Uncertainty, joint uncertainty, and the quantum uncertainty principle
UUncertainty, joint uncertainty, and the quantum uncertainty principle
Varun Narasimhachar, ∗ Alireza Poostindouz, † and Gilad Gour ‡ Department of Mathematics and Statistics and Institute for Quantum Science and Technology,University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4 (Dated: March 10, 2016)Historically, the element of uncertainty in quantum mechanics has been expressed through math-ematical identities called uncertainty relations, a great many of which continue to be discovered.These relations use diverse measures to quantify uncertainty (and joint uncertainty). In this pa-per we use operational information-theoretic principles to identify the common essence of all suchmeasures, thereby defining measure-independent notions of uncertainty and joint uncertainty. Wefind that most existing entropic uncertainty relations use measures of joint uncertainty that yieldthemselves to a small class of operational interpretations. Our notion relaxes this restriction, reveal-ing previously unexplored joint uncertainty measures. To illustrate the utility of our formalism, wederive an uncertainty relation based on one such new measure. We also use our formalism to gaininsight into the conditions under which measure-independent uncertainty relations can be found.
INTRODUCTION
Revealing one of the most striking features of quan-tum mechanics, Heisenberg [1] showed that the out-comes of certain pairs of measurements on a quantumsystem can never be predicted simultaneously with cer-tainty—regardless of how the system is prepared. Heisen-berg’s original statement of what he called the “indeter-minacy” principle concerned potential measurements ofthe position and the momentum of a quantum particle.Many later works [2–6] lent quantitative rigor to Heisen-berg’s original idea and generalized it, both in the num-ber and type of measurements involved and in the mea-sures used to quantify joint uncertainty. At the sametime, Heisenberg himself set off another chain of researchon a related concept: measurement-induced disturbanceand so-called noise-disturbance relations [7–10].Pioneered by Hirschman [4], many works [11–19] haveused entropies to quantify uncertainty, culminating ina recent surge of quantum information-theoretic treat-ments of the uncertainty principle [20–31]. An impor-tant contribution of these recent works is the formulationof uncertainty relations applicable on a quantum systemcorrelated with a quantum memory; such relations areused to strengthen the security proofs of cryptographictasks [32, 33]. These are in addition to existing applica-tions of uncertainty relations in quantum cryptography[34–36], the study of quantum nonlocality [37, 38], andcontinuous-variable quantum information processing [39–42].In all of these areas, the primary ingredient is the con-cept of the uncertainty of a variable, as well as that ofthe joint uncertainty of several variables. The aim of thispaper is to clarify these concepts from an information-theoretic perspective. In the literature, the uncertaintyof a variable has almost always been discussed in terms of ∗ [email protected] † [email protected] ‡ [email protected] measures that quantify “the amount of uncertainty”, e.g.the Shannon entropy and its extended family of R´enyientropies, geometric norm-based measures such as thequadratic variance, etc. In most cases, there is a clearoperational meaning for such measures, rendering themwell-suited to the particular application wherein they areused. Similarly, measures of the joint uncertainty of morethan one variable have been constructed either by consid-ering operational tasks that involve all the variables, orby combining single-variable uncertainty measures math-ematically. In the present work we extract the com-mon thread beneath the operational descriptions of allsuch (single or joint) uncertainty measures, resulting insome basic operational axioms that are independent ofthe measure used to quantify uncertainty, and that de-fine the essence of our concept of uncertainty.These axioms are motivated by information-theoreticprinciples that are intended to be as objective as pos-sible. Considering the challenges inherent in such a re-quirement, we restrict the generality of our treatmentin the following ways. Firstly, we restrict to notionsof uncertainty applied to classical random variables. Inparticular, this class of variables includes the classicaloutcomes of quantum-mechanical measurements. Sec-ondly, we avoid measures of uncertainty that explicitlyinvolve the values of a variable, and instead consider onlysuch measures that depend on the variable’s probabilitydistribution . This necessitates a restriction to discretevariables; in fact, we consider only finite-dimensionalvariables. We make some tentative suggestions for thetreatment of discrete and continuous infinite-dimensionalcases, but leave the actual extension for future work. Fi-nally, in comparing the uncertainties of different variables(which a measure of uncertainty should naturally be ex-pected to enable), we will require the compared variablesto represent the same type of physical quantity. For ex-ample, a comparison between the uncertainties in twodifferent length variables will be possible within