un-certainty region is outer-approximated as R ( f , g) ⊆ R ( f , g)with R ( f , g) :=

{( f (p), g(q)) : (p,q) ∈ R

}. In particular,

the outer-approximation is optimal R ( f , g) = R ( f , g) whenn = 2.

Note that for any given pre-testing outcome p, the majoriza-tion bounds r and t can be explicitly computed. Combiningthe Schur-concavity of the uncertainty measure g, the fine-grained outer-approximation can be simplified as

Rp ( f , g) = {( f (p) , y) | g(t) 6 y 6 g(r)} . (12)

f (p)

g(q)0

b

minρ{ f (p) + g(q)}(:= b)

f (p) + g(q) > b

g(r)

g(t)outer-approximation R ( f , g)

outer-approximation Rp ( f , g)fine-grained

uncertainty region R ( f , g)

FIG. 3: (color online) A schematic diagram depicts the uncer-tainty region R ( f , g) (magenta) with its outer-approximation R ( f , g)(cyan), and the fined-grained outer-approximation Rp ( f , g) (orange).The optimal uncertainty relations f (p) + g(q) > minρ{ f (p) + g(q)} istangent to the left lower boundaries of R ( f , g).

By running p over the probability simplex, we can explic-itly depict the whole region R( f , g) =

⋃p∈SnRp( f , g). A

schematic diagram is given in Fig. 3, which legibly explainshow our approximation method works. It is also worth men-tioning that by swapping the role of pre- and post-testing, wecan get another outer-approximation R ⊆ W. Taking theintersection R ∩ W will lead to a potentially tighter resultR( f , g) ⊆ R( f , g) ∩ W( f , g).

The extension of our results to multiple measurements isalso straightforward. For instance, let us consider a blackbox testing process consisting of one pre-testing with Mand m post-testings with Ni (i ∈ {1, . . . ,m}). If M indi-cates a probability vector p, then the outcome probabilitydistribution qi of Ni are bounded by ri ≺ qi ≺ ti, whereri and ti are obtained from Theorem 1 and 2 by replac-ing N with Ni. With this strategy, we can delineate anouter-approximation for the high-dimensional uncertainty re-gion

⋃ρ

{( f (p(ρ)) , g1(q1(ρ)), · · · , gm (qm(ρ)))

}with any un-

certainty measures f and g1, · · · , gm.Due to the generality of our approach, the approximation

is not guaranteed to work well for every uncertainty mea-sures. But we should stress that our approximation can becomputed explicitly and is valid for any eligible uncertaintymeasures, liberating us from specific forms of uncertainty re-lations. More importantly, this is the first efficient method toapproximating uncertainty regions in general, which can beused as a benchmark for future works.

Detailed comparisons of our results with previously knownresults, such as MU entropic uncertainty relations [40], UUR[13], and DSMUR [15, 16], are given in the Supplemental Ma-terial [69]. In terms of the qubit case, we find that MU bound

5

is tight only when one of the orders of the Renyi entropies isinfinite, i.e., (α, β) = (1/2,∞) or (∞, 1/2). Otherwise, MUbound can be improved. We also give an explicit exampleshowing that our result can outperform the bound given in[16] despite the spectrum of density operator is utilized whenconstructing their bound. Compared to the universal uncer-tainty relations in [13–16] which can be used to provide lowerbounds for f (p ⊗ q) and f (p ⊕ q), our universal uncertaintyregion works more generally for any arbitrary qualified mea-sure of joint uncertainties.

Fundamental Limits for Joint Uncertainties.— As anotherillustration of the generality of our framework, we study thejoint uncertainties given by the most general measure J :Rn × Rn → R for a pair of probability vectors (p,q) ∈ R [63].Such a measure includes the usual forms f (p)+g(q), f (p)g(q),f (p ⊗ q) and f (p ⊕ q) as special cases. To capture the es-sential properties of a measure of joint uncertainties, it hasbeen argued in [63] that J should meet the following pos-tulates : (i) Non-negativity: J(p,q) > 0; (ii) Monotonic-ity under randomly relabelling: J(D1p,D2q) > J(p,q) forall doubly stochastic matrices D1 and D2. The characteriza-tion of the joint uncertainties J(p,q) is crucial in the studyof quantum information and quantum measurements, lead-ing to a plethora of applications [23–32]. In particular, anystate-independent lower bound b of J(p,q) leads to a uncer-tainty relation J(p,q) > b while any state-independent up-per bound a of J(p,q) gives us a reverse uncertainty relationa > J(p,q). A natural question is to ask how to find a andb for joint uncertainties J(p,q) in general. Let us now applyour results to provide an answer.

First, by associating Hardy-Littlewood-Polya theorem[109], which states that two probability vectors x ≺ y if andonly if x = Dy for some doubly stochastic matrix D, withour right-majorization relation (p, r) ≺R (p,q) ≺R (p, t), weobtain J(p, r) > J(p,q) > J(p, t) for each p. Again let prun over all probability simplex, the joint uncertainties mustbe confined as maxp∈Sn J(p, r) > J(p,q) > minp∈Sn J(p, t).Swapping the role of pre- and post-testings, we can drivea left-majorization relation (u,q) ≺L (p,q) ≺L (v,q) foreach q. Then it follows that maxq∈Sn J(u,q) > J(p,q) >minq∈Sn J(v,q). Finally, taking the intersections, we have thestate-independent bounds a > J(p,q) > b with

a := min{

maxp∈Sn

J(p, r),maxq∈Sn

J(u,q)},

b := max{

minp∈SnJ(p, t),min

q∈SnJ(v,q)

}. (13)

In particular, the bounds a and b are tight for the qubit case.More remarkably, our method works for all quantified jointuncertainty measures and provides strong supports for findingtheir fundamental limits.

