Holevo bound of entropic uncertainty in Schwarzschild spacetime

The European Physical Journal C, Jul 2018

For a pair of incompatible quantum measurements, the total uncertainty can be bounded by a state-independent constant. However, such a bound can be violated if the quantum system is entangled with another quantum system (called memory); the quantum correlation between the systems can reduce the measurement uncertainty. On the other hand, in a curved spacetime, the presence of the Hawking radiation can reduce quantum correlation. The interplay of quantum correlation in the curved spacetime has become an interesting arena for studying uncertainty relations. Here we demonstrate that the bounds of the entropic uncertainty relations, in the presence of memory, can be formulated in terms of the Holevo quantity, which limits how much information can be encoded in a quantum system. Specifically, we considered examples with Dirac fields with and without spin, near the event horizon of a Schwarzschild black hole, the Holevo bound provides a better bound than the previous bound based on the mutual information. Furthermore, as the memory moves away from the black hole, the difference between the total uncertainty and the new lower bound remains a constant, not depending on any property of the black hole.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1140%2Fepjc%2Fs10052-018-6026-3.pdf

Holevo bound of entropic uncertainty in Schwarzschild spacetime

Eur. Phys. J. C Holevo bound of entropic uncertainty in Schwarzschild spacetime Jin-Long Huang 2 Wen-Cong Gan 0 1 Yunlong Xiao 5 Fu-Wen Shu 0 1 Man-Hong Yung 3 4 0 Center for Relativistic Astrophysics and High Energy Physics, Nanchang University , No. 999 Xue Fu Avenue, Nanchang 330031 , China 1 Department of Physics, Nanchang University , No. 999 Xue Fu Avenue, Nanchang 330031 , China 2 Department of Physics, Southern University of Science and Technology , Shenzhen 518055 , China 3 Shenzhen Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology , Shenzhen 518055 , China 4 Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology , Shenzhen 518055 , China 5 Department of Mathematics and Statistics and Institute for Quantum Science and Technology, University of Calgary , Calgary, AB T2N 1N4 , Canada For a pair of incompatible quantum measurements, the total uncertainty can be bounded by a stateindependent constant. However, such a bound can be violated if the quantum system is entangled with another quantum system (called memory); the quantum correlation between the systems can reduce the measurement uncertainty. On the other hand, in a curved spacetime, the presence of the Hawking radiation can reduce quantum correlation. The interplay of quantum correlation in the curved spacetime has become an interesting arena for studying uncertainty relations. Here we demonstrate that the bounds of the entropic uncertainty relations, in the presence of memory, can be formulated in terms of the Holevo quantity, which limits how much information can be encoded in a quantum system. Specifically, we considered examples with Dirac fields with and without spin, near the event horizon of a Schwarzschild black hole, the Holevo bound provides a better bound than the previous bound based on the mutual information. Furthermore, as the memory moves away from the black hole, the difference between the total uncertainty and the new lower bound remains a constant, not depending on any property of the black hole. Jin-Long Huang, Wen-Cong Gan and Yunlong Xiao contributed equally. 