Additivity of Entropic Uncertainty Relations

Additivity of entropic uncertainty relations

René Schwonnek

Institut für Theoretische Physik, Leibniz Universität Hannover, Germany March 28, 2018

We consider the uncertainty between two bound cABC... gives the central estimate in many ap- pairs of local projective measurements per- plications like: entropic steering witnesses [1–4], un- formed on a multipartite system. We show certainty relations with side-information [5], some se- that the optimal bound in any linear uncer- curity proofs [6] and many more. tainty relation, formulated in terms of the When speaking about uncertainty, we consider so Shannon entropy, is additive. This directly im- called preparation uncertainty relations [7–14]. From plies, against naive intuition, that the minimal an operational point of view, a preparation uncer- entropic uncertainty can always be realized by tainty describes fundamental limitations, i.e. a trade- fully separable states. Hence, in contradic- off, on the certainty of predicting outcomes of sev- tion to proposals by other authors, no entan- eral measurements that are performed on instances glement witness can be constructed solely by of the same state. This should not be confused [15] comparing the attainable uncertainties of en- with its operational counterpart named measurement tangled and separable states. However, our re- uncertainty[16–20]. A measurement uncertainty rela- sult gives rise to a huge simplification for com- tion describes the ability of producing a measurement puting global uncertainty bounds as they now device which approximates several incompatible mea- can be deduced from local ones. surement devices in one shot. Furthermore, we provide the natural gener- The calculations in this work focus on uncertainty alization of the Maassen and Uffink inequality relations in a bipartite setting. However, all results for linear uncertainty relations with arbitrary can easily be generalized to a multipartite setting by positive coefficients. an iteration of statements on bipartitions. The basic measurement setting, which we consider for bipartitions, is depicted in Fig.1. We consider a pair

Introduction XA A 1 Uncertainty and entanglement are doubtless two of 0 the most prominent and drastic properties that set YA apart quantum physics from a classical view on the λ pXAB ρAB world. Their interplay contains a rich structure, {1, 0} ⊕ which is neither suﬃciently understood nor fully dis- µ p covered. In this work, we reveal a new aspect of this XB YAB structure: the additivity of entropic uncertainty rela- 1 tions. 0 For product measurements in a multipartition, we B YB show that the optimal bound cABC... in a linear uncertainty relation satisﬁes Figure 1: Basic setting of product measurements on a biparti- arXiv:1801.04602v4 [quant-ph] 27 Mar 2018 tion: pairs of measurements XA,XB or YA,YB are applied to cABC··· = cA + cB + cC + ..., (1) a joint state ρAB at the respective sides of a bipartition. One bit of information is transmitted for communicating whether where cA, cB, cC ,... are bounds that only depend on the X or the Y measurements are performed. The weights local measurements. This result implies that minimal (λ, µ) denote the probabilities corresponding to this choice. uncertainty for product measurements can always be realized by uncorrelated states. Hence, we have an of measurements, XAB = XAXB and YAB = YAYB, example for a task which is not improved by the use to which we will refer as the global measurements of of entanglement. (tensor) product form. Each of those global measure- We will quantify the uncertainty of a measurement ments of product form is implemented by applying by the Shannon entropy of its outcome distribution. local measurements at the respective sides of a bipar- For this case, the corresponding linear uncertainty tition between parties denoted by A and B. Hereby,