Conclusions.— We have proposed a new information prin-ciple, fully characterizing the complementarity between quan-tum measurements. Compared with the standard uncertaintyprinciple, the CIP does not depend on a single quantifier of

uncertainty, but instead contain all the information in the out-come probability distributions. Hence, the CIP is more infor-mative and captures the essence of complementarity betweenmeasurements in quantum physics. This principle provides usan effective method in determining the boundary of the un-certainty region with any non-negative Schur-concave func-tions and establishes the limits for all joint uncertainties. Ourmethod can, in principle, be applied to positive operator val-ued measures (POVMs) without modification.

Conventional studies on general quantum resource theories(QRTs) [110] consider quantum state transformation with theassumption that we have full knowledge of the density oper-ator of the resource state, for which a complete tomographyis required [111]. In our Supplemental Material [69], we fur-ther discuss an application of our result in a more practicalscenario where only partial knowledge of the resource state isneeded.

The connection between the results presented here andother areas, including entanglement [23–26] and EPR steeringdetection [27–32], moderate deviation of majorization-basedresource interconversion [112] and quantum key distribution[34–36], are particularly important in the context of quantuminformation. We left these directions for future work. The ex-perimental demonstration of this paper, performed in a pho-tonic system, is currently in progress. The experimental dataand our theoretical results fit well [113].

Acknowledgments.—We would like to thank Xin Wang,Yuan Yuan, Guo-Yong Xiang, Nengkun Yu, Jie Xie, LijianZhang, Zhihao Ma, Gaurav Saxena, Carlo Maria Scandolo,and Barry C. Sanders for fruitful discussions. Y. X., and G.G. acknowledge financial support from the Natural Sciencesand Engineering Research Council of Canada (NSERC). K.F.was supported by the University of Cambridge Isaac NewtonTrust Early Career grant RG74916.

∗ Electronic address: mathxiao123@gmail.com† Electronic address: kf383@cam.ac.uk‡ Electronic address: gour@ucalgary.ca

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9

Supplemental Material:Complementary Information Principle and Universal Uncertainty Regions Regions

This supplemental material provides a more detailed analysis and proofs of the results in the main text. We may reiterate someof the steps in the main text to make the supplemental material more explicit and self-contained.

PROOF OF THEOREM 1

Before proceeding, it is worth noting that our majorization inequalities r ≺ q ≺ s is based on the following restrictedconditions

〈u j| ρ |u j〉 = c j, ∀ j ∈ {1, 2, . . . , n} . (1)

This means we are considering the possible range (under majorization) of probability vector q, generated from performing thequantum measurement N on a state ρ with ρ ∈ S (M,p) := { ρ > 0 | 〈u j| ρ |u j〉 = c j,∀ j }.

The following lemma gives a useful characterization of the partial sum of the k largest values in a probability vector.

Lemma 1. For any probability vector p with non-increasing order (p↓j)nj=1, we have the partial sum

∑kj=1 p↓j = maxIk

∑s∈Ik

ps

where the maximization on the r.h.s. is taken over all possible subsets Ik of {1, · · · , n} with cardinality k.

Proof. The result is straightforward from the definition of (p↓j)nj=1. �

Theorem 1. Let M = {|u j〉}nj=1 and N = {|v`〉}n`=1 be the measurements of pre- and post-testing respectively. If the outcome

probability distribution from M is given by p = (c j)nj=1, then any outcome q from N is bounded as (p, r) ≺R (p,q) ≺R (p, s) with

s = (S 1, S 2, . . . , S n) := (s1, s2 − s1, . . . , sn − sn−1) and sk = maxIk

maxρ

{TrNIk ρ

∣∣∣ ρ ∈ S (M,p)}, (2)

r = (R1,R2, . . . ,Rn) := (r1, r2 − r1, . . . , rn − rn−1) and rk = minρ

{γ∣∣∣ γ > TrNIk ρ,∀Ik, ρ ∈ S (M,p)

}, (3)

where Ik denotes a subset of {1, · · · , n} with cardinality k and NIk :=∑`∈Ik|v`〉〈v` | is a partial sum of N with index Ik.

Proof. Suppose q := (qk)nk=1 = (〈vk | ρ |vk〉)n

k=1 is an outcome probability generated from the measurement N on a quantum stateρ ∈ S (M,p). Denote the non-increasing order of q as (q↓j)

nj=1. By Lemma 1 and the definitions of qs and NIk , we have

k∑j=1

q↓j = maxIk

∑s∈Ik

qs = maxIk

∑s∈Ik

Tr|vs〉〈vs| ρ = maxIk

TrNIk ρ. (4)

Relaxing ρ to all possible ρ ∈ S (M,p), we have

maxIk

TrNIk ρ 6 maxρ∈S (M,p)

maxIk

TrNIk ρ =: sk. (5)

Note that sk =∑k

j=1 S k 6∑k

j=1 S ↓k . Thus we have∑k

j=1 q↓j 6∑k

j=1 S ↓k . By the definition of majorization, we have q ≺ s.Similar to Eq. (5), we have

maxIk

TrNIk ρ > minρ∈S (M,p)

maxIk

TrNIk ρ =: rk (6)

Now we are going to show that Rk is in the nonincreasing order, i.e., Rk > Rk+1, or equivalently 2rk > rk−1 + rk+1. Suppose theminimum of rk is taken at ρ. Denote q = (〈v j| ρ |v j〉)n

j=1 with the nonincreasing ordering(

q↓j)n

j=1. Then we have

2rk = 2 maxIk

TrNIk ρ = 2k∑

j=1

q↓j =

k−1∑j=1

q↓j + q↓k

+

k+1∑j=1

q↓j − q↓k+1

> k−1∑j=1

q↓j +

k+1∑j=1

q↓j , (7)