1 Introduction Traditionally, uncertainty principle in quantum mechanics has been formulated in terms of variance [ 1–15 ], while in the context of both classical and quantum information sciH (M1|B) + H (M2|B) ≥ U1, (1) where U1 ≡ − log c1 + H ( A) − I( A : B) [ 26 ], with H (ρ) ≡ −tr(ρ lnρ) is von Neumann entropy for density matrix ρ and I( A : B) = H ( A)+ H (B)− H ( A, B) is the quantum mutual information. However, as the correlation between two parties changes, is this mutual information I( A : B) a good measure to quantify how much the uncertainty will change? Another candidate is the well-known Holevo quantity (or Holevo bound) [ 27 ], which has a wide range of applications [ 28– 30 ]. Reference [ 28 ] proves a universal bound on the quantum channel capacity for two distant systems. Holevo quantity is used to connect mutual information and an upper bound for vacuum-subtracted entropy of signal ensemble. In Ref. [ 29 ], authors adopt Holevo information to quantify distinguishability of black hole microstates by measurements performed on subregion A of a Cauchy surface. Their calculation is based on the assumption that the vacuum conformal block dominates in the entropy calculation. Reference [ 30 ] added a correction term when this assumption fails. In these articles quantum channel is considered from a global state to a state restricted in a subregion, while in this paper quantum channel is from two parties ρAB to one party ρB . This channel depends on Unruh effect and measurements performed by Alice. Using Holevo quantity, we generalized entropic uncertainty with quantum memory. Similar improvement has considered before in [ 31,32 ]. The only difference is that they considered Holevo quantity on Bob’s ensemble, which is prepared after Alice optimizes her measurement choices. Instead, our lower bound is dependent on Alice’s measurement choice. Let J (B|M1) ≡ H (B) − j p1j H (ρB|u1j ) be the Holevo quantity for Bob about Alice’s M1 measurement outcomes. The new uncertainty relation says that H (M1|B) + H (M2|B) ≥ U2, (2) where U2 ≡ − log c1 + H ( A) − J (B|M1) − J (B|M2). The new entropic uncertainty relation has the property that the difference (which we denote as Δ2 below) between entropic uncertainty (LHS of (2)) and the new uncertainty lower bound U2 only depends on incompatible measurements M1, M2 and ρA, and independent of quantum memory, Δ2M1 M2 = H (M1) + H (M2) + log c1 − H ( A). (3) That is to say, when we change quantum memory ρB , the change of entropic uncertainty can be completely reflected by the change of lower bound U2. This is a remarkable result which means that the new lower bound U2 can capture the characteristic how the entropic uncertainty would behave corresponding to different quantum memory ρB , since the difference between U2 and LHS is always a constant. While for U1 the difference between LHS and U1 may increase or decrease as the quantum memory ρB changes. Thus Holevo quantity J (B|M1) + J (B|M2), as a new measure of correlation, rather than previous measure I( A : B), describes the underlying quantity between the quantum memory and entropic uncertainty. Additionally, the terms −J (B|M1) − J (B|M1) in the RHS of (2) can be further lower bounded according to enhanced Information Exclusion Relations [33]. While entropic uncertainty can be reduced by quantum correlation, quantum correlation can be affected by Hawking radiation in curved spacetime [ 34–39 ]. It has been shown in [ 34–39 ] that Hawking radiation can reduce mutual information I( A : B) and thus increase entropic uncertainty. It is interesting to investigate how Hawking radiation would modify the new uncertainty relation (2). To demonstrate this, we calculated the simplest black hole: Schwarzschild black hole, and the examples we considered are Dirac fields states. In this case quantum states are superposition of vacuum state and excited states of Dirac fields. We believe similar results can hold for other black holes and fields. In order to compare with results in [ 39 ], spinless field states with basis |0 and |1 are also calculated. In Schwarzschild spacetime, we consider the case in which the quantum memory ρB hovers near the event horizon outside the black hole and the measured system ρA is free falling. It has been known