Accepted in Quantum 2018-03-20, click title to verify 1 the variables XA,XB and YA,YB will refer to those tropy, tells us that the uncertainty of a global mea- local measurements applied to the respective sides. surement is simply the sum of the uncertainty of the We only consider projective measurements, but be- local ones. In the same way any trade-off on the global side this we impose no further restrictions on the in- uncertainties can be deduced from local ones. dividual measurements. So the only property that If the measured state is classically correlated, measurements like XA and XB have to share is the i.e a convex combination of product states [22], ad- common label ’X’, besides this, they could be non- ditivity of local uncertainties does not longer hold. commuting or even defined on Hilbert spaces with More generally, whenever we consider a concave un- different dimensions. certainty measure [23], like the Shannon entropy, the The main result of this work is stated in Prop.1 in global uncertainty of a single global measurement is Sec.3. In that section, we also collect some remarks smaller than the sum of the local uncertainties. Intu- on possible and impossible generalizations and the itively this makes sense because a correlation allows construction of entanglement witnesses. The proof of to deduce information on the potential measurement Prop.1 is placed at the end of this paper, as it relies outcomes of one side given a particular measurement on two basic theorems stated in Sec.4 and Sec.5. outcome on the other. However, a linear uncertainty Thm.1, in Sec.4, clarifies and expands the known relation for a pair of global measurements is not af- connection between the logarithm of (p, q)-norms and fected by this, i.e a trade-off will again be saturated entropic uncertainty relations. As a special case of by product states. This is because the uncertainty re- this theorem we obtain Lem.1 which states the nat- lation between two measurements, restricted to some ural generalization of the well known Maassen and convex set of states, will always be attained on an Uffink bound [21] to weighted uncertainty relations. extreme point of this set. Thm.2, in Sec.5, states that (p, q)-norms, in a certain However, if measurements are applied to an entan- parameter range, are multiplicative, which at the end gled state, more precisely to a state which shows leads to the desired statement on the additivity of EPR-steering [24–26] with respect to the measure- uncertainty relations. ments XAB and YAB, it is in general not clear how Before stating the main result, we collect, in Sec.1, a trade-off between global uncertainties relates to the some general observations on the behavior of uncer- corresponding trade-off between local ones. Just have tainty relations for product measurements with re- in mind that steering implies the absence of any lo- spect to different classes of correlated states. Fur- cal state model, which is usually proven by showing thermore, in Sec.2, we will motivate and explain the that any such model would violate a local uncertainty explicit form of linear uncertainty relations used in relation. this work. In principle one would expect to obtain smaller uncertainty bounds by also considering entangled states, and there are many entanglement witnesses known 1 Uncertainty in bipartitions based on this idea (see also Rem.3 in the following section). All uncertainty relations considered is this paper are state-independent. In practice, finding a state- independent relation leads to the problem of jointly 2 Linear uncertainty relations minimizing a tuple of given uncertainty measures, here the Shannon entropy of XAB and YAB, over all We note that there are many uncertainty measures, states. This minimum, or a lower bound on it, then most prominently variances [8, 10]. Variance, and gives the aforementioned trade-off, which then allows similar constructed measures [17, 27], describe the to formulate statements like: "whenever the uncer- deviation from a mean value, which clearly demands tainty of XAB is small, the uncertainty of YAB has to to assign a metric structure to the set of measure- be bigger than some state-independent constant" . ment outcomes. From a physicist’s perspective this Considering the measured state, ρAB, it is natural makes sense in many situations [11] but can also cause to distinguish between the three classes: uncorrelated, strange behaviours in situations where this metric classically correlated and non-classical correlated. In structure has to be imposed artificially [28]. However, regard of the uncertainty in a corresponding global from the perspective of information theory, this seems measurement, states in these classes share some com- to be an unnecessary dependency. Especially when mon features: uncertainties with respect to multipartitions are con- If the measured state is uncorrelated, i.e a prod- sidered, it is not clear at all how such a metric should uct state ρAB = ρA ⊗ ρB, the outcomes of the lo- be constructed. Hence, it can be dropped and a quan- cal measurements are uncorrelated as well. Hence, tity that only depends on probability distributions of the uncertainty of a global measurement is completely measurement outcomes has to be used. We will use determined by the uncertainty of the local measure- the Shannon entropy. It fulfills the above require- ments on the respective local states ρA and ρB. More- ment, does not change when the labeling of the mea- over, in our case, the additivity of the Shannon en- surement outcomes are permuted, and has a clear op-

Accepted in Quantum 2018-03-20, click title to verify 2 erational interpretation [29, 30]. Remarkably, Claude Shannon himself used the term ’uncertainty’ as an in- tuitive paraphrase for the quantity today known as ’entropy’ [29]. Historically, the decision to call the Shannon entropy an ’entropy’ goes back to a sugges- tion John von Neumann gave to Shannon, when he was visiting Weyl in 1940 (there are, at least, three versions of this anecdote known [31], the most popular is [32]). Because we are not interested in assigning values to measurement outcomes, a measurement, say X, is sufficiently described by its POVM elements, {Xi}. So, given a state ρ, the probability of obtaining the i-th outcome is computed by tr(ρXi). The respective probability distribution of all outcomes is de- x noted by the vector pρ. Within this notation the Shannon entropy of a X measurement is given by H(X|ρ) := − P px log px . As we restrict our- i ρ i ρ i selves to non-degenerate projective measurements, all Figure 2: Uncertainty set for measurements performed on necessary information on a pair of measurements, X a qubit. Any linear uncertainty relation, (2), with weights and Y , is captured by a unitary U that links the mea- (λ, µ), gives the description of a tangent to the uncertainty surement basis. We will use the convention to write set. All attainable pairs of entropies lie above this tangent. U as transformation from the {Xi} to the {Yi}-basis, † i.e. we will take U such that Yi = UXiU holds. Our basic objects of interest are optimal, state- above game against Alice and Bob, simultaneously. independent and linear relations. This is, for fixed Thereby, Alice and Bob share a common coin, and, therefore, apply measurements with the same labels weights λ, µ ∈ R+ we are interested in the best constant c(λ, µ) for which the linear inequality (XAB or YAB). The obvious question that arises in this context is if Eve gets an advantage in this si- λH(X|ρ) + µH(Y |ρ) ≥ c(λ, µ) (2) multaneous game by using an entangled state or not. Prop.1 in the next section answers the above ques- holds on all states ρ. tion negatively, which is somehow unexpected as in Such a relation has two common interpretations: principle the possible usage of non-classical correla- On one hand one can consider a guessing game, see tions enlarges Eve’s strategies. For example: Eve also [33]. On the other, a relation like (2) can be could have used a maximally entangled state, adjusted interpreted geometrically as in Fig.2. such that all measurements Alice and Bob perform Linear uncertainty: a guessing game are maximally correlated. In this case the remaining For the moment, consider a player, called Eve, who uncertainty Eve has, would only be the uncertainty plays against an opponent, called Alice. Depen- on the outcomes of one of the parties. However, the dent on a coin throw, in each round, Alice performs marginals of a maximally entangled state are maxi- measurement XA or YA on a local quantum state. mally mixed. Hence, Eve still has a serious amount of Thereby the weights λ and µ are the weights of the uncertainty (log d), which turns out to be not small coin and the l.h.s. of (2) describes the total uncer- enough for beating a strategy based on minimizing tainty Eve has on Alice’s outcomes in each round. To the uncertainty of the local measurements individu- be more precise, up to a (λ, µ)-dependent constant, ally. For the case of product-MUBs in prime square the l.h.s of (2) equals the Shannon entropy of the dimension [34], it turns out that the minimal uncer- XA YA outcome distribution λpρ ⊕ µpρ . tainty realizable by a maximally entangled state ac- Eve’s role in this game is to first choose a state ρ, tually equals the optimal bound. observe the coin throw, wait for the measurements Linear uncertainty: the positive convex hull to be performed by Alice, and then ask binary ques- The second interpretation comes from considering the tions to her opponent in order to get certainty on the set of all attainable uncertainty pairs, the so called outcomes. Thereby, the Shannon entropy sum on the uncertainty set l.h.s of (2) (with logarithm to the base 2) equals the expected amount questions Eve has to ask using an U = {(H(X|ρ),H(Y |ρ)) |ρ is a quantum state} . (3) optimal strategy based on a fixed ρ. Hence, the value c(λ, µ) denotes the minimal amount of expected ques- In principle this set contains all information on the un- tions, attainable by choosing an optimal ρ. certainty trade-off between two measurements. More For a bipartite setting, Fig.1, a second player, precisely, the white space in the lower-left corner of say Bob, joins the game. Here, Eve will play the a diagram like Fig.2 indicates that both uncertain-

