1 Introduction

Traditionally, uncertainty principle in quantum mechanics has been formulated in terms of variance [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15], while in the context of both classical and quantum information sciences, it is more natural to use entropy to quantify uncertainties. The first entropic uncertainty relation (EUR) for position and momentum was given in [16], which can be shown to be equivalent to Heisenberg’s original relation [17]. Later Deutsch [18] formulated entropic uncertainty relation for any pair of observables with bounded spectrums. An improvement of Deutsch’s entropic uncertainty relation was subsequently conjectured by Kraus [19] and later proved by Maassen and Uffink [20]. Recently, more improvements have been formulated [21,22,23,24,25]. However, if the measured system \(\rho _A\) is prepared with a quantum memory \(\rho _B\), correlation between \(\rho _A\) and \(\rho _B\) will decrease the entropic uncertainties of \(\rho _A\). The entropic uncertainty relation in the presence of quantum memory was proposed by Berta et al.:

$$\begin{aligned} H(M_{1}|B)+H(M_{2}|B)\ge U_1 , \end{aligned}$$
(1)

where \(U_1\equiv -\log c_{1}+H(A)-\mathcal {I}(A: B)\) [26], with \(H(\rho )\equiv -\text {tr}(\rho \text { ln} \rho )\) is von Neumann entropy for density matrix \(\rho \) and \(\mathcal {I}(A: B)=H(A)+H(B)-H(A, B)\) is the quantum mutual information.

However, as the correlation between two parties changes, is this mutual information \(\mathcal {I}(A: B)\) 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 [28,29,30]. 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 \(\rho _{AB}\) to one party \(\rho _B\). 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 \(\mathcal {J}(B|M_1) \equiv H(B)-\sum _j p^1_j H(\rho _{B|u^1_{j}})\) be the Holevo quantity for Bob about Alice’s \(M_1\) measurement outcomes. The new uncertainty relation says that

$$\begin{aligned} H(M_{1}|B)+H(M_{2}|B)\ge U_2 , \end{aligned}$$
(2)

where \(U_2\equiv -\log c_{1}+H(A)-\mathcal {J}(B|M_1)-\mathcal {J}(B|M_2)\). The new entropic uncertainty relation has the property that the difference (which we denote as \(\varDelta _2\) below) between entropic uncertainty (LHS of (2)) and the new uncertainty lower bound \(U_2\) only depends on incompatible measurements \(M_1, M_2\) and \(\rho _A\), and independent of quantum memory,

$$\begin{aligned} \varDelta _2^{M_1 M_2}=H(M_1)+H(M_2)+\log c_1-H(A) . \end{aligned}$$
(3)

That is to say, when we change quantum memory \(\rho _B\), the change of entropic uncertainty can be completely reflected by the change of lower bound \(U_2\). This is a remarkable result which means that the new lower bound \(U_2\) can capture the characteristic how the entropic uncertainty would behave corresponding to different quantum memory \(\rho _B\), since the difference between \(U_2\) and LHS is always a constant. While for \(U_1\) the difference between LHS and \(U_1\) may increase or decrease as the quantum memory \(\rho _B\) changes. Thus Holevo quantity \(\mathcal {J}(B|M_1)+\mathcal {J}(B|M_2)\), as a new measure of correlation, rather than previous measure \(\mathcal {I}(A: B)\), describes the underlying quantity between the quantum memory and entropic uncertainty. Additionally, the terms \(-\mathcal {J}(B|M_1)-\mathcal {J}(B|M_1)\) 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 [34,35,36,37,38,39]. It has been shown in [34,35,36,37,38,39] that Hawking radiation can reduce mutual information \(\mathcal {I}(A: B)\) 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 \(\left| {0} \right\rangle \) and \(\left| {1} \right\rangle \) are also calculated. In Schwarzschild spacetime, we consider the case in which the quantum memory \(\rho _B\) hovers near the event horizon outside the black hole and the measured system \(\rho _A\) is free falling. It has been known that the entanglement between \(\rho _A\) and \(\rho _B\) would degrade when \(\rho _B\) get closer to the event horizon due to Unruh effect on \(\rho _B\) [34], so the lower bound for entropic uncertainty would increase [39]. Because \(\varDelta _2\) is independent of \(\rho _B\), \(U_2\) 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).

Fig. 1
figure 1

Entropic uncertainty (LHS) and lower bound \(U_2\) using accessible information for black hole with mass \(M_0\) and \(M'\). \(\varDelta _2^1=\varDelta _2^2=\varDelta _2^3=\varDelta _2^4\) where \(\varDelta _2^1\) and \(\varDelta _2^2\) correspond to different relative distance \(R_0=r_0/2M\) from event horizon. As shown above, at event horizon \(R_0=1\). It is sufficient to conduct experiments near one Schwarzschild black hole with mass \(M_0\) to obtain LHS. For any other Schwarzschild black holes with mass \(M'\), we can precisely predict its LHS using LHS for \(M_0\), \(\varDelta _2\) and \(U_2\) for \(M'\)