our for-malism, but not one between a length uncertainty and amass uncertainty.The crux of this paper are the following axioms: (1) a r X i v : . [ qu a n t - ph ] M a r One’s knowledge about a variable cannot increase un-der any processing without addition of new informationabout the variable; (2) The uncertainty in a variablerepresenting a physical observable is invariant under thesymmetries of the observable; and (3) The joint uncer-tainty of several variables is a valid concept even with-out an underlying operational description that combinesthose variables.The first two axioms are inspired by earlier approaches[24–27] to measure-independent notions of uncertainty,wherein the connection between uncertainty and a math-ematical concept called majorization [43] was utilized.Majorization is a hierarchy among probability distribu-tions, induced by the action of a class of transformationscalled doubly stochastic maps. In this paper, by findinga mathematical characterization of mechanisms that canincrease a variable’s uncertainty, we gain an operationalunderstanding of why, and to what extent, majorizationplays a role in characterizing uncertainty.First, we find that for variables with unrestricted sym-metries, uncertainty-increasing mechanisms are associ-ated with the set of all doubly stochastic maps, leading tothe emergence of majorization as the relation determin-ing uncertainty. A function that quantifies uncertaintymust then possess the property of never decreasing un-der any doubly stochastic maps. On the other hand, withrestricted symmetries, only certain sub-classes of doublystochastic matrices feature. The resulting hierarchy isthen different from majorization, and a measure of un-certainty is required to be non-decreasing only under therestricted classes of doubly stochastic maps. This opensup more options for functions that can serve as uncer-tainty measures for variables with restricted symmetries.Another element of novelty in our work lies in the thirdof our axioms, concerning joint uncertainty. In the con-text of physics, we can rephrase this axiom in terms of ex-periments: Suppose that we are interested in quantifyingthe joint uncertainty of several experiments, e.g. in con-nection with the quantum uncertainty principle, wherethe several experiments are different quantum measure-ments. One approach would be to construct new exper-iments that combine the original experiments in someway. For example, consider the following combined ex-periments constructed from a given set of experiments:(a) all the original experiments are performed indepen-dently; (b) all the apparatuses are set up, but only oneof the experiments is chosen at random and performed.The uncertainty in the outcome of such a combinedexperiment would quantify the joint uncertainty of theconstituent experiments. But we see that there are dif-ferent ways to combine experiments, which all capturedifferent aspects of the joint uncertainty. In this paperwe argue that the richness of joint uncertainty is not cap-tured even by considering all such combined experiments.The most general notion of joint uncertainty is devoidof the particulars of such combinations, and allows allthe component experiments to be, in principle, counter-factual. To illustrate this, we consider an extensively- studied type of quantum uncertainty relations: the so-called preparational uncertainty relations. For ease ofexplanation, let’s consider a two-measurement prepara-tional uncertainty relation, which has the generic form J ( p ( ρ ) , q ( ρ )) ≥ c, (1)where J is a measure of the joint uncertainty of twovariables, and p ( ρ ) and q ( ρ ) are the expected outcomeprobability distributions of a pair of measurements per-formed on a quantum state represented by the densityoperator ρ (our arguments can be extended to more thantwo measurements). We show that most existing prepa-rational uncertainty relations can be subjected to one ofthe specific operational interpretations (a) and (b) men-tioned above. To show that these two interpretations areunnecessarily restrictive, we construct joint uncertaintymeasures that cannot be interpreted either way. We goon to derive an uncertainty relation based on one suchmeasure, which is a relation nontrivially different from allthe ones discovered in the past. The main purpose of de-riving this new relation is to demonstrate the possibilitiesopened up by our joint uncertainty axiom.Another contribution of this paper is a deeper un-derstanding of so-called universal uncertainty relations found in [24–27]: pairs of vectors ( u , v ) such that J ( u , v )provides a nontrivial bound [like the c of Eq. (1)] for awhole class of measures, J ∈ J . We find that no univer-sal relations exist if J includes all possible measures; how-ever, restricting to specific operational frameworks [usingthe (1) and (2) types of combined experiments discussedin the previous paragraph] is what makes the nontrivialuniversal relations found in [24–27] possible.Even though we focus on preparational uncertainty re-lations in quantum mechanics, in principle our notionscan be applied to any situation where probability-baseduncertainty measures of classical variables are relevant.We summarize the possible applications in the conclu-sion, along with open problems. I. WHAT IS UNCERTAINTY?
