Entropic uncertainty relations for multiple measurements assigned with biased weights

The entropic way of formulating Heisenberg's uncertainty principle not only plays a fundamental role in applications of quantum information theory but also is essential for manifesting genuine nonclassical features of quantum systems. In this paper we investigate R\'{e}nyi entropic uncertainty relations (EURs) in the scenario where measurements on individual copies of a quantum system are selected with nonuniform probabilities. In contrast with EURs that characterize an observer's overall lack of information about outcomes with respect to a collection of measurements, we establish state-dependent lower bounds on the weighted sum of entropies over multiple measurements. Conventional EURs thus correspond to the special cases when all weights are equal, and in such cases, we show our results are generally stronger than previous ones. Moreover, taking the entropic steering criterion as an example, we numerically verify that our EURs could be advantageous in practical quantum tasks by optimizing the weights assigned to different measurements. Importantly, this optimization does not require quantum resources and is efficiently computable on classical computers.


I. INTRODUCTION
Heisenberg's uncertainty principle [1] is a fundamental concept in quantum mechanics which underlies one of the most important nonclassical features of quantum physics-quantum observables can be incompatible such that no observer has precise knowledge about them simultaneously.As a consequence, an observer's ability to predict (or certainty about) outcomes of measuring incompatible observables is inherently limited.
We emphasize that, as is pointed out in [54,55], EURs like Eq. ( 1) are thus far formulated in a restricted form considering that they describe lower bounds on simply entropy sums, whereas the most general entropic way of expressing the uncertainty principle should be with {w θ } being arbitrary positive weights.Conceptually speaking, requiring the weights {w θ } to be equal is unnecessary since the l.h.s. of Eq. ( 2) well captures the presence of uncertainty regardless of the weights, i.e., it is positive for observables with no common eigenstate.Realizing this, weighted EURs (WEURs) for multiple measurements in terms of the collision entropy as well as for two projective measurements in terms of the Shannon entropy have been established, respectively, in Refs.[54] and [55].
In this paper, inspired by a recent work [54] on complementarity relations, we obtain upper bounds (4) on the weighted sum of IC over multiple outcome probability distributions induced by general measurements.We establish lower bounds on the weighted sum of Rényi entropies [56] for multiple generalized measurements, i.e., positive-operator-valued measures (POVMs).Compared with previous EURs, our WEURs are generally stronger and apply to versatile measurement scenarios.
This paper is structured as follows.In Sec.II, we introduce general upper bounds on the IC of probability distributions induced by performing general sets of measurements on quantum systems.In Sec.III, we propose WEURs for multiple measurements assigned with positive weights.In Sec.V, we take the steering test as an example to show numerically that our WEURs are advantageous in practical quantum tasks.Finally, we draw a brief conclusion in Sec.VI.

II. PRELIMINARY
Each quantum measurement is described by a set of positive semi-definite operators (POVM effects) M = {M i } that satisfy i M i = ½, with ½ being the identity operator.For example, the POVM description of measuring a nondegenerate observable consists of rank-1 projectors onto its eigenbasis, called rank-1 projective measurements.Throughout the rest of this article, we frequently consider the measurement scenario where an observer chooses, according to the value of a classical random variable θ sampled from some probability distribution {w θ } ( θ w θ = 1, w θ > 0), to perform one of a set of measurements {M θ } on individual copies of a quantum system.We denote by p i|θ the probability of obtaining the ith outcome when performing the θth measurement M θ = {M i|θ } i on the state ρ, which is p i|θ = Tr(M i|θ ρ) according to Born's rule.
In Ref. [54], the authors proposed an upper bound on the average information gain on quantum systems in individual trials of measurements, i,θ Here, I com (ρ) = Tr(ρ 2 ) − 1/d is the operationally invariant measure of complete information content contained in d-dimensional quantum states [57].ĝ denotes the largest eigenvalue of the weighted average of view operators ĝ = θ w θ Ĝ(M θ ) [54] (see also Appendix.A), which is state-independent and depends only on the measurement scenario.In fact, the weighted average information gain given as the l.h.s. of Eq. (3) quantifies how much the state ρ can be discriminated from the completely mixed state through the respective outcome statistics {p i|θ } and {Tr(M i|θ )/d}.The r.h.s. of Eq. ( 3), on the other hand, limits one's ability to gain information about quantum systems in different measurement scenarios.Interestingly, Eq. ( 3) naturally explains the origin of wave-particle duality in two-way interferometers as exclusion relations between information gains in complementary measurements [54].
To establish WEURs for l-outcome ETE-POVMs from the certainty relation (3), let us cast Eq. ( 3) into an equivalent inequality for convenience, i,θ It is worth mentioning that Eq. ( 4) becomes a tight equality for an arbitrary complete set of design-structured measurements with equal weights.Moreover, 1 Θ ≤ ĝ ≤ 1 holds for a number Θ of rank-1 projective measurements, regardless of the weights {w θ } [54].In particular, ĝ = 1 Θ is saturated by random measurements in one of Θ MUBs, and for arbitrary nondegenerate observables with one or more common eigenstates there is ĝ = 1.Following Ref. [54], we will call the total measurement exclusivity, which takes value in the range X tot ∈ [0, Θ − 1] for rank-1 projective measurements.The exclusivity may be interpreted as a complement of overlap between multiple measurements, which tends to be larger for measurement bases that are less overlapped (closer to being mutually unbiased).