Accepted in Quantum 2018-03-20, click title to verify 3 ties cannot be small simultaneously. In this context, a Remark 1 (Product states). Assume that cA(λ, µ) state-independent uncertainty gives a quantitative de- and cB(λ, µ) are optimal constants, and φA and φB scription of this white space. Unfortunately, it turns are the states that saturate the corresponding uncer- out that computing U can be very hard, because the tainty relations (4). Then the product state φAB := whole state-space has to be considered. Here a lin- φA ⊗ φB saturates (5), due to the additivity of the ear inequality, like (2), gives an outer approximation Shannon-entropy. However, this does not imply that of this set. More precisely, if c(λ, µ) is the optimal all states that saturate (4) have to be product states. constant in (2), this inequality describes a halfspace Examples for this, involving MUBs of product form, bounded from the lower-left by a tangent on U. This are provided in [34]. tangent has the slope µ/λ. The points on which this tangent touches the boundary of U corresponds to Remark 2 (Minkowski sums of uncertainty regions). states which realize equality in (2). Those states are Prop.1 shows how the uncertainty set UAB, of the called minimal-uncertainty states. Given all those product measurement, relates to the uncertainty sets tangents, i.e. c(λ, µ) for all positive (λ, µ), we can UA and UB of corresponding local measurements: For intersect all corresponding halfspaces and get a con- the case of an optimal cAB(λ, µ), and fixed (λ, µ), vex set which we call the positive convex hull of U, equality in (5) can always be realized by product denoted by U in the following. Geometrically, the states (see Rem.1). In an uncertainty diagram, like positive convex hull can be constructed by taking the Fig.3, those states correspond to points on the lower- convex hull of U and adding to it all points that have left boundary of an uncertainty set, and, in general, bigger uncertainties then, at least, some point in U. they produce the finite extreme points of the positive If U is convex, like in the example above, U contains convex hull of an uncertainty set. the full information on the relevant parts of U. If U is not convex, U still gives a variety of state independent uncertainty relations, but there is still place for finding improvements, see [34].

3 Additivity, implications and applications

We are now able to state our main result

Proposition 1 (Additivity of linear uncertainty relations). Let cA(λ, µ) and cB(λ, µ) be state-independent lower bounds on the linear entropic uncertainty for local measurements XA,XB and YA,YB, with weights (λ, µ). This means we have that Figure 3: Uncertainty sets of local measurements can be combined by the Minkowski sum: Uncertainty sets (green and λH(XA|ρA) + µH(YA|ρA) ≥ cA(λ, µ) yellow) for two pairs of local measurements on Qubits and the uncertainty set of the corresponding global measurements λH(XB|ρB) + µH(YB|ρB) ≥ cB(λ, µ) (4) (blue). holds on any state ρA from B(HA) and ρB from B(HB). Let XAB and YAB be the joint global mea- For product states we have the additivity of the surements that arise from locally performing XA,XB Shannon entropy, which gives and Y ,Y respectively. Then A B H(X |φ ⊗ φ ) H(X |φ ) H(X |φ ) AB A B = A A + B B H(Y |φ ⊗ φ ) H(Y |φ ) H(Y |φ ) λH(XAB|ρAB) + µH(YAB|ρAB) ≥ cA(λ, µ) + cB(λ, µ) AB A B A A B B (5) (7) This implies that we can get every extreme point of holds for all states ρAB from B(HA ⊗ HB). Further- U by taking the sum of two extreme points of U more, if cA and cB are optimal bounds, then AB A and U B. Due to convexity the same holds for all cAB(λ, µ) := cA(λ, µ) + cB(λ, µ) (6) points in U AB and we can get this set as Minkowski sum [35]. is the optimal bound in (5), i.e. linear entropic uncertainty relations are additive. U AB = U AB U B (8) The proof of this proposition is placed at the end For convex uncertainty regions, arising from local of Sec.5. We will proceed this section by collecting measurements, this is depicted in Fig.3. For this some remarks related to the above proposition: example, it is also true that UAB itself is given as