We will now develop a notion of uncertainty that canbe applied to finite-dimensional classical variables. Inparticular, we seek a general method of comparing theuncertainties of two variables, in such a way that thecomparison gives the same verdict independent of thefunction used to measure uncertainty. For our purpose,it will be sufficient to be able to compare physically simi-lar variables, that is, variables representing the same un-derlying physical quantity; for example, comparing theuncertainty of a length with that of another length. Wewill not concern ourselves with how a comparison can bemade between dissimilar variables.Consider an experiment where Alice is about to roll a(possibly biased) die, whose faces she calls “1”, “2”. . . ,“6”. The eventual outcome of the roll will be a value x ∈ { , . . . , } , but since we don’t know x a priori, werepresent it as a random variable X ≡ { ( x, p x ) } . Whatis Alice’s minimum uncertainty about X prior to the ex-periment? We could answer this question in differentways, some of which might appeal to the particular la-bels that Alice uses to call her outcome. For example,the difference between the largest and smallest possibleoutcomes that have a non-zero probability is an uncer-tainty indicator, and it depends on the choice of labels. Inprinciple, Alice could relabel her die’s faces to, say, “a”,“b”, etc., without changing the essential physical natureof the experiment. We will require our notion of un-certainty to make no distinction between two physicallyidentical experiments that differ only in the outcome la-bels. In other words, we will consider uncertainty to be aproperty of just the distribution p X , measured possiblyby some real-valued function U ( p X ). In fact, an evenstronger restriction follows. Let Y be a random variableobtained by merely relabeling the different values of X .The probability distribution p Y of Y must then necessar-ily contain the same values as p X , possibly differing onlyin their order. Therefore, the effect of any relabeling on p X is as though the original labels were just permutedamongst themselves: p X (cid:55)→ M ( π ) p X . In this sense, per-mutations, although not the only possible way to relabeloutcomes, still capture the effect of arbitrary relabelings,as far as our notion of uncertainty is concerned.If, instead of a die-roll outcome, X were a physicalproperty, e.g. the energy of a quantum harmonic oscil-lator, arbitrary permutations could result in loss of thevariable’s physical meaning. To avoid this, we would haveto restrict the permutations, e.g. to only shifts in the en-ergy. In general, the restricted class of reorderings is thegroup G of symmetries of the observable underlying X ,with each symmetry g corresponding to a change in one’sreference frame . For finite-dimensional observables, G isa subgroup of the group of all permutations.Our first requirement from a measure U of uncertaintyis that it be invariant under the symmetry group G of theunderlying observable. This immediately leads to the fol-lowing: Two variables X and Y , both representing thesame observable, are equally uncertain if their distribu-tion vectors are related by some g ∈ G . If X is theoutcome of a certain experiment, the random variables Y that are equally uncertain to X include relabeled (un-der G ) versions of X (which are perfectly correlated with X ); the outcomes of other runs of the same experimentwith the same apparatus (which may be correlated with X if the apparatus has a memory); outcomes of the sameexperiment performed on independent but identical ap-paratuses (uncorrelated with X ); and in general any Y representing the same observable, with p Y = M ( g ) p X forsome g ∈ G .Thus far, we have found a way to tell when the uncer-tainties of two variables are equal. Now we will developa method of determining when and how the uncertaintyof one variable can be said to be more, or less, than thatof another. To this end, we will first identify certainty- FIG. 1. Bob tries to estimate his uncertainty about Alice’svariable X after it has been corrupted by channel T into Y . nonincreasing transformations: processes that take anygiven variable X to an equally- or more-uncertain one, ˜ X ,by virtue of a “randomizing” or “forgetting” mechanism.Thereafter, we will use the following rule to compare theuncertainties of two variables X and Y (arbitrary butwith the same underlying physical observable): Y is atleast as uncertain as X if some uncertainty-increasingtransformation of X results in a variable ˜ X that has thesame probability distribution as Y up to the symmetriesof the underlying observable.In order to identify the certainty-nonincreasing trans-formations, we will now construct a couple of extendedversions of the “Alice rolls a die” thought experiment.First, consider the modified experiment depicted in Fig. 1[44]: After rolling her die, Alice sends the outcome x toher collaborator Bob (who doesn’t even know the biasdistribution of Alice’s die) via some classical channel [45]given by the column-stochastic matrix T ≡ ( T y | x ). Herelet’s pause to reflect upon the uncertainty in the output Y of the channel. The channel could transmit x perfectly,or with some added noise. In these cases the output Y isequally or more uncertain than X . On the other hand,the channel could also completely ignore x and outputsome constant value, in which case the uncertainty of Y could be less than that of X . In fact, the processingmight result