III. WEIGHTED ENTROPIC UNCERTAINTY RELATIONS
The Rényi entropies are generalizations of the Shannon entropy defined as below [56] where p = (p 0 , p 1 , • • • ) can be any probability distribution and the parameter α > 0 and α = 1.The Shannon entropy H( p) = − i p i log p i = lim α→1 H α ( p) is thus recovered in the limit α → 1. Rényi entropies has many essential significance in cryptography and information theory.As a noteworthy example, the minimum entropy terizes the number of random bits that can be extracted from a random variable obeying the distribution p.For more discussions on the basic properties and applications of Rényi entropies, we recommend the review [28].
Our discussions on WEURs revolve around the relationship between the IC of probability vectors and the The IC-entropy diagrams and convex estimations Q1 (Eq.( 9); dot-dashed blue line) and Q3 (Eq.( 6); dashed red line) of their lower boundaries, respectively for the Shannon entropy (blue region) and the Rényi 3-entropy (orange region).
corresponding Rényi entropies.The IC of a probability vector p refers to the probability that two independent random variables drawn from p take the same value, that is, c( p When l = 3, the IC-entropy diagrams-the ranges of the map p → c( p), H α ( p) -are plotted in Fig. 1 for α = 1, 2, 3 respectively.We can see they intersects at the green points {(1/3, log 3), (1/2, 1), (1, 0)} on the curve (c, − log c) of the Rényi 2-entropy.Similar results hold also for general l ≥ 2. This is because H α ( p) is monotonic decreasing of α except when the nonzero probabilities of p is uniform, Let us first consider the case α ≥ 2. To estimate the lower boundary of the IC-entropy diagram we adopt the function [53] where . This estimation function (see the dashed red line in Fig. 1 for the case α = 3) is optimal when α = 2 or +∞, as well as when l = 2, and it remains a good estimation in other situations, especially when l is large.Additionally, Eq. ( 6) is convex with respect to c (see a detailed proof in Appendix.B), combined with the IC bound (4) we immediately have the theorem below.
Theorem 1. Suppose {M θ } are l-outcome ET-POVMs to be performed on the state ρ with selection probabilities {w θ }, and ĝ is the average view operator.When α ≥ 2 the average Rényi α-entropy satisfies Interestingly, the uncertainty lower bound given as the r.h.s. of Eq. ( 7) decreases monotonically with the quantity ĝ • I com (ρ).Meanwhile, for fixed measurements and selection probabilities, q α reaches its minimum at all pure states, thus becoming state-independent in the sense that it is an uncertainty lower bound valid for all quantum states in the Hilbert space considered.
Corollary 1.For rank-1 projective measurements on individual qubits in the state ρ Equation ( 8) is compared in Fig. 2 with the corresponding numerical optimal entropic lower bound for three nondegenerate observables with equal weights.As shown, q 2 is very strong, especially when the exclusivity is around X tot = 2 (MUBs) or X tot = 0 (compatible bases).
As for the Shannon entropy, the corresponding lower boundary of the IC-entropy diagram consists of concave curves each joining a pair of neighbor points with coordinates on the set [53,58] (see the green points in Fig. 1).Substituting these curves for line sections we arrive at the estimation function (see the dot-dashed blue line in Fig. 1) Here n = ⌊1/c⌋ denotes the round-down of 1/c to the nearest integer.Similar to the estimation function (6) for α ≥ 2, equation ( 9) is convex with respect to c. Combining Eq. ( 9) with the IC bound (4), the following theorem is then obvious.
Theorem 2. Suppose {M θ } are l-outcome ETE-POVMs to be performed on the state ρ with selection probabilities {w θ }, and ĝ is the associated average view operator.The average Shannon entropy satisfies Corollary 2. For rank-1 projective measurements on individual qubits in the state ρ If the inverse of 1/l + ĝ • I com (ρ) happens to be an integer, say n, then q 1 = log n would be the best uncertainty bound that can be obtained from the IC bound (4).But this is not the case in general.Luckily, when restricted to ETE-POVMs with equal weights, we can make full use of Eq. ( 4) to derive improved EURs.
Theorem 3. Suppose {M θ } Θ θ=1 is a set of l-outcome ETE-POVMs to be performed on individual quantum systems in the state ρ.Then, the sum of Shannon entropies satisfies Here, with c = 1/l + 1 Θ Ĝtot • I com (ρ) being the average IC, n = ⌈1/c⌉ and k . We refer to Appendix.C for a detailed proof of Theorem 3.
For a complete set of d + 1 MUBs (CMUBs) in ddimensional Hilbert space, note that the corresponding total view operator must satisfy Ĝtot = 1 (see also Appendix.A).Substituting I com (ρ) ≤ 1 − 1/d into Eq.( 12) then leads us to the strong entropic bound previously obtained in Refs.[47,59,60].This immediately indicates that Eq. ( 12) would be strong for approximately CMUBs as q S is continuous with respect to measurements.Corollary 3.For Θ rank-1 projective measurements on individual qubits in the state ρ where Both q 1 (11) and q S (12) decreases monotonically with the operationally invariant information I com (ρ) contained in the quantum systems to be measured, and achieve their state-independent minimum at pure states.For two rank-1 projective measurements (Θ = 2), Eq. ( 13) reduces to q S ≥ h bin ( . This recovers an earlier bound reported in Ref. [48], which is known to be tighter than q MU (1).For rank-1 projective measurements onto three MUBs (Θ = 3, Ĝtot = 1), q S = 2 + S(ρ) is known to be tight [61], where S(ρ) = −Tr(ρ log ρ) is the von Neumann entropy.We emphasize here that Eq. ( 12) is a general result valid beyond the aforementioned simple cases.In Fig. 2, the state-independent forms of q 1 and q S are compared with the respective numerical optimal uncertainty bounds for three random nondegenerate observables of qubits.As depicted, both of them are tight for MUBs (X tot = 2) and remain to be strong when X tot < 2.
Comparison between the state-independent forms of bounds qS, qLMF, qSCB, qRPZ and the numerical optimal bound B1 for 750 randomly generated sets of three bases in H2.(c) Comparison between state-independent entropic bounds for 750 randomly generated sets of four bases in H2.(d) Entropic bounds for four bases in H3 defined by Eq. ( 16).
We present in Fig. 3(a) numerical comparisons between q S (13), q LMF (14) and the simply constructed bound q SCB (15) on the sum of Shannon entropies over three single-qubit observables.As shown, q S ≥ q SCB holds for all the four sets of observables considered.Additionally, although q LMF is relatively weaker for pure states [S(ρ) = 0; I com (ρ) = 0.5], it can be stronger than q SCB and q S for mixed states.It is worth noting that q LMF > q S may hold for mixed states in the special case when two of the three observables are approximately compatible while the third one is complementary to them [see Fig. 3(d)-3].But more generally, q LMF tends to be weaker than q S , especially for observables that are close to being mutually complementary or commutable [see Fig. 3(d In fact, for Θ MUBs in H d , the state-independent forms of Eqs.(14,15) are q LMF = log d and This already demonstrates that Eqs.(10,12) would be stronger than q LMF and q SCB , at least for Θ ≥ √ d +1 bases in H d that are sufficiently close to being mutually unbiased.In Fig. 3(b), we move on to take into consideration the famous state-independent bound q RPZ obtained from the direct sum majorization relations by Rudnicki et al. [34].As depicted, both q S and q RPZ are never weaker than the state-independent forms of q LMF and q SCB for arbitrary three bases in H 2 .Moreover, q S attains the optimal bound B 1 for three MUBs in H 2 (X tot = 2), q LMF = 1 < q SCB = 1.5 < q RPZ ≈ 1.8 < q S = 2 , and remains to be stronger than q RPZ for approximately MUBs.On the other hand, for bases that are close enough to be compatible (X tot 1), q RPZ is stronger in general.
When considering four bases in H 2 , we can see from Fig. 3(c) that q S > q RPZ tends to hold for measurement bases with large exclusivity and, roughly speaking, q RPZ > q S holds for bases that are close enough to be compatible.In Fig. 3(d), we make a further comparison between q S and the bounds q RPZ and q SCB for Θ = 4 bases in H 3 , including the computational basis and three unitary transformations of it below, Here E is diagonal with eigenvalues {1, e i2π/3 , e i2π/3 }, and F denotes the discrete Fourier transform, i.e, F jk = 1 √ 3 e i2πjk/3 .Again, q S = 4 is tight [62] for MUBs (β = π/4), in which case it is stronger than q RPZ .

