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Additivity of Entropic Uncertainty Relations

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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 ba- sic measurement setting, which we consider for bi- partitions, 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 un- certainty 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 un- certainty 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]).
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