in information in a fundamentally differ-ent form from X . For example, Alice could just sendthe parity of her die outcome to Bob, in which case Y doesn’t even represent the same underlying observable as X . Therefore, we cannot make a general statement aboutnon-increase of certainty under an arbitrary channel.However, instead of the uncertainty of Y itself, we canconsider the following question: How much informationdoes Y contain about x ? Since Y results from processing X with the possible addition of noise or irrelevant infor-mation, it cannot tell us more about x than X does. Inorder to lend mathematical rigor to this statement, wemust extract from Y some variable that has the samephysical meaning as X , so that we can treat them bothon an equal footing. Now let’s return to Alice and Bob’sexperiment: Bob, who knows T but not p X , now triesto recover x from the channel output y , which is a pri-ori distributed according to q Y = T p X . Since this gameis being designed to analyze uncertainty about x , Bob’saim in his recovery task is not to maximize his chancesof guessing x correctly, but rather to faithfully accountfor the uncertainty that Y contains about x . Suppose hesees an instance Y = y . This could have resulted froma particular X = x (cid:48) with conditional probability T y | x (cid:48) .Without knowing the prior p X , Bob’s rational guess forthe likelihood that X = x (among all the possible x (cid:48) ) isgiven by R x | y = T y | x (cid:80) x (cid:48) T y | x (cid:48) . (2)The resulting distribution of Bob’s recovered variable(call it ˜ X ) is given by the composite action of T and R on p X : p rec˜ X = RT p X =: D rec p X . (3)Since Y could contain irrelevant information, its uncer-tainty cannot be interpreted as “uncertainty about x ”.On the other hand, X directly represents x , while ˜ X re-sults from extracting out of Y precisely all the informa-tion it contains about x . Therefore, these two variablesboth represent the same physical observable as x , andtheir uncertainties directly quantify uncertainty about x .This equal physical footing also ensures that their uncer-tainties can be compared under our rules. This compari-son tells us that the uncertainty of ˜ X cannot be less thanthat of X . The cumulative transformation that takes X to ˜ X is therefore a certainty-nonincreasing transfor-mation . It can be verified easily that for any column-stochastic T , with the corresponding R [46] constructedas in (2), the matrix D rec = RT is doubly stochastic.The (necessarily degenerative) evolution of the informa-tion about some entity (like x ), when the representationof this information is subjected to any classical process-ing (represented by the action of the channel), is alwaysvia such matrices, whose collection we call D rec .We saw that, after the action of a generic channel, theuncertainty of the final variable Y doesn’t have a consis-tent hierarchical relationship with that of the initial vari-able X . In order to draw a consistent rule of certaintynon-increase, we had to consider a recovery transforma-tion from Y to ˜ X . But there are certain special trans-formations that always result in certainty non-increase,even without the addition of a recovery transformation.In fact, we already saw an example: symmetry trans-formations of the underlying physical observable. In thedie-roll example, symmetry transformations include non-identity permutations, which can easily be shown to beoutside of the die’s D rec class, yet result in final vari-ables Y with the same physical meaning and (consis-tently) no less uncertain than X . We will find a familyof such certainty-nonincreasing transformations by con-sidering another thought experiment, depicted in Fig. 2:Before rolling her die, Alice will toss a coin; she will then FIG. 2. Extended die-roll experiment: Alice relabels her diebased on a coin toss, then rolls the relabeled die. The outcomeof this experiment is more uncertain than a simple die roll. relabel the die’s faces with a permutation that is deter-mined by the outcome of this coin toss, and then roll thedie. The random choice of relabeling makes the outcome Y of this modified experiment more uncertain than X .In general, if a variable X is transformed by applying a g ∈ G chosen at random under a distribution t ≡ ( t g ),the resulting variable Y is distributed as q Y = D sym p X ,where D sym = (cid:80) g ∈ G t g M ( g ) . Since each M ( g ) is a per-mutation, every possible D sym is doubly stochastic. Wedenote by D sym the set of all such D sym matrices. Ifthe observable’s symmetry group G includes all permu-tations, then by Birkhoff’s theorem [47, 48] D sym is theset of all doubly stochastic matrices, but a restricted G results in a corresponding shrinkage of D sym .The characterization of the classes D rec and D sym is aninteresting problem that we leave for future work. Whilethe latter class depends on the symmetry group of the ob-servable, the former depends only on the dimensionality.For a variable with complete permutation symmetry, asnoted above, D sym contains all doubly stochastic matri-ces, in particular all of D rec . But under restricted sym-metries, each class can contain members not belongingto the other. For instance, take a 3-dimensional variablewhose symmetry group is the (order-3) group of cyclicpermutations of the components. The two nontrivial per-mutations are transformations contained (by design) in D sym , but not in D rec . On the other hand, the matrix . .