10

where the second equality follows from Lemma 1. Using Lemma 1 again, we have

k−1∑j=1

q↓j = maxIk−1

TrNIk−1 ρ > minρ∈S (M,p)

maxIk−1

TrNIk−1 ρ = rk−1, (8)

k+1∑j=1

q↓j = maxIk+1

TrNIk+1 ρ > minρ∈S (M,p)

maxIk+1

TrNIk+1 ρ = rk+1. (9)

Combining Eqs. (7-9), we have 2rk > rk−1 + rk+1, that is, Rk is in the nonincreasing order. Together with Eqs. (4) and (6), weobtain

∑kj=1 q↓j > rk =

∑kj=1 Rk =

∑kj=1 R↓k , which implies q � r. Finally, we note that

rk = minρ∈S (M,p)

maxIk

TrNIk ρ = minρ∈S (M,p)

min{γ | γ > TrNIk ρ,∀Ik

}= min

ρ

{γ∣∣∣ γ > TrNIk ρ,∀Ik, ρ ∈ S (M,p)

}, (10)

where the rightmost is a semidefinite program. �

Remark 1 Note that we have shown r is in nonincreasing order. But this is not necessarily the case for s since there exist casessuch that 2sk < sk−1 + sk+1. That is why we need an additional flatness procedure for the upper bound in general. However, forthe qubit and qutrit case, we can show that the vector s is always in nonincreasing order.

Proposition 1. For n = 2, 3, the vector s is in nonincreasing order.

Proof. Recall that sk = maxρ∈S (M,p)

maxIk

TrNIk ρ. We first show that 2s1 > s2. Suppose the optimal state of s2 = maxρ∈S (M,p)

maxI2

TrNI2 ρ

is taken at ρ and the optimal index set I2 = {1, 2}. We have

s2 = Tr(|v1〉〈v1| + |v2〉〈v2|) ρ 6 2 maxI1

TrNI1 ρ 6 2 maxρ∈S (M,p)

maxI1

TrNI1 ρ = 2s1. (11)

Furthermore, when n = 3, we have s3 = 1. Suppose the optimal state of s1 is taken at ρ and optimal index is taken at I1 = {1}.Then we have

s3 + s1 = 1 + Tr|v1〉〈v1| ρ = Tr(|v1〉〈v1| + |v2〉〈v2|) ρ + Tr(|v1〉〈v1| + |v3〉〈v3|) ρ 6 2 maxρ∈S (M,p)

maxI2

TrNI2 ρ = 2s2, (12)

which completes the proof. �

Remark 2 We show that when n = 4, there exists an explicit example that s is not in nonincreasing order. Consider themeasurements

M =

−0.4703 −0.3508 −0.6040 −0.53940.8392 0.0745 −0.4820 −0.24040.0627 0.2180 0.5452 −0.80700.2657 −0.9077 0.3249 −0.0051

, N =

0.4960 0.1166 −0.8579 −0.0654−0.3299 −0.2437 −0.2898 0.86470.6759 −0.6566 0.2886 0.16960.4339 0.7042 0.3109 0.4682

(13)

and the probability vector p = (1/4, 1/4, 1/4, 1/4). By running our SDP algorithm, we obatin the result s =

(0.9047, 0.0424, 0.0529, 0.0001) where the second element is strictly smaller than the third one.

PROOF OF THEOREM 2

In order to prove the tightness of bounds r and t we first recall all the related concepts

Definition 1 (Poset). A partial order is a binary relation “≺” over a set L satisfying reflexivity, antisymmetry, and transitivity.That is, for all x, y, and z in L, we have

(i) Reflexivity: x ≺ x,

(ii) Antisymmetry: If x ≺ y and y ≺ x, then x = y,

(iii) Transitivity: If x ≺ y and y ≺ z, then x ≺ z.

11

Note that without the antisymmetry, “≺” is just a preorder. Let us now define the set of all n-dimensional probability vectors as

Pn =

p = (p1, . . . , pn) | p j ∈ [0, 1],n∑

j=1

p j = 1, p j > p j+1

, (14)

with components in non-increasing order. Accordingly, majorization is a partial order over Pn, i.e. 〈Pn,≺〉 is a poset.

Definition 2 (Lattice). A poset 〈L,≺〉 is called a join-semilattice, if for any two elements x and y of L, it has a unique leastupper bound (lub,supremum) x ∨ y satisfying

(i) x ∨ y ∈ L,

(ii) x ≺ x ∨ y and y ≺ x ∨ y.

On the other hand, 〈L,≺〉 is called a meet-semilattice, if for any two elements x and y of L, it has a unique greatest lower bound(glb,infimum) x ∧ y satisfying

(i) x ∧ y ∈ L,

(ii) x ∧ y ≺ x and x ∧ y ≺ y.

〈L,≺〉 is called a lattice if it is both a join-semilattice and a meet-semilattice, and denote it as a quadruple 〈L,≺,∧,∨〉.

Definition 3 (Bounded Lattice). A lattice 〈L,≺,∧,∨〉 is called bounded, if it has a top, denoted by > and a bottom, denoted by⊥ which satisfy

(i) x ≺ > for all x ∈ L,

(ii) ⊥ ≺ x for all x ∈ L.

By definition,Pn is bounded, since for any probability vector p belongs to the setPn we have u ≺ p ≺ l where u := (1/n, . . . , 1/n)and l := (1, 0, . . . , 0).

Definition 4 (Complete Lattice). A lattice 〈L,≺,∧,∨〉 is called complete, if for any subset S ⊂ L, it has a greatest element,denoted by > and a least element, denoted by ⊥ which satisfy

(i) x ≺ >, for all x ∈ S and x ≺ y for all x ∈ S ⇒ > ≺ y,

(ii) ⊥ ≺ x, for all x ∈ S and y ≺ x for all x ∈ S ⇒ y ≺ ⊥.