that the entanglement between ρA and ρB would degrade when ρB get closer to the event horizon due to Unruh effect on ρB [ 34 ], so the lower bound for entropic uncertainty would increase [ 39 ]. Because Δ2 is independent of ρB , U2 is always a constant away from LHS. In other words, when quantum memory gets closer to the event horizon, the correlation between it and measured system is decreased, and the decreased correlation is equal to increased entropic uncertainty (Fig. 1). From an experiment point of view [ 40,41 ], a proper uncertainty game [17] can be conducted to measure Bob’s uncertainty LHS about Alice’s measurement outcomes for a particular Schwarzschild black hole with mass M0. U2 and Δ2 can be calculated for this black hole from Rindler decomposition. Since the difference Δ2 between LHS and U2 is independent of mass of black hole M , energy ω of quantum state and the relative distance R0 of quantum memory from event horizon, Δ2 for different ρB in different black hole backgrounds can be obtained. Fixing the energy of mode ω, for an arbitrary Schwarzschild black hole with different mass M , we can predict Bob’s entropic uncertainty accurately without conducting any other new experiments. This article is organized as follows. In Sect. 2 we propose to use Holevo χ -quantity as a part of lower bound for entropic uncertainty relation with quantum memory, and prove that the difference between two sides of this inequality is independent of quantum memory B. Then in Sect. 3 by calculating different examples in Schwarzschild spacetime with Dirac field states, we demonstrate that the new lower bound U2 is tighter than previous bound U1, and more importantly, the Holevo quantity J (B|M1) + J (B|M2) serves as a better correlation measure to reveal how quantum memory would change entropic uncertainty. 2 Entropic uncertainty relation and information exclusion principle Entropic uncertainty relation proved by Maassen and Uffink [ 20 ] is (we use base 2 log throughout this paper), H (M1) + H (M2) ≥ log where M1 = {|u j } and M2 = {|vk } are two orthonormal bases on d-dimensional Hilbert space HA, and H (M1) = − j p j log p j is the Shannon entropy of the probability distribution { p j = u j |ρA|u j } for state ρA of HA (similarly for H (M2) and {qk = vk |ρA|vk }). The number c1 is the largest overlap among all c jk = | u j |vk |2 (≤ 1) between M1 and M2. When measured system is a mixed state, EUR (4) can be improved as [ 26 ] H (M1) + H (M2) ≥ log + H ( A), where H ( A) characterize the amount of uncertainty increased by the mixedness of A. However, if the measured system A is prepared with a quantum memory B, then the entropic uncertainties in the presence of memory are H (M1|B) + H (M2|B), where H (M1|B) = H (ρM1 B ) − H (ρB ) is the conditional entropy with ρM1 B = j (|u j u j | ⊗ I )(ρAB )(|u j u j | ⊗ I ) (similarly for H (M2|B)). Then the difference between H (M1) + H (M2) and H (M1|B) + H (M2|B), i.e. H (M1) + H (M2) − H (M1|B) − H (M2|B), (7) reveals the uncertainty decrease due to the correlations between measured system A and quantum memory B. At the heart of information theory lies the mutual information, Shannon’s fundamental theorem [42, Chapter 12] states that the mutual information corresponding to a measurement is the average amount of error-free data which may be gained through the measurement of system. Information is a natural tool and concept in communications and physics, in an operation of measurement or communication, one may seek to maximize the gained information. This kind of optimization is trivial for classical systems [ 43 ]. (4) (5) (6) 1 c1 , 1 c1 (8) (9) (10) One route to generalize mutual information is motivated by replacing the classical probability distribution by the density matrices of quantum systems., e. g., I( A : B) = H ( A) + H (B) − H ( A, B). Here, H ( A) stands for the von Neumann entropy of quantum state A and H ( A, B) denotes the information of combined system. On the other hand, the quantum memory B, after the measurement corresponding to |u j (M1) has been performed, becomes ρB|u j = u j |ρAB |u j /Tr( u j |ρAB |u j ), with probability p j = Tr( u j |ρAB |u j ). Similarly, we can define qk and ρB|vk for measurement M2. Here H (ρB|u j ) is the missing information about quantum memory. The entropies H (ρB|u j ) with weighted probability p j leads to a second quantum generalization of mutual information J (B|M1) = H (B) − p j H (ρB|u j ). j This quantity reveals the information gained about the quantum memory through the measurement M1. The difference between I( A : B) and J (B|M1) is related to quantum discord [ 44 ]. For any quantum systems, the quantity H (M1)+ H (M2)− H (M1|B) − H (M2|B) describes the uncertainty decrease according to the extra quantum memory, while on the same time J (B|M1) + J (B|M2) is the increase of information content of observables due to quantum memory. What is the relation between the values of uncertainty decrease and information increase in the presence of quantum memory? Actually, we can rewrite ρM1 B = j (|u j u j | ⊗ I )(ρAB )(|u j u j | ⊗ I ) as ρM1 B = j p j u j u j ⊗ ρ Bj , where ρ Bj is density matrix for B if Alice measurement outcome is j . According to joint entropy theorem [ 45 ], we have H (ρM1|B ) = H (ρM1 B ) − H (ρB ) = H (M1) − J (B|M1), thus H (M1) + H (M2) − H (M1|B) − H (M2|B) = J (B|M1) + J (B|M2), (11) for incompatible observables M1 and M2. Through this unified equation, we have shown that the increase of information content of quantum observables in the presence of quantum memory equals to the decrease of quantum uncertainties due to the extra quantum memory. Now these two fundamental concepts in quantum theory and information theory have been unified. ¢ H i + The entropic uncertainty relation now reads that H (M1|B) + H (M2|B) ≥ U2, where U2 ≡ − log c1 + H ( A) − J (B|M1) − J (B|M2). The left hand side (LHS) minus right hand side (RHS) equals to Δ2M1 M2 = H (M1) + H (M2) + log c1 − H ( A) which is independent of quantum memory B. Actually, Δ2 is the difference between LHS and RHS of (5). Put another way, in the presence of quantum memory, the amount of uncertainty decrease equals to the amount of decrease of the lower bound U2. This fact has been revealed in (11). In next section, we will apply this Holevo quantity generalized entropic uncertainty relation to the cases with Dirac field states in Schwarzschild spacetime. Note that the quantity J (B|Mi ) (i = 1, 2) is related to the optimal bound of the Holevo–Schumacher–Westmoreland (HSW) channel capacity [ 45–49 ], i.e., H+ R I I I I I here ε denotes a quantum channel, and { p j , ρ j } is a ensemble decomposition for the density matrix ρ. When we set ε = id, the HSW channel capacity degenerates to the Holevo quantity. However, if the particle (quantum memory) is prepared to be entangled with a measuring system ρA, then the HSW channel capacity CHSW can be generalized to A J + J p j ε(ρB|u j )⎠ − p j ε(ρB|u j )⎠ − j j ⎫ p j H ε(ρB|u j ) ⎬ , ⎫ p j H ε(ρB|u j ) ⎬ , ⎭ ⎭ (13) with measurement M1 perform on measured system A and all ρB|u j satisfy the condition p j ρB|u j = ρB . Similarly, j we define the generalized HSW (GHSW) quantities for mea max min surement M2, i.e. CGHSW(M2) and CGHSW(M2). The maximal value of GHSW is related with the asymptotic rate at which classical information can be transmitted over a quantum channel ε per channel use in the presence of quantum memory [ 45–48 ]. On the other hand, the minimal value of GHSW plays an important role in generalized uncertainty min relations, especially the sum form CGHSW(M1, M2), +H qk ε(ρB|vk ) − qk H ε(ρB|vk ) . (14) k Based on CGmHinSW(M1, M2), we obtain the following relation: max U2 = mρAaBx {− log c1 + H ( A) − J (B|M1) − J (B|M2)} ρAB min = − log c1 + H ( A) − CGHSW(M1, M2)|ε=id. (15) 3 Generalized entropic