50 0 . . is in D rec , but not in D sym . Therefore, the structure ofthe union of these classes cannot be reduced to either oneof the classes. This example can be generalized naturallyto higher dimensions.Due to our restriction to uncertainty comparison be-tween physically-similar variables, the “sym” and “rec”classes of doubly stochastic matrices together suffice asmechanisms of uncertainty increase. In principle, anyfunction U ( p X ) meant to measure the uncertainty of X is required to increase under both these matrix classes.But the “sym” class is more important that the “rec”:the former is based on the natural symmetries of an ob-servable, and therefore the constraints that it induces onuncertainty measures are inviolable. On the other hand,“rec”, even though it is an essential ingredient in thestrictest information-theoretic definition of uncertainty,could be ignored in natural situations where information-processing is not involved. Functions that respect the“sym” constraints, but violate the “rec” ones, neverthe-less turn out to be useful indicators of uncertainty. Basedon these considerations, we define: Definition 1. A measure of uncertainty of a variable X is a function U of the distribution p ≡ p X of the variable,satisfying U ( D p ) ≥ U ( p ) ∀ D ∈ D sym ; (4) U ( D p ) ≥ U ( p ) ∀ D ∈ D rec . (5) Here the class “ sym ” is determined by the symmetries ofthe variable’s underlying physical observable. A functionthat satisfies (4), but not (5), will be considered a weakmeasure of uncertainty . If the symmetry group G of a finite-dimensional X contains all permutations, then functions that satisfy (4)are called Schur-concave functions [43]. Examples of suchfunctions are the entropies of Shannon, R´enyi, and Tsal-lis. Now, Hardy et al. [49] proved that the existence ofa doubly stochastic D such that q = D p is equivalentto the binary relation p (cid:31) q , read “ p majorizes q ” [43],which for a general d -dimensional vector space is definedas follows. Define p ↓ and q ↓ as the same vectors withtheir components arranged in nonincreasing order. Then, p (cid:31) q if, and only if, k (cid:88) i =1 p ↓ i ≥ k (cid:88) i =1 q ↓ i ∀ k ∈ { , . . . , d } . (6)The “completely certain” and “completely uncertain”distributions e ≡ (1; 0 . . . ; 0) and u ≡ (1 /d ; 1 /d . . . ; 1 /d )satisfy e (cid:31) p (cid:31) u , ∀ p .If a variable has restricted symmetries, then the un-certainty hierarchy of its distributions becomes differentfrom the majorization hierarchy. All Schur-concave func-tions still remain valid uncertainty measures. But inaddition, by virtue of the reduction in the class D sym ,some non–Schur-concave functions could also qualify tobe weak measures of uncertainty [i.e., may violate condi-tion (5)]. For example, for a finite-dimensional variable X whose symmetries are cyclic permutations, it can beeasily shown that the variance of X is only a weak uncer-tainty measure. Generalizing the classes D sym and D rec for discrete-infinite and continuous variables may not bestraightforward, and we leave it for future work. We ex-pect it to be possible to achieve such a generalizationby considering parametrized families of symmetries (e.g.Lorenz transformations) and convex combinations (inte-grals) over different parameter assignments. II. JOINT UNCERTAINTY
The uncertainty of the outcomes of individual experi-ments cannot provide a complete description of the quan-tum uncertainty principle, since most uncertainty rela-tions are lower bounds on measures of the joint uncer-tainty of the outcomes of at least two measurements.For clarity of discussion, here we will restrict to pairs ofexperiments, each with a finite number of possible out-comes; extension to more experiments is straightforward.To motivate our definition of joint uncertainty, considerthe following hypothetical scenarios involving the jointuncertainty of a coin-toss outcome, X , and a die-roll out-come, Y : Example 1 : Perform the combined experiment compris-ing an independent and simultaneous performance of both the original experiments [Fig. 3 (a)]. The outcome is Z ≡ ( X, Y ), which has | X || Y | = 12 possible values, dis-trubuted as p Z = p X ⊗ p Y . Therefore, U ( p X ⊗ p Y ), for U any single-variable uncertainty measure (in the sense ofDef. 