V. ENTROPIC STEERING CRITERION
In this section we discuss the applications of our EURs in steering detection.The concept of steering [63,64] dates back to Einstein, Podolskey, and Rosen's [65] prominent observation that quantum correlation allows one to predict the outcome of measuring one particle based on the measurement performed on a distant particle.One possible explanation for this kind of correlation between distant measurement outcomes is the local hidden state (LHS) model.To illustrate, consider a bipartite state shared between Alice and Bob, then Bob's local state after Alice's measurement on subsystem A is said to admit a LHS model if it can be decomposed as where λ denotes the value of an assumed hidden variable subject to some probability distribution π(λ), p A (a|θ, λ) denotes Alice's probability to obtain the ath outcome when she chooses to perform the measurement labeled by θ, and {ρ λ } are local hidden states on Bob's side independent of Alice's measurements.Thus, Eq. ( 17) essentially describes a particular form of correlation between Bob's local states {σ a|θ } and Alice's measurement outcomes {(a, θ)}.What's interesting is quantum states can be steerable and violate steering inequalities [66] such that they do not admit a LHS model.In Schroödinger's words, "the steering forces Bob to believe that Alice can influence his particle from a distance" [67].From a modern point of view, steering signifies the presence of entanglement, but not necessarily Bell nonlocality [63].
Steering has many applications in quantum information processing (see the reviews [63,64] and references therein for details), wherein a key point is to detect when and how much a certain steering inequality can be violated, or Bob's ability to predict his local measurement outcomes conditioned on Alice's measurement results.To this end, we utilize the conditional Rényi entropy [68] to quantify Bob's uncertainty about (inability to predict) his local measurement outcomes where p B|A (i|j) is the conditional probability and • denotes the α-norm.As a straightforward generalization of the entropic steering criterion [42,69], if Alice is unable to convince Bob that the state ρ AB is entangled by performing the local measurements {M A θ } on subsystem A we have Here q α ({M B θ , w θ }) denotes the state-independent entropic bounds given as the r.h.s. of Eqs.(7,10) when Bob chooses to perform the measurement M B θ with probability w θ on subsystem B. Violation of Eq. ( 18) necessarily implies Bob's local uncertainty can be reduced given Alice's measurement results, so that the state ρ AB is steerable from Alice to Bob.
the commutativity of observables to generalized measurements.White noise robustness is a commonly used measure of incompatibility.It refers to the critical value η * of η (0 ≤ η ≤ 1) below which a set of noisy measurements {ηE i|θ + (1 − η) 1 d ½ d } become jointly measurable (η = 0 corresponds to trivial measurements, which are always compatible).Next we study how much white noise could corrupt the incompatibility of Alice's measurements to keep inequality (18) saturated.Following Refs.[42,69], we exploit the equivalence [72,73] between incompatibility and steering of a maximally entangled state, that is, whenever Alice chooses incompatible measurements she will succeed in convincing Bob that the state shared between them is entangled.
As a simle example, we then utilize Eq. ( 18) to estimate the incompatibility of single-qubit observables.Suppose now the maximally entangled state ρ AB = 1 √ 2 (|00 − |11 ) is shared between Alice and Bob, and in each round of the test, if Alice chooses to measure the observable σ, Bob measures its noisy counterpart ησ + (1 − η) 1 2 ½ 2 .For simplicity, we consider three observables in the form {cos We present in Fig. 4 some numerical results about the threshold value η equ (η opt ) of η for Bob's measurements with equal (optimal) weights to violate Eq. (18).As shown, Eq. ( 18) yields η equ ≈ 0.577 for the case of three complementary observables (β 1 = β 2 = 0, X tot =2).This value coincides with the noise robustness η * = 1 √ 3 .More generally we have η equ > η opt , therefore optimizing the weights in Eq. ( 18) leads to superior performance.We would like to remark that the performance of Eq. (18) has been underestimated in the preceding analysis, as we restricted Alice's measurement choice to be the noiseless counterpart of Bob's measurement for simplicity.