uncertainty relations in Schwarzschild spacetime 3.1 Dirac field in Schwarzschild spacetime We first review the definition of proper accelerated observer’s vacuum states in Schwarzschild spacetime. A Schwarzschild black hole in Schwarzschild coordinates is given by ds2 = − 1 − 2M r dt 2 + 2M 1 − r −1 dr 2 + r 2d 2 where M is the mass of black hole. Near the event horizon, the metric has similar structure as Rindler horizon in flat spacetime [ 37 ]. The Penrose diagram of the Schwarzschild spacetime is plotted in Fig. 2. To compare two lower bound in uncertainty relation, the field states can be considered as bosonic field states or Dirac field states. In this thesis we choose Dirac field states. However, to make comparison with results in [ 39 ], spinless field states with basis |0 and |1 are also calculated. (16) Dirac field state is considered here instead of bosonic state because there is at most one particle for each spin in one mode due to Pauli’s exclusion principle [ 37 ]: For spinless state spanned by {|0 , |1 }, the Hartle– Hawking vacuum |0 and its first excitation |1 can be expressed in Rindler basis as [ 37,39 ] † σωi I = cI,ωi ,σ |0 I , † σωi IV = dIV,ωi ,σ |0 IV , † † † † pωi I = cI,ωi ,↑cI,ωi ,↓ |0 I = −cI,ωi ,↓cI,ωi ,↑ |0 I , † † † † pωi IV = dIV,ωi ,↑dIV,ωi ,↓|0 IV = −dIV,ωi ,↓dIV,ωi ,↑|0 IV , (17) where pωi represents a pair of spin states in the mode with frequency ωi , σ = ↑ or ↓, and cI,ωi ,σ , dI†V,ωi ,σ are † create operators for particle and anti-particle, respectively. Thus, for each mode, a Dirac particle has four basis states: |0 , |↑ , |↓ , | p . The vacuum corresponding to free falling observer is called Hartle–Hawking vacuum |0 H, which is analogous to Minkowski vacuum. The vacuum corresponding to proper accelerated observer is called Boulware vacuum |0 R, which is analogous to the Rindler vacuum. There is another Boulware vacuum |0 R¯ in region IV. Vacuum is made of different frequency modes |0H ≡ i |0ωi H and similarly for first excitation |1H ≡ i |1ωi H . The relation between different notation is |0 R ↔ |0 I , |0 R¯ ↔ |0 IV , |0 A,B ↔ |0 H . Just like the case in Rindler spacetime, vacuum and excited state for different observer are related by Bogoliubov transformation [ 37,38,50 ] 0ωi H = (cos qd,i )2 0ωi R 0ωi R¯ + sin qd,i cos qd,i ↑ωi R ↓ωi R¯ + ↓ωi R ↑ωi R¯ + (sin qd,i )2 pωi R pωi R¯ , and for one particle state of Hartle–Hawking vacuum ↑ωi H = cos qd,i ↑ωi R 0ωi R¯ + sin qd,i pωi R ↑ωi R¯ , ↓ωi H = cos qd,i ↓ωi R 0ωi R¯ − sin qd,i pωi R ↓ωi R¯ , (20) with tan qd,i = exp Ω − 2 1 − 1/R0 , where R0 = r0/RH = r0/2M , Ω = ω/ TH = 8π ω M and ω is the mode frequency measured by Bob, just the same as above. Rindler approximation is only valid in vicinity of event horizon as mentioned above, i.e. R0 − 1 1 [ 37 ]. (18) (19) (21) (22) (23) (24) (25) (26) (27) For each pair of them, after calculating their eigenvectors, we can find the incompatible term − log c1 = − log maxi1,i2 | uiM1 1 |uiM2 2 |2 is always log 83 . Thus in the following discussion, without loss of generality, we choose σx and σy only. 3.3 Setup In this section we detail the uncertainty game between Alice and Bob. Firstly Bob sends Alice a quantum state A, entangled with his quantum memory B. In this stage, both of them free falling towards the black hole. Then Alice remains free falling into the black hole. But Bob locates at a fixed distance r0 outside the event horizon. At this stage Alice measures her quantum system with measurement either M1 or M2, then sends her measurement choice to Bob through a classical communication channel. The goal of this game is for Bob to reduce his uncertainty about Alice’s measurement outcomes. We assume that Alice has a detector which only detects mode with frequency k and Bob has a detector sensitive to mode 0ωi H = [1 + exp(−Ω + [1 + exp(Ω 1ωi H = 1ωi I 1ωi I V 1 1 − 1/R0]− 2 0ωi I 0ωi I V 1 1 − 1/R0]− 2 1ωi I 1ωi I V where