By embedding the majorization “≺”, the quadruple 〈Pn,≺,∧,∨〉 forms a complete lattice. We remark that the result of com-pleteness follows directly from the work presented in [1], and the algorithm in finding p ∧ q and p ∨ q (also known as flatnessprocess) was first introduced in [2]. Only recently, the completeness of majorization lattice has been applied to the study ofoptimal common resource [3] .

The main progress over the main text is a fine-grained approach in studying the feasible set of probability vector q conditionedon p; that is

Q :={q := (Tr(|vk〉〈vk | ρ))k | ρ ∈ S (M,p)

}, (15)

which forms a convex subset of Pn and our aim is to find its infimum and supremum. By appealing to the definition of lattice, itis straightforward to show that the unique infimum ∧∧∧

Q and supremum ∨∨∨Q of the set Q always exist whenever Q is a finite subset

of Pn. However, the feasible set Q will not be a finite subset of Pn in general. A crucial property to solve this problem is thecompleteness of majorization lattice, and several useful definitions are needed. Let Rn

+ and Rn+.> be defined as

Rn+ =

{p = (p1, . . . , pn) ∈ Rn | p j > 0

}, and Rn

+,> ={p = (p1, . . . , pn) ∈ Rn | p j > p j+1 > 0

}. (16)

A vector x ∈ Rn is weakly majorized by y ∈ Rn, denoted by x ≺w y, if∑k

j=1 x↓j 6∑k

j=1 y↓j for all 1 6 k 6 n. Here the down-arrow notation denotes that the components of the corresponding vector are ordered in an nonincreasing order. Based on thesedefinitions, a stronger version of completeness has been proved in [1]

Lemma 2 (Infimum). Let S ⊂ Rn+ be a nonempty set. Then there exists a unique vector ∧∧∧

S ∈ Rn+,> such that

12

⊥

a ∧ b

d ∧ e

a

b

c

d

e

a ∨ b

b ∨ c

d ∨ e

>

FIG. 1: (color online) A schematic diagram of lattice structure 〈L,≺,∧,∨〉, consisting of a partially order set (poset) 〈L,≺〉 and two binaryoperations ∧, and ∨. For any two elements of L, there exists a unique greatest lower bound (glb, ∧) and a unique least upper bound (lub, ∨)under ≺. Here L := {⊥, a ∧ b, d ∧ e, a, b, c, d, e, a ∨ b, b ∨ c, d ∨ e,>}, and the notation “←” stands for “≺”. Clearly, the quadruple 〈L,≺,∧,∨〉is a bounded lattice, that is for any x ∈ L we have ⊥ ≺ x ≺ >. However, this lattice is not complete since by defining the subset S as{a ∧ b, a, b, c, a ∨ b, b ∨ c}, its greatest lower bound and the least upper bound do not exist.

(i) x ∈ S ⇒ ∧∧∧S ≺w x,

(ii) y ≺w x for all x ∈ S ⇒ y ≺w∧∧∧S.

Lemma 3 (Supremum). Let the set of upper bounds of S ⊂ Rn+ be nonempty. Then there exists a unique vector ∨∨∨

S such that

(i) x ∈ S ⇒ x ≺w∨∨∨S,

(ii) x ≺w y for all x ∈ S ⇒ ∨∨∨S ≺w y.

Therefore, the completeness of majorization lattice follows from lemma 2 and lemma 3 immediately.

Proposition 2. Majorization lattice is complete: for any subset S of Pn, it has both a infimum and supremum in Pn.

Proof. Taking any subset S of Pn we get that S ⊂ Pn ⊂ Rn+. The existence of (1, 0, . . . , 0) implies that the set of upper bounds

of S is nonempty. Therefore, S must have a infimum ∧∧∧S and supremum ∨∨∨

S. We now prove that both ∧∧∧S and ∨∨∨

S belongs toPn. Since Pn is bounded by (1/n, 1/n, . . . , 1/n) and (1, 0, . . . , 0), hence for any x ∈ S we have

(1/n, 1/n, . . . , 1/n) ≺ x ≺ (1, 0, . . . , 0). (17)

Therefore

(1/n, 1/n, . . . , 1/n) ≺∧∧∧S ≺ x ≺

∨∨∨S ≺ (1, 0, . . . , 0). ∀x ∈ S (18)

which is equivalent to say ∧∧∧S, ∨∨∨S ∈ Pn. We thus prove the completeness of majorization lattice. �

From the completeness of Pn, we can always find the unique infimum ∧∧∧Q and supremum ∨∨∨

Q for the feasible set Q. Now weare in position to prove that ∧∧∧

Q = r and ∨∨∨Q = t.

Theorem 2. For pre-testing with outcome probability distribution p = (c j)nj=1, the outcome q of post-testing is bounded by r and

t, and they are tight under majorization; That is ∧∧∧Q = r ≺ q ≺ t =

∨∨∨Q. (19)

Proof. We start by proving ∧∧∧Q = r. From the definition of the infimum ∧∧∧

Q we have r ≺ ∧∧∧Q. Let ∧∧∧

Q := (x1, . . . , xn), then it isimmediate to observe that ∧∧∧

Q ≺ q for any q ∈ Q; that is

k∑s=1

xs 6 maxIk

TrNIk ρ, ∀k ∈ {1, . . . , n} and ∀ρ ∈ S (M,p) (20)

13

which leads to

k∑s=1

xs 6 minρ∈S (M,p)

maxIk

TrNIk ρ = rk, ∀k ∈ {1, . . . , n} (21)

and hence one has ∧∧∧Q ≺ r. Since both ∧∧∧

Q and r (see the proof of theorem 1) are arranged in nonincreasing order, ∧∧∧Q = r holds.