VI. CONCLUSION
In this work, we have derived Rényi EURs for multiple measurements assigned with positive weights from upper bound (4) on the respective IC of outcome probability distributions.On one hand, our results extend a series of independent investigations [35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53] on EURs for designstructured measurements to WEURs for much more general measurements.On the other hand, we verified both analytically and numerically that our Shannon EURs are generally stronger than the generalizations, q LMF (14) and q SCB (15), of Maassen and Uffink's famous EUR (1) to multiple bases.Our bound q S (12) can also outperform the strong direct-sum majorization bound q RPZ [34] for certain measurements, especially for approximately mutually unbiased bases (MUBs).Taking the steering test as an example, we also demonstrated numerically that WEURs could achieve better performance in practical applications simply by optimizing the weights assigned to different measurements.Crucially, the optimization process can be readily accomplished on a classical computer without incurring any additional quantum costs.
Our investigation provides new insights into the interpretation of quantum uncertainty from an entropic perspective and we expect it to inspire future in-depth researches on the applications of WEURs in quantum information theory.To improve our results, future works will take into consideration other eigenvalues of the average view operator to tighten the IC bound (4).Exploring new applications of WEURs also presents an intriguing avenue for future investigation.
} is the solution to the equations p a + (l − 1)p b = 1 and Let z = p b /p a ∈ [0, 1], then the r.h.s. of Eq. (B2) shares the same sign as the following term We proceed to show Q α (l, c) is convex with respect to c when α ≥ 2. Let us begin with the function below Based on Eq. (B7) we have Considering that p a + (l − 1)p b = 1, now we know that p a increases with c and p b decreases with c.Consequently, p a − p b increases with c.According to Eq. (B7) we then have d 2 p a /dc 2 < 0 and The above proves A is convex with respect to c.When α ≥ 2, according to Eq. (B7) we have B is non-increasing means −1 2 −B is non-decreasing, while the term in the square braket is obviously non-increasing due to the monotonicity of p b and p a − p b .Thus d 2 B/dc 2 > 0, which, combined with (B9), completes the proof that Q α is convex with respect to c.