1), serves as a joint uncertainty measure of X and Y . Most measures considered in the literature on thequantum uncertainty principle, e.g. the sum of Shannonentropies of the individual outcome distributions, can beinterpreted through such a combined experiment. Example 2 : This time we first toss a second coin to makea choice between the actions “toss the coin” (resulting inoutcome X ) and “roll the die” (leading to Y ), and thenperform only the chosen action [Fig. 3 (b)]. The outcome Z of this experiment has | X | + | Y | = 8 possible values,whose uncertainty (modulo the uncertainty in the choiceof action) is also a manifestation of the joint uncertaintyof ( X, Y ). In this case, if the choice coin is unbiased, p Z = (cid:0) p X ⊕ p Y (cid:1) and therefore we get measures ofthe form U (cid:0) p X ⊕ p Y (cid:1) . The measures of joint uncer-tainty proposed in [37] can be interpreted through sucha combined experiment.As these scenarios illustrate, there could be differ-ent ways in which experiments could be combined intoone super-experiment, the uncertainty of whose outcomesthen reflects an aspect of the joint uncertainty of ( X, Y ).But the essence of joint uncertainty is not quite cap-tured by any one of these joint experiments. In fact,some joint uncertainty measures, such as the functions H α ( p X ) + H β ( p Y ) (where H α and H β are R´enyi en-tropies) [12], and even Heisenberg’s ∆ x ∆ p , cannot beinterpreted as the uncertainty of any single combinedexperiment. The quantum uncertainty principle appliesalso to cases with several potential measurements , eacha potential (actual or counterfactual) experiment in itsown right.These considerations indicate that the notion of jointuncertainty is not bound to the concept of combined ex-periments. What, then, are the essential properties of ameasure of joint uncertainty? Firstly, the pairs ( X, Y )that have the least joint uncertainty are ones where bothdistributions are completely certain. The most jointly-uncertain pairs, on the other hand, are the ones where
FIG. 3. Two possible operational combinations of a coin-toss experiment and a die-roll experiment: (a) Both experiments areperformed simultaneously and independently; (b) Only one of the two experiments is performed, based on a random choice. both variables are completely uncertain. Furthermore,all the measures of the joint uncertainty of (
X, Y ) arereal-valued functions of the distributions p ≡ p X and q ≡ q Y , and must reduce to the measures of single-variable uncertainty (as in Def. 1) if one of the vectors p and q is kept fixed. This brings us to the followingdefinition: Definition 2.
A measure of joint uncertainty of twovariables X and Y is a real-valued function J of ( p , q ) ≡ ( p X , q Y ) , such that J ( D p , D q ) ≥ J ( p , q ) (7) for all doubly stochastic matrices D , D in the respective“ sym ” and “ rec ” classes of both variables. As in thesingle-variable case, we will call functions satisfying (7)for the “ sym ” class, but not for the “ rec ” class, weakmeasures of joint uncertainty . It can be verified that this definition applies to entropicjoint uncertainty measures of the form f ( p )+ g ( p ), where f and g are single-variable uncertainty measures. Thevast majority of the literature on entropic uncertaintyrelations uses such measures. Note that if the symmetrygroups of both variables are the respective full permu-tation groups, then D and D can be any two doublystochastic matrices of appropriate dimensions. In thiscase, the relation in (7) states that J is monotonic un-der the direct product relation “ Ï ” defined by:( p , q ) Ï ( p , q ) ⇔ ( p (cid:31) p and q (cid:31) q ) . III. THE QUANTUM-MECHANICALUNCERTAINTY PRINCIPLE
The “uncertainty principle” of quantum mechanics isactually a collection of identities known as uncertainty re-lations (UR’s), all concerning the uncertainties of individ-ual quantum-mechanical measurements, as well as jointuncertainties of sets of two or more (actual or counter-factual) measurements. Broadly, there are three differentoperational contexts of UR’s: different measurements ap-plied on the same quantum state (either counterfactuallyor by preparing many copies of the same state); simulta-neous (approximate) execution of several measurements; and sequential execution of several measurements. Thenotions that we developed in the last two sections can beapplied in all of these contexts, since they all include in-stances of finite-dimensional classical variables. But herewe will focus on the first type of situation, where differentmeasurements are considered on identical preparations.Furthermore, we restrict to UR’s that involve only theprobability distributions of measurement outcomes, andnot the “values” assigned to the outcomes.Since these UR’s involve only the probabilities of out-comes, a positive-operator–valued measure (POVM) de-scription of measurements is adequate in the formal-ism. Consider the case of two POVM’s