On the other hand, in order to prove that t is indeed the supremum ∨∨∨Q, we need only to prove that t ≺ ∨∨∨

Q. Suppose now ∨∨∨Q

has the form (y1, . . . , yn), and assume, by contradiction, that there exists an index l such that

l∑s=1

ys < tl, (22)

where tk :=∑k

s=1 Ts (k = 1, . . . , n). Without loss of generality we can assume l is the smallest index for which above inequality

holds, and then we havel−1∑s=1

ys > tl−1, which implies yl < a.

If l < i, the summations of ∨∨∨Q with the upper index smaller than l + 1 yields

tl = sl = maxIl

maxρ

{TrNIl ρ

∣∣∣ ρ ∈ S (M,p)}6

l∑s=1

ys < tl = sl. ∀l ∈ {1, . . . , i − 1} (23)

Hence, under the standing hypothesis on l, we must have l > i.Consider now l > j − 1, from the flatness process we have tl = sl. Therefore, for index l it holds that

tl = sl = maxIl

maxρ

{TrNIl ρ

∣∣∣ ρ ∈ S (M,p)}6

l∑s=1

ys < tl = sl, ∀l ∈ { j, . . . , n} (24)

that is a contradiction. Thus we conclude that l ∈ {i, . . . , j − 1}.Finally, for all l = i, i + 1, . . . , j − 1, one has that

t j = s j = maxI j

maxρ

{TrNI j ρ

∣∣∣ ρ ∈ S (M,p)}6

j∑s=1

ys =

l∑s=1

ys +

j∑s=l+1

ys < tl +

j∑s=l+1

ys. (25)

For l < s 6 j, we also have a > yl > . . . > y j. As an immediate consequence, we obtain∑ j

s=l+1 ys < ( j − l)a. Hence

j∑s=1

ys < tl + ( j − l)a = t j. (26)

So that t j < t j, which is clearly a contradiction. Therefore, t ≺ ∨∨∨Q. Consequently, t,∨∨∨Q ∈ Pn implies that t =

∨∨∨Q, which

completes the proof. �

EXAMPLE 1: UNCERTAINTY RELATION BETWEEN TWO MEASUREMENTS

In this part, we derive the analytical result of our majorization bounds for two measurements on a qubit state. Without loss ofgenerality, we consider two sets of qubit measurements

M = {|u1〉, |u2〉} with |u1〉 = |0〉, |u2〉 = |1〉,N = {|v1〉, |v2〉} with |v1〉 = cos θ|0〉 − sin θ|1〉, |v2〉 = sin θ|0〉 + cos θ|1〉, θ ∈ [0, π/2].

(27)

Proposition 3. Suppose the qubit measurements M, N are defined above. If the outcome probability distribution under themeasurement M is p = (λ, 1 − λ) with λ ∈ (0, 1/2), then the outcome q under the measurement N is bounded as r ≺ q ≺ s withs = (s1, 1 − s1), r = (r1, 1 − r1) and

s1 =

(√λ sin θ +

√1 − λ cos θ

)2, θ ∈

[0, π4

),

(√1 − λ sin θ +

√λ cos θ

)2, θ ∈

[ π4 ,

π2],

r1 =

(√

1 − λ sin θ −√λ cos θ)2, cot(2θ) ∈

(−∞,− 2

√λ(1−λ)

1−2λ

),

12 , cot(2θ) ∈

[−

2√λ(1−λ)

1−2λ , 2√λ(1−λ)

1−2λ

],(√

λ sin θ −√

1 − λ cos θ)2, cot(2θ) ∈

(2√λ(1−λ)

1−2λ ,∞).

(28)

14

Proof. The proof can be given by straightforward calculations. According to Theorem 1, we have

s1 = max{k1, k2}, with ks = maximizeρ

Tr|vs〉〈vs| ρ (29)

subject to ρ > 0, 〈0|ρ|0〉 = λ, 〈1|ρ|1〉 = 1 − λ. (30)

Since |vs〉 are real vectors, the optimal solution can be always taken at a real operator ρ. From the condition (30), let ρ =

λ|0〉〈0| + (1 − λ)|1〉〈1| + x(|0〉〈1| + |1〉〈0|). We have

k1 = maximizex

(1 − 2λ) sin2 θ + λ − x sin 2θ (31)

subject to −√λ(1 − λ) 6 x 6

√λ(1 − λ). (32)

Thus k1 =(√

1 − λ sin θ +√λ cos θ

)2. Similarly, we have k2 =(√λ sin θ +

√1 − λ cos θ

)2. Then we have

s1 =

(√λ sin θ +

√1 − λ cos θ

)2, θ ∈

[0, π4

),(√

1 − λ sin θ +√λ cos θ

)2, θ ∈

[ π4 ,

π2].

(33)

Again based on Theorem 1, we have

r1 = minimizeρ

γ (34)

subject to γ > Tr|v1〉〈v1| • ρ, γ ≥ Tr|v2〉〈v2| • ρ, (35)ρ > 0, 〈0|ρ|0〉 = λ, 〈1|ρ|1〉 = 1 − λ. (36)

Let ρ = λ|0〉〈0| + (1 − λ)|1〉〈1| + x(|0〉〈1| + |1〉〈0|). We have

r1 = minimizex

γ1(x) ∨ γ2(x) (37)

subject to −√λ(1 − λ) 6 x 6

√λ(1 − λ), (38)

with γ1(x) = (1 − 2λ) sin2 θ + λ − x sin 2θ and γ2(x) = (1 − 2λ) cos2 θ + λ + x sin 2θ and γ1(x) ∨ γ2(x) denotes the maximumfunction between γ1 and γ2. Note that γ2(x) − γ1(x) = 2 sin 2θ

[x − (λ − 1

2 ) cot 2θ]. If (λ − 1

2 ) cot(2θ) 6 −√λ(1 − λ), we have

γ2 > γ1 for all feasible x, then

r1 = γ2

(−

√λ(1 − λ)

)=

(√λ sin θ −

√1 − λ cos θ

)2. (39)

If (λ − 12 ) cot(2θ) >

√λ(1 − λ), we have γ2 6 γ1 for all feasible x, then

r1 = γ1

( √λ(1 − λ)

)=

(√1 − λ sin θ −

√λ cos θ

)2. (40)

If −√λ(1 − λ) 6 (λ − 1

2 ) cot(2θ) 6√λ(1 − λ), the value r1 is taken at the intersection point of γ1(x) = γ2(x) and we will always

have r1 = 12 .