Appendix C: Shannon EURs
To prove Theorem 3, here we mainly utilize the method for constructing Shannon EURs from IC introduced in Ref. [53].Some intermediate conclusions in the first subsection have been obtained also in Refs.[58,60].
The properties above is enough to derive the best lower bound on the Shannon entropy for multiple probability distributions based only on an upper bound on the average IC (see the next subsection).We present below the detailed proof of property (4).The proof is straight forward, for c ∈ .

(C9)
Let 0 ≤ u = sn(n − 1) < 1, the term in square brackets is The r.h.s. of Eq. (C10) increases monotonically with u, obviously it's nonnegative.To show Eq.(C7) we only need Similar to Eq. (C10), the term in square brackets of Eq. ( C13) is positive To see why Eq. (C17) is indeed the optimal lower bound, i.e., D Θ (l, c tot ) = h Θ (l, c tot ), observe first the property (ii) of h(c) ensures that the set of nonnegative numbers {c 1 , • • • , c Θ } satisfying θ c θ = c tot and h Θ (l, c tot ) = θ h(c θ ) contains at most one element that is not an inverse of an integer.Without losing generality, suppose c

Figure 4 .
Figure 4. Estimations of the incompatibility of three qubitobservables based on Eq. (18) when α = +∞ (minimum entropy).For three complementary observables (Xtot = 2), Eq. (18) yields ηequ ≈ 0.577 when the weights w1 = w2 = w3 are equal, which well estimates the corresponding incompatibility η * = 1 B5) where A = − log p a , B = − log p 2 a + (l − 1) 2 α p 2 b .The coefficient of B is obviously positive.As for A, non-negative too.Next we show A and B are convex with respect to c, respectively.Differentiate the equations p a + (l − 1)p b = 1 and p 2 a + (l − 1)p 2 b = c leads us to