A ≡ { Π a } a and B ≡ { Γ b } b . For a quantum state ρ , measurement A leads to outcome probability distribution p ( ρ ) where p a ( ρ ) = Tr [Π a ρ ], and B to q ( ρ ) with q b ( ρ ) = Tr [Γ b ρ ].For a so-called incompatible pair of POVM’s ( A , B ), thereis no ρ that results in both p ( ρ ) and q ( ρ ) completely cer-tain, leading to the existence of a “minimal joint uncer-tainty”. Many UR’s are statements to this effect: J ( p ( ρ ) , q ( ρ )) ≥ c ∀ ρ, (8)where J is a measure of joint uncertainty, and 0 We could construct various uncertainty relations usingthe aforementioned recipe, with the given pair ( A , B ) and different measures J . Every relation is stated in terms ofa lower bound like the c of (8), which in turn depends on J . In general, for a given J it might be hard to computesuch a bound. But suppose there were a fixed pair ( u , v )of distribution vectors, such that J [ p ( ρ ) , q ( ρ )] ≥ J ( u , v ) ∀ ρ, J . (11)If there were such a pair, then for any given J we wouldmerely have to compute J ( u , v ), immediately yielding abound. In this sense, finding such a pair would amount tofinding a plethora of uncertainty relations; therefore, sucha pair can be said to constitute a universal uncertaintyrelation for the pair ( A , B ) [24, 25].As it turns out, a nontrivial pair satisfying (11) neverexists for any given ( A , B ), because the clause “ ∀J ” in(11) includes all single-uncertainty measures of p and q alone, leading necessarily to the trivial choice ( u , v ),where u (cid:31) p ( ρ ) and v (cid:31) q ( ρ ) for all ρ . Such a ( u , v )would be unhelpful in that it wouldn’t impose joint re-strictions on ( p , q ). In order to avoid this triviality, wecan relax the condition “ ∀J ”, and instead require theinequality in (11) to only hold for some restricted classof J ’s.Here we consider again the two restricted combined-experiment scenarios that we discussed in Section II. Inthe first scenario, both A and B are carried out indepen-dently of each other on copies of the same state ρ . The re-stricted class of joint uncertainty measures then consistsof functions of the form J ( p ( ρ ) , q ( ρ )) = U ( p ( ρ ) ⊗ q ( ρ )).In Ref. [24, 25], it was shown that for any given A , B thereexists a distribution vector u ≡ ω ( A , B ) such that thepair ( u , e ) forms a universal uncertainty relation underthis restricted class of joint uncertainty measures.Similarly, following Example 2 of Section II, we canconsider a combination wherein we first pick, at random,only one of the two measurements A and B , and thenperform that one. The joint uncertainty measures con-sidered here are of the form U ( p ( ρ ) ⊕ q ( ρ )). A nontriv-ial ( u , v ) for this restricted class can be found using themethods in [26, 27].It might be possible to unify the spirit of the above twoclasses of universal relations into a larger class, by includ-ing all measures of joint uncertainty that are symmetric in the two (or more) distributions: J ( p , q ) = J ( q , p ).This requirement avoids the case of trivial relations re-sulting from the requirement ( u , v ) Ï ( p , q ), but weleave it open whether a nontrivial ( u , v ) can be found.Another way of unifiying several classes of universal re-lations, each with its respective ( u i , v i ), is by boundingany measure J as follows: J ( p ( ρ ) , q ( ρ )) ≥ min j ∈{ ,...,m } J ( u j , v j ) ∀ ρ, J . (12)An interesting open problem is whether there exists afinite integer m such that minimizing over all j ≤ m provides a nontrivial bound for all nontrivial joint uncer-tainty measures.Universal uncertainty relations are a powerful toolinasmuch as they generate a variety of uncertainty re-lations, but the bounds they yield may not be tight. Be-sides, there are joint uncertainty measures that may notlend themselves to inclusion in a class that admits a non-trivial universal relation, but nevertheless do provide anontrivial uncertainty relation. An example is the mea-sure J ( p , q ) = 1 − p ↓ · q ↓ , for which we found a UR inthe previous section. V. CONCLUSION In this paper, we identified the most basic, measure-independent elements of the concept of uncertainty asapplicable to finite-dimensional classical variables. Webased our analysis on an information-theoretic study ofmechanisms of uncertainty increase: randomly-chosensymmetry transformations; and classical processing viachannels (followed by recovery). Corresponding to these,we identified two classes of doubly stochastic matrices, D sym and D rec . Uncertainty measures in the strictestsense must be monotonically non-decreasing under boththese classes.We then took a similar information-theoretic approachto the concept of joint uncertainty of several variables,resulting in the principle that the most basic featuresof joint uncertainty measures must not depend on spe-cific operational combinations of the variables. We thenconsidered quantum uncertainty relations (UR’s) of thepreparational uncertainty type, where past works havealways considered specific operational combinations. Ap-plying our new notion of joint uncertainty not only re-sulted in a unified understanding of a large class of UR’s,but also opened up the possibility of deriving