Accepted in Quantum 2018-03-20, click title to verify 4 Corollary 1 (Generalization to multipartite measurements). Assume parties A1 ...An that lo-

cally perform measurements, XA1 ,...,XAn or ~ YA1 ,...,YAn , with weights λ = (λ1, . . . , λn). In anal- ~ ~ ogy to (4), let cA1 (λ), . . . , cAn (λ) denote optimal lo- ~ cal bounds and let cA1···An (λ) be the optimal bound

corresponding to product measurements XA1...An and

YA1...An . We have

n ~ X ~ cA1...An (λ) = cAi (λ) (9) i=1 This follows by iterating (6). Remark 4 (Generalization to three measurements). The generalization of Prop.1 to three measurements, say XAB, YAB and ZAB, fails in general. The following counterexample was provided by O. Gühne [40]: For both parties we consider local measurements deduced from the three Pauli operators on a qubit and take all weights equal to one. In short hand nota- Figure 4: Multiparite setting: Additivity of entropic uncer- tion we write X = σ ⊗ σ , Y = σ ⊗ σ , and tainty relations also holds if a pair of global product mea- AB X X AB Y Y Z = σ ⊗ σ . In this case, the minimal local un- surements for many local parties is considered. AB Z Z certainty sum is attained on eigenstates of the Pauli operators. If such a state is measured, the entropy Minkowski sum of local uncertainty sets. However, for one of the measurements is zero and maximal for we have to note, this behavior cannot be concluded the others. Hence, the local uncertainty sum is always from Prop.1 alone. bigger than 2 [bit]. Therefore we have

Remark 3 (Relation to existing entanglement wit- H (σX ⊗ σX |φA ⊗ φB) + nesses). A well know method for constructing non- H (σY ⊗ σY |φA ⊗ φB) + linear entanglement witnesses is based on computing H (σZ ⊗ σZ |φA ⊗ φB) ≥ 4 (10) the minimal value of a functional, like the sum of uncertainties [36–38], attainable on separable states. for all product states. In contrast to this a Bell state, Given an unknown quantum state, the value of this say Ψ−, will give the entropy of 1[bit], for all above functional is measured. If the measured value under- measurements. Hence we have, goes the limit set by separable states, the presence of − entanglement is witnessed. For uncertainty relations H σX ⊗ σX |Ψ + − based on the sum of general Schur concave functionals H σY ⊗ σY |Ψ + this method was proposed in [4], including Shannon − H σZ ⊗ σZ |Ψ = 3 4. (11) entropy, i.e. the l.h.s. of (5), as central example. Our result Prop.1 shows that this method will not work for Shannon entropies, because there is no en- 4 Lower bounds from (p, q)-norms tangled state that undergoes the limit set by separable states. We note that there is no mathematical contra- The quite standard technique for analyzing a linear diction between Prop.1 and [4]. We only show that uncertainty relation is to connect it to the (p, q)- the set of examples for the method proposed in [4] is norm (see (12) below) of the basis transformation U. empty. Thereby, the majority of previous works in this ﬁeld For uncertainty relations in terms of Shannon, Tsal- is concentrating only on handling the case of equal lis and Renyi entropies a similar procedure for con- weights λ = µ = 1, which is connected to the (p, q)- structing witnesses was proposed by [37, 39]. Here norm for the case 1/p + 1/q = 1. However, for the explicit examples for states, that can be witnessed to purpose of this work, i.e. for proving Prop.1, we have be entangled, were provided. Again, our proposition to extend this connection to arbitrary (λ, µ). We will Prop.1 is not in contradiction to this work because do this by Thm.1 on the next page. in [37, 39] observables with a non-local degeneracy A historically important example for the use of the where considered. connection between (p, q)-norms and entropic uncertainties, is provided by Bialynicki-Birula and Myciel- Prop.1 can easily be generalized to a multipartite ski [41]. They used Beckner’s result [42], who com- setting, see Fig.4: puted the (p, q)-norm of the Fourier-Transfromation,