R0 = r0/RH = r0/2M , Ω = ω/ TH = 8π ω M and ω is the mode frequency measured by Bob. 3.2 Incompatible measurements For spinless field state, normal 2-dimensional Pauli matrices are used. σx = |0 1| + 1| |0 , σy = −i |0 1| + i |1 0| , σz = |0 0| − |1 1|. For Dirac field state, 4-dimensional Pauli matrices are utilized for measurements [ 51 ] σy ≡ 21 ⎛⎜⎜⎝ i √000 3 −i200√i3 i−√0023i − U2 (X,Y) U1 U1 U2 (X,Y) with respect to R0 = r0/2M when Ω = ω/ TH = 10, 30. In the following calculation, TH is Hawking temperature and ω is the frequency of the mode. The relative distance of Rob to event horizon R0 ≤ 1.05 is assumed thus Rindler approximation can be hold. Our calculation for U1 agrees with bound in [ 39 ]. 1.02 1.03 Distance R0 1.04 1.05 Fig. 3 Given Ω = ω/ TH = 10 or 30, U2xy is always better than U1 ω. Therefore, the states corresponding to mode ω must be specified in Boulware basis. Since the static observer cannot access the modes beyond the horizon, the lost information reduces the entanglement between A and B, therefore modifies the uncertainty bound. 4 Results 4.1 Spinless field states 4.1.1 Bell state A Bell state in Hilbert space spanned by {|0 , |1 } can be expressed as 1 |Ψ H = √2 (|0 A |0 B + |1 A |1 B ). Figure 3 depicts EUR lower bound for (28) with both U1 and U2x y . The figure depicts the uncertainty bound U1 and U x y 2 1.6 1.4 d n ou1.2 B tyn1.0 itr a ce0.8 n U 0.6 0.4 30 30 10 10 1.02 1.03 Distance R0 Fig. 4 The left figure depicts the uncertainty bound U1 and U xy 2 with respect to R0 = r0/2M when Ω = ω/ TH = 10, 30. The right figure shows the gap between =leftHsi(dXe|Ran)d+riHgh(tYs|iRde):−ΔU12x=y. H (X |R) + H (Y |R) − U1 and Δxy 2 4.2 Dirac field states 4.2.1 A bell-like state We consider a Bell-like state 1 |Ψ H = √2 (|0 A |0 B + |↑ A |↓ B ) . We depict its EUR lower bound for both U1 and U2x y and the difference between H (M1| B) + H (M2| B) in Fig. 4. 4.2.2 W state Consider the case when Alice, Bob and Charlie initially shared a W state from perspective of inertial frame, 1 W = √ (|00 ↑ + |0 ↑ 0 3 + | ↑ 00 ) The entropy uncertainty game is only between Alice and Bob, so Charlie has been traced. We depict EUR lower bound for Alice and Bob when Alice free falls into the black hole and Bob hovers near the event horizon. Both U1 and U x y and 2 their difference with H (M1| B) + H (M2| B) are shown in Fig. 5. U2XY U1 U2XY U1 (29) (30) (28) U2XY U1 U2XY U1 1.00 1.01 1.04 1.05 1.00 1.04 1.05 nd1.70 u o B y it n ta1.65 r e c n U 1.60 10 30 1.01 30 10 1.02 1.03 Distance R0 Given Ω = ω/ TH = 10 or 30, U xy is always better than U1. For 2 different Ω and R0, Δ2 is constant while Δ1 decreases as Ω and R0 increase 2.0 d 1.9 n u oB1.8 y itn 1.7 a tr ce 1.6 n U1.5 1.02 1.03 Distance R0 In all examples we calculated, U2 is tighter than U1. When Bob gets closer to the horizon, his uncertainty about Alice’s state gradually increases for both U1 and U2x y . In addition, the figures shows that for a particular bound U1 or U2, when Ω = ω/ TH is larger, the uncertainty bound is lower. This is evident since fixing the mode energy ω, the larger Ω is, the lower Hawking temperature TH is, which results in more correlation which can reduce the uncertainty. Besides, there is no surprise that Δ2xy = H (M1) + H (M2) − (− log c1) is constant as it is only influenced by the choice of measurements M1, M2 and measured system ρA, not by the quantum memory ρB . We can see from these figures that Δ1 is not always constant but can decrease or increase when R0 increase. This fact indicates that U2 is better than U1 in the sense that, for U2 when the correlation decreases, the amount of increased uncertainty always equals to the amount of decreased correlation. 