�

Remark 3 Note that for the probability vectors (λ, 1 − λ) and (1 − λ, λ), we will have exactly the same result of s1 and r1.

To illustrate the connection of our uncertainty regions with previously known results, we consider the most well-knownMaassen and Uffink’s (MU) entropic uncertainty relation [4], which states that the Renyi entropies of measurement outcomes ofM and N for any quantum state ρ satisfy

Hα(M) + Hβ(N) > −2 log c1, with c1 := maxj,k|〈u j|vk〉|, (41)

where Hα(M) and Hβ(N) are the Renyi entropies of the probability distributions p and q with order α and β respectively, and the

orders α and β satisfy 1/α + 1/β = 2. Consider the measurements M = {|0〉, |1〉} and N =

{12 |0〉 −

√3

2 |1〉,√

32 |0〉 +

12 |1〉

}. Then the

maximum overlap c1 between M and N is√

3/2 and hence MU relation holds for 1/α+ 1/β = 2 with the lower bound log (4/3).

15

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

(a) 1/α + 1/β = 1 (b) 1/α + 1/β = 2 (c) 1/α + 1/β = 3

(α, β) =(

2c ,

2c

)(α, β) =

(∞, 1

c

)(α, β) =

(1c ,∞

)MU bound

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

FIG. 2: (colour online) Uncertainty regions with different parameters 1/α + 1/β = c; The red regions are uncertainty regions R(Hα,Hβ

)with

orders (α, β) = (2/c, 2/c), the green regions stand for uncertainty regions with orders (α, β) = (∞, 1/c), the blue regions represent uncertaintyregions with orders (α, β) = (1/c,∞) and the dotted lines describe MU bound Hα(M) + Hβ(N) > log (4/3). Here, the horizontal axis representsthe numerical value of Hα and vertical axis stands for the numerical value of Hβ. (a) Uncertainty regions with 1/α + 1/β = 1; (b) Uncertaintyregions with 1/α + 1/β = 2; (c) Uncertainty regions with 1/α + 1/β = 3.

In this qubit case we have R ( f , g) = R ( f , g), and from Proposition 3 we can study a family of uncertainty regions R(Hα,Hβ

)with 1/α + 1/β = c for any c > 0. We first consider a family of uncertainty regions R

(Hα,Hβ

)with 1/α + 1/β = 1 in Fig. (2a)

where MU bound losses its efficacy while our framework provides a full description for uncertainty regions with different orders.In the case 1/α + 1/β = 2 or 3, MU bound is optimal for all allowable orders α and β. However, it is clearly not optimal for aspecific pair of (α, β) except for the two extreme cases (α, β) = (1/2,∞) and (∞, 1/2). In Fig. (2b), for instance, the uncertaintyregion with (α, β) = (2/c, 2/c) is depicted in red, where the optimal uncertainty relation lower bound is given by a tangent linewith slope −1 on the lower-left boundary which outperforms the MU bound.

Actually the MU bound for qubit measurements is only tight in two extreme cases in general. For the qubit measurements Mand N defined in (27), the Maassen-Uffink bound is given by −2 log c1 with c1 = cos θ if θ ∈

[0, π4

)and c1 = sin θ if θ ∈

[ π4 ,

π2].

When comparing with the Maassen-Uffink bound, we only need to consider the majorization upper bound s, which correspondsto the lower-half boundary in the region plots. For the lower-half boundary

(Hα(p),Hβ(s)

), consider the zero point on the

horizontal axis, i.e., Hβ(s) = 0. We have λ = cos2 θ, when θ ∈ (0, π/4) and λ = sin2 θ, when θ ∈ (π/4, π/2). Only when α→ ∞,Hα(p) → H∞(p) = − log max{λ, 1 − λ} = −2 log c1. That is, the zero point on the horizontal axis reaches the MU bound if andonly if α = ∞. Similar argument holds for β. Thus the MU bound for qubit measurements is only tight in the two extreme cases(α, β) = (1/2,∞) or (∞, 1/2). These observations indicate that our framework is generally more informative than the MU bound.

EXAMPLE 2: UNCERTAINTY RELATION BETWEEN MULTIPLE MEASUREMENTS

Since the proof of the MU bound mainly relies on a version of Riesz theorem [5], it would be difficult to generalize their resultto multiple measurements or general α, β beyond the relation 1/α + 1/β = 2, even for the qubit case. As an demonstration ofour framework to multiple measurements with general uncertainty measures, we consider the triple measurements M, N (used

in example 1) and T =

{ √2

2 |0〉 +√

22 |1〉,

√2

2 |0〉 −√

22 |1〉

}. Denote w as the outcome probability distribution from measurement T .

Then its Renyi entropy of order γ is given by Hγ(T ) := Hγ(w). From Proposition 3, for any given probability p = (λ, 1− λ) withλ ∈ (0, 1/2), the majorization upper bound for N and T are respectively given by tN = (tN

1 , 1 − tN1 ) and tT = (tT

1 , 1 − tT1 ) with

tN1 =

(√3√

1 − λ +√λ)2

4, tT

1 =

(√1 − λ +

√λ)2

2. (42)

For given α, β, γ, the optimization

b := minimizeλ

Hα(p) + Hβ(tN) + Hγ(tT ) subject to 0 < λ <12, (43)

leads to a state-independent lower bound for the uncertainty relation Hα(M)+Hβ(N)+Hγ(T ) ≥ b. Implementing this optimizationvia Mathematica function “NMinimize”, we obtain the numerical results listed in the following table.