Accepted in Quantum 2018-03-20, click title to verify 5 for proving the corresponding uncertainty relation, (2) for λ = µ = 1. Hence, the functional depen- between position and momentum, conjectured by dence of kUkq,p on (p, q) in the limit (p, q) → (2, 2) Hirschmann [43]. Also Maassen and Uffink [21] took gives the optimal bound c(1, 1), in (2). For the this way for proving their famous relation. Our result case of the L2(R)-Fourier transformation the norm p 1/p p 1/q gives a direct generalization of this, meaning we will kUF kq,p = p / q was computed by Beckner recover the Maassen and Uffink relation at the end [42], leading to c(1, 1) = log(πe). However, to the of this section as special case of (50). Albeit, before best of our knowledge, computing kUkq,p, for general stating our result, we will start this section by shortly U and (p, q), is an outstanding problem, and presum- reviewing the previously known way for connecting ably very hard [47, 48]. Albeit, for special choices of (p, q)-norms with linear uncertainty relation, see also (p, q) this problem gets treatable, see [49] for a list of [44, 45] for further details: those. The known cases include p = q = 2, p = ∞ or The (p, q)-norm, i.e the lp → lq operator norm, of q = ∞ such as p = 1 or q = 1. a basis transformation U is given by The central idea of Maassen’s and Uffink’s work [21] is to show that the easy case of (p = 1, q = ∞), here kUφkq kUkq,p := sup . (12) we have kUk1,∞ = maxij |Uij|, gives a lower bound on φ∈H kφkp c(1, 1). More precisely, they show that, for 1 ≤ p ≤ 2 and on the line 1/p + 1/q = 1, the r.h.s. of (18) Here, the limit of kUkq,p for (p, q) → (2, 2) goes to 1. However, when p and q are fixed on the curve approaches c(1, 1) from below. Note that this is far from being obvious. Explicitly, for p ≤ 2 ≤ q we have 1/p + 1/q = 1, the leading order of kUkq,p around (p, q) = (2, 2) recovers the uncertainty relation (2) in Hq/2(Y |φ) ≥ H(Y |φ) and Hp/2(X|φ) ≤ H(X|φ), the case of equal weights λ = µ = 1/2, see [41, 43]. so one term approaches the limit from above and More precisely, taking the negative logarithm of the other approaches the limit from below. Whereas (12) gives Maassen and Uffink showed, using the Riesz-Thorin interpolation [50, 51], that the infφ of the sum of both − log kUkq,p = inf log kφkp − log kUφkq . (13) approaches the limit from below. φ∈H The following Theorem, Thm.1, extends the above Here, we can identify the squared modulus of the com- to the case of arbitrary (λ, µ). Notably, we have to ponents of φ as probabilities of the X and Y measure- take (p, q) from curves with 1/p + 1/q 6= 1, those are ment outcomes depicted in Fig.5. In contrast to Maassen and Uffink, the central inequality we use is the ∞-norm versions of 2 X |(φ)i| = hφ| Xi |φi = (pφ )i the Golden Thompson inequality (see [52–54] and the 2 Y blog of T.Tao [55] for a proof and related discussions). |(Uφ)i| = hφ| Yi |φi = (pφ )i (14) and substitute Theorem 1. Let c(λ, µ), with λ, µ ∈ R+, be the optimal constant in the linear weighted entropic uncer- X 2 Y 2 kφkp = kpφ kp/2 and kUφkq = kpφ kq/2 . tainty relation (15) c(λ, µ) := inf λH (X|ρ) + µH (Y |ρ) . (19) ρ By this, (13) gives a linear relation in terms of the α α-Renyi entropy [46], Hα(p) = 1−α log(kpkα). Here Then: we get (i) c(λ, µ) is bounded from below by −N log (ωN (λ, µ))

2 − p 2 − q 2 † inf Hp/2(X|φ) − Hq/2(Y |φ) = − log kUkq,p . with ωN (λ, µ) = sup x Uy (20) φ∈H p q d x∈Br (C ) d (16) y∈Bs(C ) 2N 2N If we evaluate this on the curve 1/p + 1/q = 1, for and r = s = (21) p ≤ 2 ≤ q, we can use N + 2λ N + 2µ 2 − p 1 1 q − 2 where = − = , (17) p p q q Br(Ω) := {x ∈ Ω| 1 ≥ kxkr} denotes the unit r-norm Ball on Ω. which can be employed to (16), in order to get (ii) For λ, µ ≤ N/2 we can write −1 1 1 2 inf Hp/2(X|φ) + Hq/2(Y |φ) = − log kUkq,p . kUφk 0 kUφk 0 φ∈H q p r s ωN (λ, µ) = sup = sup (22) d kφk d kφk (18) φ∈C s φ∈C r 2N 2N Here, the limit (p, q) → (2, 2), in the l.h.s of (18), with r0 = s0 = (23) N − 2λ N − 2µ gives the limit from the Renyi to the Shannon entropy. This gives the l.h.s. of the uncertainty relation (24)

Accepted in Quantum 2018-03-20, click title to verify 6 (iii) For µ, λ ∈ R+\{0}, we have such as the respective statement for H (Y |ρ) and B(q). If we employ this rewriting to c(λ, µ), we ob- c(λ, µ) = lim −N log (ωN (λ, µ)) (25) tain the minimal entropy sum as a minimization over N→∞ a parametrized eigenvalue problem, namely