5 Conclusion In this article, we calculated examples with spinless field states and Dirac field states in Schwarzschild spacetime, demonstrating that uncertainty relation generalized by Holevo quantity not only has a tighter lower bound, but reveals how the quantum memory would influence the entropic uncertainty as well. The second result has implications in experiments. It is sufficient to conduct experiments near one Schwarzschild black hole with mass M0 to obtain LHS. For any other Schwarzschild black holes with mass M , we do not need experiments and can precisely predict its LHS by only using LHS for M0, Δ2 and U2 for M . Acknowledgements Fu-Wen Shu was supported in part by the National Natural Science Foundation of China under Grant no. 11465012. Man-Hong Yung was supported by the Guangdong Innovative and Entrepreneurial Research Team Program (Grant no. 2016ZT06D348), Natural Science Foundation of Guangdong Province (Grant no. 2017B030308003), and the Science, Technology and Innovation Commission of Shenzhen Municipality (Grants no. ZDSYS20170303165926217 and no. JCYJ20170412152620376). Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Funded by SCOAP3. 1. W. Heisenberg , Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik . Z. Phys . 43 , 172 ( 1927 ) 2. E.H. Kennard , Zur Quantenmechanik einfacher Bewegungstypen. Z. Phys . 44 , 326 ( 1927 ) 3. H. Weyl , Gruppentheorie und Quantenmechanik (Hirzel, Leipzig, 1928 ) 4. H.P. Robertson , The uncertainty principle . Phys. Rev . 34 , 163 ( 1929 ) 5. E. Schrödinger , Ber. Kgl. Akad. Wiss. Berlin 24 , 296 ( 1930 ) 6. L. Maccone , A.K. Pati , Stronger uncertainty relations for all incompatible observables . Phys. Rev. Lett . 113 , 260401 ( 2014 ) 7. J.L. Li , C.F. Qiao , Reformulating the quantum uncertainty relation . Sci. Rep . 5 , 12708 ( 2015 ) 8. Y. Xiao , N. Jing , X. Li-Jost , S.-M. Fei , Weighted uncertainty relations . Sci. Rep . 6 , 23201 ( 2016 ) 9. Q.-C. Song , C.-F. Qiao , Stronger Schrödinger-like uncertainty relations . Phys. Lett. A 380 , 2925 ( 2016 ) 10. A.A. Abbott , P.-L. Alzieu , M.J.W. Hall , C. Branciard , Tight stateindependent uncertainty relations for qubits . Mathematics 4 , 8 ( 2016 ) 11. Y. Xiao , N. Jing , B. Yu , S.-M. Fei , X. Li-Jost , Strong variance-based uncertainty relations and uncertainty intervals . arXiv:1610.01692 12. R. Schwonnek , L. Dammeier , R.F. Werner , State-independent uncertainty relations and entanglement detection in noisy systems . Phys. Rev. Lett . 119 , 170404 ( 2017 ) 13. Y. Xiao , C. Guo , F. Meng , N. Jing , M.-H. Yung , Incompatibility of observables as state-independent bound of uncertainty relations . arXiv:1706.05650 14. Q.C. Song , J.L. Li , G.X. Peng , C.F. Qiao , A stronger multiobservable uncertainty relation . Sci. Rep . 7 , 44764 ( 2017 ) 15. Z.X. Chen , J.L. Li , Q.C. Song , H. Wang , S.M. Zangi , C.F. Qiao , Experimental investigation of multi-observable uncertainty relations . Phys. Rev. A 96 , 062123 ( 2017 ) 16. I. Białynicki-Birula , J. Mycielski , Uncertainty relations for information entropy in wave mechanics . Commun. Math. Phys. 44 , 129 ( 1975 ) 17. P.J. Coles , M. Berta , M. Tomamichel , S. Wehner , Entropic uncertainty relations and their applications . Rev. Mod. Phys . 89 , 015002 ( 2017 ) 18. D. Deutsch , Uncertainty in quantum measurements . Phys. Rev. Lett . 50 , 631 ( 1983 ) 19. K. Kraus , Complementary observables and uncertainty relations . Phys. Rev. D 35 , 3070 ( 1987 ) 20. H. Maassen , J.B.M. Uffink , Generalized entropic uncertainty relations . Phys. Rev. Lett . 60 , 1103 ( 1988 ) 21. P.J. Coles , M. Piani , Improved entropic uncertainty relations and information exclusion relations . Phys. Rev. A 89 , 022112 ( 2014 ) 22. Y. Xiao , N. Jing , S.-M. Fei , X. Li-Jost , Improved uncertainty relation in the presence of quantum memory . J. Phys. A 49 , 49LT01 ( 2016 ) 23. Z. Puchała , Ł. Rudnicki, K. Z˙ yczkowski, Majorization entropic uncertainty relations . J. Phys. A 46 , 272002 ( 2013 ) 24. Ł. Rudnicki , Z. Puchała , K. Z˙ yczkowski, Strong majorization entropic uncertainty relations . Phys. Rev. A 89 , 052115 ( 2014 ) 25. S. Friedland , V. Gheorghiu , G. Gour, Universal uncertainty relations . Phys. Rev. Lett . 