16

Parameters (α, β, γ) (1, 1, 1) (1, 2, 3) (1, 1,∞) (∞, 1,∞) (∞,∞,∞)min Hα(M) + Hβ(N) + Hγ(T ) 1.15898 0.957202 0.903285 0.50165 0.474238

TABLE I: Numerical results for uncertainty relation Hα(M) + Hβ(N) + Hγ(T ) consisting of three measurements M, N and T . The parameters(α, β, γ) in the list are only chosen for demonstration. They could be any possible combinations of α, β and γ in general.

EXAMPLE 3: COMPARISON WITH DIRECT-SUM AND TENSOR-PRODUCT BOUNDS

In this part we compare our results with the majorization uncertainty relations formulated in [6–9]. Explicit examples areprovided to demonstrate how the obtained results in this work outperform previous ones.

Given quantum measurements M = {|u j〉}nj=1 and N = {|vk〉}

nk=1 with n possible classical outcomes, denote their joint set as

R := (|r1〉, |r2〉, · · · , |r2n〉) = (|u1〉, · · · , |un〉, |v1〉, · · · , |vn〉) which contains 2n elements. Define xk := maxIk ‖∑

s∈Ik|rs〉〈rs| ‖∞,

where the maximization is taken over all possible index subset Ik of {1, 2, · · · , 2n} with cardinality k. The direct-sum upperbound is defined as [7]

w⊕ := (x1, x2 − x1, · · · , x2n − x2n−1) = (x1, x2 − x1, · · · , xn+1 − xn, 0, 0, . . . , 0︸ ︷︷ ︸n − 1 times

), (44)

where the equation holds since xk = 2 for k > n + 1 [6, 7]. Similarly, the direct-product upper bound is defined as [6]

w⊗ :=14

(x2

2, x23 − x2

2, · · · , x2n+1 − x2

n, 0, 0, . . . , 0︸ ︷︷ ︸n2 − n times

). (45)

Denote the trivial lower and upper bounds as

u :=(

1n,

1n, · · · ,

1n

)and l := (1, 0, · · · , 0). (46)

In Fig. 3, we demonstrate that

u ⊕ u ≺ p ⊕ r ≺p ⊕ q ≺ p ⊕ t ≺ w⊕ ≺ l ⊕ l, (47)u ⊗ u ≺ p ⊗ r ≺p ⊗ q ≺ p ⊗ t ≺ w⊗ ≺ l ⊗ l, (48)

where the probability vectors r and t are obtained from Theorem 1. It is clear that our majorization upper bounds p⊕ t and p⊗ tprovide the tighter estimation for p ⊕ q and p ⊗ q respectively. The shaded area depicts the feasible set of Lorenz curves givenfrom Q(M,N ,p), whose boundaries are established by our majorization lower (purple) and upper (blue) bounds.

0 1 2 3 4 5 6

0

0.5

1

1.5

2

aaaaaa

0 1 2 3 4 5 6 7 8 9

0

0.2

0.4

0.6

0.8

1

aaaaaal ⊕ lw⊕

w⊕(ρ)p ⊕ tp ⊕ ru ⊕ u

(a) Direct-sum comparison.

l ⊗ lw⊗p ⊗ t

p ⊗ ru ⊗ u

(b) Direct-product comparison.

FIG. 3: (color online) Comparison of several known bounds via Lorenz curves. The example is taken as the probability p = (1, 7, 7)/15,measurements M = { (1, 0, 0), (0, 1, 0), (0, 0, 1) } and N =

{ (1/√

3, 1/√

2, 1/√

6),(1/√

3, 0,−√

2/√

3),(1/√

3,−1/√

2, 1/√

6) }

.

17

In the case that the spectrum of the quantum state ρ is known, the direct-sum majorization bound can be improved further [8].Denote

−−−→λ(ρ) and

−−−−−−−→λ(R (Ik)) as the spectrum of ρ and R (Ik) :=

∑s∈Ik|rs〉〈rs| respectively (arrange in non-increasing order). Define

yk := maxIk

−−−→λ(ρ) •

−−−−−−−→λ(R (Ik)), where the symbol • stands for inner product between vectors. Using the method presented in [9] one

can show that, for the probability vectors p and q derived from measurements M and N on ρ, we get

p ⊕ q ≺ w⊕(ρ), with w⊕(ρ) := (y1, y2 − y1, · · · , y2n − y2n−1). (49)

The same example in Fig. (3a) demonstrates that the performance of our result can be still tighter than w⊕(ρ) where−−−→λ(ρ) is taken

as the spectrum of ρ = 115

(1 0 00 7 70 7 7

). This indicates that the spectrum of a density operator is not necessary more useful than other

classical outcomes of the same state, even though the former is usually regarded as more difficult to be obtained.

PRACTICAL RESOURCE THEORIES WITHOUT COMPLETE TOMOGRAPHY

In quantum resource theories (QRTs) [10], one of the most central research topics is to study the conversion between differentresource objects under certain constraints (or free operations) [11–26, 28–33]. In the recent study of resource theories, it is usu-ally assumed that the density operator of our resource state is already known, which is not completely practical. This is becausequantum tomography is the main method used to assess the matrix representation of a quantum state, but the resources neededin achieving quantum tomography is exponential in the device size [27]. In particular, it is notoriously hard to determine thedensity matrix in high-dimensional Hilbert spaces and that is why we need to consider a more general framework in determiningquantum state transformation with only partial knowledge of the state.