0 λ/µ = 0.1 λ/µ = 1 µ/λ = 0.1 r c(λ, µ) = inf λH (X|ρ) + µH (Y |ρ) 4 ρ = inf tr (ρ (λA(p) + µB(q))) (29) p,q,ρ

Now we will turn the minimization, over ρ, into a maximization by applying the convex function e−x/N , with N ≥ 1, to the weighted sum of A and B. This will map the smallest eigenvalue of λA + µB to the − λA(p)+µB(q) 3 largest of e N and so on. In order to get back the correct value of c we will have to apply the inverse function, −N log(x), afterwards. We get − λA(p)+µB(q) c(λ, µ) = −N log sup tr ρe N . (30) p,q,ρ

Due to the positivity of the operator exponential, i.e. A and B are hermitian, the optimization over ρ is 2 equivalent to the Schatten-∞ norm. We have 0 1 s 2 − λA(p)+µB(q) Figure 5: Evaluating kUkr0,s on the depicted curves gives a c(λ, µ) = −N log sup e N (31) lower bound for c(λ, µ), (see Thm.1). Because c(λ, µ) is p,q ∞ a linear bound it is 1-homogenious in (λ, µ). Hence all information on the optimal bound c(λ, µ) can be recovered by At this point we apply the Golden-Thompson inequal- knowing it for any fixed ratio λ/µ. The thick red curve corre- ity sponds to the case 1/r0 +1/s = 1 which gives bounds c(1, 1) S+T S T from below. For s = 1 the norm kUkr0,s=1 can be computed ke kp ≤ ke e kp (32) analytically, this gives a generalization of the Masssen and Uffink bound (see Lem.1). and expand the resulting exponentials, as well as the Schatten norm. We get Proof. The starting point of this proof is a modifica- − λA(p) − µB(q) c(λ, µ) ≥ -N log sup e N e N (33) tion of a technique, used by Frank and Lieb in [56], p,q ∞ for reproving the Maassen and Uffink bound (see also   the talk of Hans Maassen [44], for a finite dimensional X λ/N µ/N = -N log sup Xipi Yjqj  (34) version). p,q ij d ∞ For probability distributions p, q ∈ B1(R+) we de-   fine the operators X λ/N µ/N = -N log  sup x Xipi Yjqj y  p,q X X |xi,|yi ij A(p) := − Xi log(pi) and B(q) := − Yi log(qi) (26) (35)

Based on this, we can further rewrite the Shannon X entropy as an optimization over a linear function in ρ −N log sup sup χixi hei|fji ξjyj  . (36) χ,x ξ,y by using the positivity of the relative entropy, i.e. we ij P P have D(p||q) = pi log(pi)− pi log(qi) ≥ 0, which P Here we can identify hei|fji = Uij, i.e. the overlaps implies − pi log(qi) ≥ H(p). We obtain are the components of U when represented in the basis H (X|ρ) = inf tr(ρA(p)), (28) X. At this point, it is straightforward to check that p d d χ ∈ BN/λ(R+) and ξ ∈ BN/µ(R+). Using the gener-

Accepted in Quantum 2018-03-20, click title to verify 7 alized Hölder inequality we can fuse some of the max- Remark 5 (The Maassen and Uﬃnk bound). For imizations above as follows: On one hand, we have the case of N = 2 and λ = µ = 1, in Thm.1, we get 0 0 1 s = r = 1 and s = r = ∞. Hence, we recover the X r r Maassen-Uﬃnk bound [21]. Explicitly, we have |χixi| ≤ kxk2kχkN/λ ≤ 1 (37)

1 † X s s ω2(1, 1) = sup x Uy = max |Uij| . (46) and |ξ y | ≤ kyk kξk ≤ 1 d ij j j 2 N/µ x∈B1(C ) d 1 1 λ 1 1 µ y∈B1(C ) for = + and = + , (38) r 2 N s 2 N Here we used that x†Uy is convex in x and y. Hence, sup is attained at the extreme points of B ( d). which means that the vectors v and w, with vi = χixi x,y 1 C Up to a phase, those extreme points have the form and wj = ξjyj, are in Br(C) and Bs(C) respectively. On the other hand, the converse is also true, i.e. (0, ··· , 0, 1, 0 ··· , 0) , i.e. they have their support only every v and w from B ( d) and B ( d) can be real- on a single site. So, choosing x and y, with support r C s C † ized by suitable choices of x, χ and y, ξ. For example, on the i − th and j − th site, will give x Uy = |Uij|. we can always set Remark 6 (Renyi-Entropies). Alternatively, the bound obtained in Thm.1 can be expressed in terms x = |v |rλ/N and χ = |v |2/rei arg(vi) (39) i i i i of Renyi-entropies: Using statement (i), (ii) and (iii) for getting x and χ from v, componentwise. For this together directly gives particular choice we can check that c(λ, µ) ≥ −N log kUkr0,s