111 , 230401 ( 2013 ) 26. M. Berta , M. Christandl , R. Colbeck , J.M. Renes , R. Renner , The uncertainty principle in the presence of quantum memory . Nat. Phys . 6 , 659 ( 2010 ) 27. A.S. Holevo , Bounds for the quantity of information transmitted by a quantum communication channel . Probl. Inf. Transm . 9 , 177 ( 1973 ) 28. R. Bousso , Universal limit on communication . Phys. Rev. Lett . 119 , 140501 ( 2017 ) 29. N. Bao , H. Ooguri , Distinguishability of black hole microstates . Phys. Rev. D 96 , 066017 ( 2017 ) 30. B. Michel , A. Puhm , Corrections in the relative entropy of black hole microstates . arXiv: 1801 .02615 31. Y. Xiao , N. Jing , X. Li-Jost , Uncertainty under quantum measures and quantum memory . Quantum Inf. Proc. 16 , 104 ( 2017 ) 32. F. Adabi , S. Salimi , S. Haseli , Tightening the entropic uncertainty bound in the presence of quantum memory . Phys. Rev. A 93 , 062123 ( 2016 ) 33. Y. Xiao , N. Jing , X. Li-Jost , Enhanced information exclusion relations . Sci. Rep . 6 , 30440 ( 2016 ) 34. I. Fuentes-Schuller , R.B. Mann , Alice falls into a black hole: entanglement in noninertial frames . Phys. Rev. Lett . 95 , 120404 ( 2005 ) 35. Q. Pan , J. Jing , Hawking radiation, entanglement and teleportation in background of an asymptotically flat static black hole . Phys. Rev. D 78 , 065015 ( 2008 ) 36. J. Wang , Q. Pan , J. Jing , Entanglement redistribution in the Schwarzschild spacetime . Phys. Lett. B 692 , 202 ( 2010 ) 37. E. Martin-Martinez , L.J. Garay , J. Leon , Unveiling quantum entanglement degradation near a Schwarzschild black hole . Phys. Rev. D 82 , 064006 ( 2010 ) 38. E. Martin-Martinez , J. Leon , Quantum correlations through event horizons: fermionic versus bosonic entanglement . Phys. Rev. A 81 , 032320 ( 2010 ) 39. J. Feng , Y.Z. Zhang , M.D. Gould , H. Fan , Uncertainty relation in Schwarzschild spacetime . Phys. Lett. B 743 , 198 ( 2015 ) 40. R. Prevedel , D.R. Hamel , R. Colbeck , K. Fisher, K.J. Resch , Experimental investigation of the uncertainty principle in the presence of quantum memory and its application to witnessing entanglement . Nat. Phys . 7 , 757 ( 2011 ) 41. C.-F. Li , J.-S. Xu , X.-Y. Xu , K. Li , G.-C. Guo , Experimental demonstration of delayed-choice decoherence suppression . Nat. Phys . 7 , 752 ( 2011 ) 42. C.L. Mallows , F.M. Reza , An introduction to information theory . Am Math Mon 71 , 108 ( 1964 ) 43. M.J.W. Hall , Information exclusion principle for complementary observables . Phys. Rev. Lett . 74 , 3307 ( 1994 ) 44. H. Ollivier , W.H. Zurek , Quantum discord: a measure of the quantumness of correlations . Phys. Rev. Lett . 88 , 017901 ( 2001 ) 45. M.A. Nielsen , I.L. Chuang , Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2015 ) 46. P. Hausladen , R. Jozsa , B.W. Schumacher , M. Westmoreland , W.K. Wootters , Classical information capacity of a quantum channel . Phys. Rev. A 54 , 1869 ( 1996 ) 47. B.W. Schumacher , M. Westmoreland , Sending classical information via noisy quantum channels . Phys. Rev. A 56 , 131 ( 1997 ) 48. A.S. Holevo , The capacity of the quantum channel with general signal states . IEEE Trans. Inf. Theory 44 , 269 ( 1998 ) 49. R.A.C. Medeiros , F.M. de Assis , Quantum zero-error capacity and HSW capacity . AIP Conf. Proc. 734 , 52 ( 2004 ) 50. P.M. Alsing , I. Fuentes-Schuller , R.B. Mann , T.E. Tessier , Entanglement of Dirac fields in non-inertial frames . Phys. Rev. A 74 , 032326 ( 2006 ) 51. D.J. Griffiths , Introduction to Quantum Mechanics (Cambridge University Press, Cambridge, 2016 )


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1140%2Fepjc%2Fs10052-018-6026-3.pdf

Jin-Long Huang, Wen-Cong Gan, Yunlong Xiao, Fu-Wen Shu, Man-Hong Yung. Holevo bound of entropic uncertainty in Schwarzschild spacetime, The European Physical Journal C, 2018, 545, DOI: 10.1140/epjc/s10052-018-6026-3