We first consider QRT of entanglement. Let |ψ〉AB =∑n

j=1√x j| j〉A| j〉B and |φ〉AB =

∑nj=1√y j| j〉A| j〉B be two bipartite pure

states, and their corresponding Schmidt vectors are defined as x := (x1, . . . , xn) and y := (y1, . . . , yn). By Nielsen’s theorem [34],the necessary and sufficient condition for state transformation under local operations and classical communication (LOCC) isgiven as

ψLOCC−−−−→ φ ⇔ x ≺ y (Entanglement). (50)

This majorization criterion for entanglement transformation is fundamental and important since it plays an important roles inboth deterministic and nondeterministic LOCC transformations, and gives rise to the concept of resource catalyst. Moreover,majorization criterion can also be found in QRT of coherence. For pure states |ψ〉 =

∑nj=1√x jeiψ j | j〉 and |φ〉 =

∑nj=1√y jeiφ j | j〉,

their probability amplitudes are denoted as x := (x1, . . . , xn) and y := (y1, . . . , yn) respectively. Similarly we have [35, 36]

ψSIO−−−→

IOφ ⇔ x ≺ y (Coherence). (51)

Here SIO stands for strictly incoherent operations and IO denotes incoherent operations. Inspired by these criteria, we wil focuson our study on QRTs for which

ψfree−−→ φ ⇔ x (ψ) ≺ x (φ) (Majorization-Based), (52)

where x (ψ) is a probability vector associated with ψ (e.g. Schmidt vector).Let us now relate our complementary information relation with majorization-based QRTs [3, 37], such as entanglement and

coherence. If the pre-testing measurement M can be implemented on our resource pure state ψ via a free operation and outcomesa probability vector p, then based on p it is possible to infer some useful results of resource conservation. More specifically,if we can derive the probability vector x (ψ) from post-testing with measurement N, then based on Theorem 2 we obtain thefollowing result.

Proposition 4. If a pure state ψ with free pre-testing indicates an outcome probability distribution p = (c j) j, then a quantumstate φ2 can be convert to ψ by free operations if x (φ2) ≺ r and ψ can be transformed into φ1 whenever t ≺ x (φ1); that is

φ2free−−→ ψ ⇐ x (φ2) ≺ r, and ψ

free−−→ φ1 ⇐ t ≺ x (φ1) . (53)

Note that there is no general answers to the question whether ψ can be convert to φ based on limited information gain fromthe pre-testing measurement; that is, there are three possible answers for a one-shot state transformation question: “yes”, “no”and “lack of information”.

18

f (p)

g(q)0

fmax

gmax

(a) Classical Range

fmax

A

B

gmaxC D

f (p)

g(q)0

(b) Uncertainty Principle

fmax

A

B

gmaxC D

f (p)

g(q)0

(c) Uncertainty Relation

fmax

A

B

gmaxC D

f (p)

g(q)0

(d) Uncertainty Region

FIG. 4: (color online) A schematic diagram depicts the classical range of [0, fmax] × [0, gmax], Heisenberg uncertainty principle, uncertaintyrelation f (p) + g(q) > b, and uncertainty region R( f , g).

VISUALIZATION OF UNCERTAINTY PRINCIPLE, UNCERTAINTY RELATIONS, AND UNCERTAINTY REGIONS

So far we have mostly talked about the rigorous mathematical formulation of uncertainty and complementarity, however avisualization that can help the reader understand the connections among them is also preferred. To start with, let us outlinethe trade-off between measurements in classical world. We still consider a protocol of black box testings, including pre-testingand post-testing, collect their probability distribution into vectors p and q. Therefore, for given non-negative Schur concavefunctions f and g, the uncertainty of p and q are bounded as

fmax := f (1/n, 1/n, . . . , 1/n) > f (p) > 0,gmax := g(1/n, 1/n, . . . , 1/n) > g(q) > 0. (54)

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Thus the classical region is the Cartesian product of [0, fmax] and [0, gmax], i.e. in classical world we have R( f , g) = [0, fmax] ×[0, gmax] with × stands for Cartesian product, as shown in Fig. 4 (a).

In quantum mechanics Heisenberg uncertainty principle refers to the situation, where for two incompatible measurements Mand N there exists no sharp value for both of them. Equivalently, when the measurement outcome of M appears with certainty,i.e. f (p) = 0, then the measurement outcome of N becomes indeterministic. That means the range of g(q), denoted by [C,D],does not contain the original point o := (0, 0). By exchanging their roles, we have 0 < [B, A] with [B, A] stands for the rangeof f (p) when g(q) = 0. The above discussion shows that uncertainty principle is equivalent to say o < [B, A] and o < [C,D],thus we obtained a picture illustrated by Fig. 4 (b). One can also ask, how about uncertainty relations? Let us emphasize thatuncertainty relation f (p) + g(q) > minρ{ f (p) + g(q)} := b is tangent to the connection line of points B and C in Fig. 4 (c).Finally, we can delineate the boundary of uncertainty region R( f , g) in Fig. 4 (d). As argued in the main text, uncertainty regionas illustrated in Fig. 4 (d) in more informative than both uncertainty principle and uncertainty relation of form f (p) + g(q) > b.

An interesting questions about uncertainty region is whether we can study the boundary of uncertainty region through all thetangent lines of R( f , g). Unfortunately, the answer is negative since the uncertainty region may not be a convex set in general.Explicit examples are given in Fig. 2 (b) and (c) in this Supplemental Material. Therefore this is another reason why theconsiderations of the statistics set R :=

⋃ρ

{(p(ρ),q(ρ))

}in Rn × Rn and our complementary information principle are needed.

∗ Electronic address: mathxiao123@gmail.com† Electronic address: kf383@cam.ac.uk‡ Electronic address: gour@ucalgary.ca

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