r(λ/N+1/2) i arg(vi) xiχi = |vi| e = inf N log kφkr − N log kUφks0 . (47) φ∈H r/r i arg(vi) = |vi| e = vi (40) Here a straightforward computation shows holds, such that we will get back v. Furthermore x ∈ 0 d d 2 − r 2 − s BN/λ(R+) and χ ∈ B2(C ) follows by writing out = λ/N and = −µ/N. (48) r s0 X N/λ X r X 2 X r xi = vi ≤ 1 and χi = vi ≤ 1. So, when we proceed as in (13), substituting the Renyi i i i i entropy in (47) gives (41) c(λ, µ) ≥ inf λHr/2(X|φ) + µHs0/2(Y |φ). (49) φ∈H If we use the above in (36), we can replace supx,χ by supv and supy,ξ by supw, in order to get the statement Lemma 1 (Generalization of the Maassen and Uﬃnk (i) with bound). Let ui denote the i-th column of the basis † transformation U that links the measurements X and ωN := sup v Uw . (42) d Y . Then, for 1 ≥ λ ≥ µ ≥ 0 and all states ρ we have v∈Br (C ) d w∈Bs(C )

0 λH(X|ρ) + µH(Y |ρ) ≥ −2λ log sup ui t . For showing the statement (ii), we take r , with 1 = i=1···d 1/r + 1/r0. If λ ≤ N/2 holds we have r0 ≥ 0 and (50) we can use the tightness of the Hölder inequality to rewrite with 2 † t = (51) sup v Uw = kUwkr0 , (43) (1 − µ/λ) v∈Br Note that for the case 1 ≥ µ ≥ λ ≥ 0 the same holds, i.e. the maximization over Br gives the dual norm of † r. Substituting w by φ = wkφk then gives if U is replaced by U , i.e. by the transformation be- s tween Y and X. kUφkr0 ωN = sup (44) Proof. The linear uncertainty bound c(λ, µ) is homo- d kφk φ∈C s geneous in (λ, µ). Hence, we can consider 0 Here the analogous rewriting applies with s given by c(λ, µ) = λc(1, µ/λ) (52) 1 = 1/s + 1/s0, if µ ≤ λ/2 holds. For showing (iii), i.e. We will apply Thm.1, with N = 2, in order to get a 2 lower bound. Here, we have s = 1+µ/λ and c = lim −N log(ωN ) , (45) N→∞ † ω2(1, µ/λ) = sup x Uy = sup |ui y| . d i=1,··· ,d it suﬃces to expand all exponentials in (31) and (33) x∈B1(C ) d y∈B ( d) up to the ﬁrst order in N. On this order the Golden- y∈Bs(C ) s C Thomson inequality is a equality. (53)

Accepted in Quantum 2018-03-20, click title to verify 8 Here the second equality stems from the same argu- Proof. We note that a related result, for pointwise d mentation as in Rem.5. The sup over Bs(C ) on the positive maps between Lebesque spaces, was discov- most right of (53), gives the norm dual to s, given by ered by Grey and Sinnamon [61]. 2 t = 1−µ/λ . All in all we have, The basic object of this proof will be the p⊗q-norm c(1, µ/λ) ≥ −2 log (ω(1, µ/λ)) which will be deﬁned immediately. The basic work of this proof is devoted to show some properties of this

= −2 log sup ui t (54) norm from which the statement directly follows. i=1···d

Let |φi ∈ H with components φ = {φij} sorted A B within the product base XAB by φij = hφ|ei ⊗ ej i Remark 7 (More than two observables). As men- and consider the norm tioned in Sec.3, the proposition Prop.1 does not generalize to three measurements. A reasoning, or at least a hint, for this can be found by carefully fol- q 1   q lowing the proof of Thm.1. In principle, the ansatz   p X X p in (29) can be generalized to more than two measure- kφk :=  |φ |  . (57) q⊗p   ij   ments as well, and all following steps work out in a i j similar way, up to (33). Here the Golden-Thompson inequality was used. It is well known, that the direct generalization of this inequality to three operators fails to hold. Hence, the technique of our proof This norm shares the following properties cannot be generalized for this case. We note that there is an ongoing work of exploring more sophisti- cated generalizations of this inequality [57–60]. How- ever, we leave relating this to entropic uncertainty for (i) kφkq⊗q = kφkq (58) future work. (ii) k(I ⊗ V φ)kr⊗q ≤ kφkr⊗p ηV (59)

(iii) kφkq⊗p ≤ kFφkp⊗q (60) 5 Additivity of bounds from multiplica- with Fφ1 ⊗ φ2 = φ2 ⊗ φ1 and p ≤ q. tivity of (p, q)-norms