a new classof preparational UR’s, namely identities that are math-ematically valid for any preparation, but cannot be in-terpreted based on any single experimental scenario. Toillustrate, we constructed a class of joint uncertainty mea-sures with this property, and derived a new UR using oneof these measures as an example. Finally, we found that so-called universal uncertainty relations cannot be foundover all possible measures of joint uncertainty. We con-nected universal relations found in past works [24–27]with specific operational interpretations of joint uncer-tainty.In cryptographic tasks we must consider the uncer-tainty of systems that could be correlated with quan-tum memories in adversarial control; our recent work[50] is a step towards developing a measure-independentnotion of such conditional uncertainty . More generally,a formalism for treating the uncertainty of quantum in-formation correlated with quantum memories is not yetdeveloped. A more complete characterization of uncer-tainty on infinite-dimensional systems is another chal-lenging future project. This could impact applicationsof squeezed states, which are ubiquitous in quantum in-formation processing with continuous variables. Yet an-other open problem is to improve our understanding ofuniversal uncertainty relations; in particular, to answerthe open questions posed at the end of Section IV re-garding stronger classes of universal relations. Finally,there is much to be understood about the classes D rec and D sym of doubly stochastic matrices. VI. ACKNOWLEDGMENTS GG is grateful for many interesting discussions withMicha(cid:32)l Horodecki, Amir Kalev, Iman Marvian, and RobSpekkens. In particular, we are grateful to Iman Mar-vian and Rob Spekkens, for pointing out to us the roleof symmetry in quantum uncertainty relations. 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Appendix A: Example uncertainty relation forprojective measurements on pure qubit states On a two-level quantum system (a qubit) in a purestate | ψ (cid:105) , consider two rank-1 projective measurements A and B , respectively defined by the orthonormal bases {| x (cid:105) , | x (cid:105)} and {| y (cid:105) , | y (cid:105)} . When A is applied on | ψ (cid:105) ,the resulting outcome distribution p ( ψ ) has the compo-nents p ( ψ ) =: p ( ψ ) = |(cid:104) x | ψ (cid:105)| and p ( ψ ) = 1 − p ( ψ );similarly, we denote the distribution of the outcomes of B q ( ψ ) ≡ ( q ( ψ ) , − q ( ψ )). We shall now find a lower boundon the minimum joint uncertainty of ( p ( ψ ) , q ( ψ )), overall pure states | ψ (cid:105) , under the measure J ( p , q ) = 1 − p ↓ · q ↓ . (A1)Note that J is a valid measure of joint uncertaintyas it satisfies the constraints in Definition 2. We canpartition the set of all pure states into two subsets S and S given by S = {| ψ (cid:105)| ( p (cid:62) . , q (cid:62) . 5) or ( p < . , q < . } ; S = {| ψ (cid:105)| ( p (cid:62) . , q < . 5) or ( p < . , q (cid:62) . } , where p and q are understood to be ψ -dependent. Thefunction J can be defined piecewise using this partitionas J ( ψ ) = (cid:26) p + q − pq, | ψ (cid:105) ∈ S ;1 − p − q + 2 pq, | ψ (cid:105) ∈ S . (A2)Modulo a global phase, | ψ (cid:105) can be parametrized as | ψ (cid:105) = cos α | y (cid:105) + e i ϕ sin α | y (cid:105) , (A3)with α ∈ [0 , π/ 2] and ϕ ∈ [0 , π ). Appropriate global phases can be added to the measurement basis vectors sothat q ( ψ ) = cos α ; p ( ψ ) = (cid:12)(cid:12) cos α cos β + sin α sin β e i ϕ (cid:12)(cid:12) , where cos ( β ) := |(cid:104) x | y (cid:105)| . The minimization of the func-tion J using its piecewise definition (A2) can be doneby separately minimizing over S and S and then find-ing the smaller of the two minima. Let us first consider S . Now, it can be verified that the ϕ dependence of thefunction is through a term of the form f ( α, β ) sin ( ϕ/ ϕ alone, and then over all α . In the cases where f ( α, β ) > φ is achieved when sin ( ϕ/ 2) = 1; if f ( α, β ) < 0, the minimim occurs when sin ( ϕ/ 2) = 0.In either case, the minimum over ϕ , as a function of α ,takes the formmin ϕ J ( ψ ) =: J ( α )= cos α + cos ( β ± α ) − α cos ( β ± α ) . Since J is even in α , the subsequent minimization over α leads to the same value regardless of whether the positiveor negative sign is used in the ± above. One can checkthat the minimum is attained at α = β/ 2, yieldingmin | ψ (cid:105)∈S J = 12 sin β. (A4)Using similar arguments, we can determine the minimumover the other partition:min | ψ (cid:105)∈S J = 12 cos β. (A5)Note that without loss of generality we can take 0 (cid:54) β (cid:54) π/ 2. Comparing the two local minima, we can expressthe global minimum succinctly asmin | ψ (cid:105) J ( p , q ) = 12 (1 − η ) , (A6)where η := max i,j |(cid:104) x i | y j (cid:105)|(cid:105)|