In this section we will provide the proof of Prop.1, i.e. the additivity of linear uncertainty relations. By using We will show the validity of (i − iii) in a moment. Thm.1 from the section before we can formulate the First notice that, if (i − iii) are valid we can easily linear uncertainty in terms of the logarithm of a (p, q)- conclude norm. At this point, it is straightforward to check that the additivity of the linear uncertainty is equivalent to the multiplicativity of the (p, q)-norm. In fact, the kU φk = kU ⊗ U φk following theorem Thm.2 provides that, for p and q AB q A B q⊗q coming from the correct range: The (p, q) norm of a = k(I ⊗ UB)(UA ⊗ I)φkq⊗q transformation which admits a product form UAB = ≤ ηBkUA ⊗ Iφkq⊗p U ⊗ U is multiplicative. A B ≤ ηBkI ⊗ UAFφkp⊗q Theorem 2 (Global bounds from local bounds). Let ≤ ηBηAkFφkp⊗p XAB and YAB be tensor-product bases of a Hilbert = ηBηAkφkp . (61) i space HAB = HA ⊗ HB, i.e. we have XAB = {XA ⊗ i i i XB}i=1,··· ,d and YAB = {YA ⊗ YB}i=1,··· ,d, such as UAB = UA ⊗ UB. Furthermore let ηA and ηB denote the optimal constants for Furthermore, if we consider states that realize equality in (55), i.e. states that belong to optimal ηA and kUAφkq ≤ ηAkφkp ∀φ ∈ HA ηB. The tensor-product of two of those states will realize, due to multiplicativity of the p-norm, equal- kUBφkq ≤ ηBkφkp ∀φ ∈ HB . (55) ity in (56) as well. Hence, (61) will prove the main If 1 ≤ p ≤ q then statement of this Theorem.

kUAB φkq ≤ ηAηBkφkp ∀φ ∈ HAB (56) Property (i) follows directly by plugging p = q in the deﬁnition of the p ⊗ q norm, here is nothing more holds with ηAηB = ηAB as optimal constant. to prove. The property (ii) follows by expressing I⊗V

Accepted in Quantum 2018-03-20, click title to verify 9 as δikVjl in X-Basis and cB and cAB to already be constants for the best linear uncertainty bound. If the additivity 1  r  r  q q X X X cAB = cA + cB (69) k( ⊗ V φ)k =  δ V φ  I r⊗q   ik jl kl  

i j lk holds we can directly conclude that the sum of lower (62) bounds on cA and cB gives a valid lower bound on 1 c as well.  r  r AB   q q Given measurements X and Y , speciﬁed by a X X X AB AB =  V φ    jl il   product unitary UAB = UA ⊗ UB, we use Thm.1 to

i j l rewrite cA, cB and cAB as the limit of logarithms of (63) (p, q)-norms. We assume λ ≤ µ both to be ﬁnite and 1 N to be suﬃciently large such that we can use Thm.1 ! r X r part (ii) (here we needed λ, µ ≤ N/2), and get = kV φikq (64) i cA = − lim log (||UA||r,s) (70) 1 N→∞  r  r   p cB = − lim log (||UB||r,s) (71) X X p  N→∞ ≤ ηV   |φij|   i j cAB = − lim log (||UAB||r,s) (72) N→∞ = η kφk . (65) V r⊗p Using r, s, as given in (20) it is straightforward to As a last step, (iii) is a direct consequence of check that λ, µ ≤ N/2 implies 1 ≤ r ≤ s. Therefore, Minkowski’s inequality / lp-triangle inequality (see we can use Lem.2 and get [62]), i.e. if p ≥ 1 : cAB = − lim log (||UA||r,s||UB||r,s) N→∞ 1 1 p ! p ! p X X X X p = − lim log (||UA||r,s) + log (||UB||r,s) N→∞ axy ≤ |axy| (66) y x x y = cA + cB . (73) So, if 1 ≤ q/p we can use this inequality as follows

q 1   q   p X X p kφk =  |φ |  Outlook and conclusion q⊗p   ij   i j In this work we showed that linear uncertainty rela- q 1 1   q/p p tions between product type measurements in multi- p

X X p partions are additive. Prop.1 gives some clear struc- =  |φij|    ture to the problem of computing entropic uncertainty i j bounds. Especially in the context of quantum-coding

p 1   p in cryptography, this result might turn out to be use- ! q X X p q ful, because now it is possible to compute uncertainty ≤ |φ | p = k φk (67)  ij  F p⊗q bounds in the limit of infinite system sizes for block- j i coding schemes [6, 63, 64]. and show the validity of (iii). The generalization of the Maassen and Uffink bound for arbitrary weights (λ, µ), provided in Lem.1, Lemma 2 (Multiplicativity of the (p, q)-norm). can also be directly employed in a multipartite setting For 1 ≤ p ≤ q, the (p, q)-norm of a product unitary in order to obtain valid state-independent uncertainty UAB = UA ⊗ UB is multiplicative, i.e. we have relations for this case. However, this bound is easy computable, it is only a lower bound and presumably ||UAB||q,p = ||UA||q,p||UB||q,p. (68) only tight in high symmetrical cases (see [34] for a characterization of tightness for the usual Maassen Proof. This is a direct consequence of Thm.2. Using and Uffink bound). The more general problem of the definition of the (p, q)-norm we can parse ηA = providing a ’good’ method for computing the optimal ||UA||q,p, ηB = ||UB||q,p and ηAB = ||UAB||q,p, if we bound cAB remains open. We note that there are only consider ηA, ηB and ηAB to be optimal bounds. few and special cases, including angular momentum Proof of Prop.1 and mutual unbiased bases, where this optimal bound is actually known. Thereby, the cases where the op- Proof. For proving Prop.1 it suffices to proof the ad- timal bound can be computed analytically are even ditivity of the optimal case, i.e. we will consider cA, fewer [34, 65, 66] and the known numerical methods

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