Full size image

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 \(M_0\). \(U_2\) and \(\varDelta _2\) can be calculated for this black hole from Rindler decomposition. Since the difference \(\varDelta _2\) between LHS and \(U_2\) is independent of mass of black hole M, energy \(\omega \) of quantum state and the relative distance \(R_0\) of quantum memory from event horizon, \(\varDelta _2\) for different \(\rho _B\) in different black hole backgrounds can be obtained. Fixing the energy of mode \(\omega \), 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 \(\chi \)-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 \(U_2\) is tighter than previous bound \(U_1\), and more importantly, the Holevo quantity \(\mathcal {J}(B|M_1)+\mathcal {J}(B|M_2)\) 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),

$$\begin{aligned} H(M_{1})+H(M_{2})\ge \log \frac{1}{c_{1}}, \end{aligned}$$
(4)

where \(M_{1}=\{|u_{j}\rangle \}\) and \(M_{2}=\{|v_{k}\rangle \}\) are two orthonormal bases on d-dimensional Hilbert space \(\mathcal {H}_{A}\), and \(H(M_{1})=-\sum _{j}p_{j}\log p_{j}\) is the Shannon entropy of the probability distribution \(\{p_{j}=\langle u_{j}|\rho _{A}|u_{j}\rangle \}\) for state \(\rho _{A}\) of \(\mathcal {H}_{A}\) (similarly for \(H(M_{2})\) and \(\{q_{k}=\langle v_{k}|\rho _{A}|v_{k}\rangle \}\)). The number \(c_{1}\) is the largest overlap among all \(c_{jk}=|\langle u_{j}|v_{k}\rangle |^{2}\) (\(\le 1\)) between \(M_{1}\) and \(M_{2}\).

When measured system is a mixed state, EUR (4) can be improved as [26]

$$\begin{aligned} H(M_{1})+H(M_{2})\ge \log \frac{1}{c_{1}}+H(A), \end{aligned}$$
(5)

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

$$\begin{aligned} H(M_{1}|B)+H(M_{2}|B), \end{aligned}$$
(6)

where \(H(M_{1}|B)=H(\rho _{M_1B})-H(\rho _B)\) is the conditional entropy with \(\rho _{M_{1}B}=\sum _{j}(|u_{j}\rangle \langle u_{j}|\otimes I)(\rho _{AB})(|u_{j}\rangle \langle u_{j}|\otimes I)\) (similarly for \(H(M_{2}|B)\)). Then the difference between \(H(M_{1})+H(M_{2})\) and \(H(M_{1}|B)+H(M_{2}|B)\), i.e.

$$\begin{aligned} H(M_{1})+H(M_{2})-H(M_{1}|B)-H(M_{2}|B), \end{aligned}$$
(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].

One route to generalize mutual information is motivated by replacing the classical probability distribution by the density matrices of quantum systems., e. g.,

$$\begin{aligned} \mathcal {I}(A: B)=H(A)+H(B)-H(A, B). \end{aligned}$$
(8)

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 \(|u_{j}\rangle \) (\(M_{1}\)) has been performed, becomes

$$\begin{aligned} \rho _{B|u_{j}}=\langle u_{j}|\rho _{AB}|u_{j}\rangle /\mathrm {Tr}(\langle u_{j}|\rho _{AB}|u_{j}\rangle ), \end{aligned}$$
(9)

with probability \(p_{j}=\mathrm {Tr}(\langle u_{j}|\rho _{AB}|u_{j}\rangle )\). Similarly, we can define \(q_{k}\) and \(\rho _{B|v_{k}}\) for measurement \(M_{2}\). Here \(H(\rho _{B|u_{j}})\) is the missing information about quantum memory. The entropies \(H(\rho _{B|u_{j}})\) with weighted probability \(p_{j}\) leads to a second quantum generalization of mutual information

$$\begin{aligned} \mathcal {J}(B|M_1)=H(B)-\sum \limits _{j}p_{j}H(\rho _{B|u_{j}}). \end{aligned}$$
(10)

This quantity reveals the information gained about the quantum memory through the measurement \(M_{1}\). The difference between \(\mathcal {I}(A: B)\) and \(\mathcal {J}(B|M_1)\) is related to quantum discord [44].

For any quantum systems, the quantity \(H(M_{1})+H(M_{2})-H(M_{1}|B)-H(M_{2}|B)\) describes the uncertainty decrease according to the extra quantum memory, while on the same time \(\mathcal {J}(B|M_1)+\mathcal {J}(B|M_2)\) 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 \(\rho _{M_{1}B}=\sum _{j}(|u_{j}\rangle \langle u_{j}|\otimes I)(\rho _{AB})(|u_{j}\rangle \langle u_{j}|\otimes I)\) as \(\rho _{M_1 B}=\sum _j p_j \left| {u_j} \right\rangle \left\langle {u_j} \right| \otimes \rho ^j_B\), where \(\rho ^j_B\) is density matrix for B if Alice measurement outcome is j. According to joint entropy theorem [45], we have \(H(\rho _{M_1| B})=H(\rho _{M_1 B})-H(\rho _B)=H(M_1)-\mathcal {J}(B|M_1)\), thus

$$\begin{aligned}&H(M_{1})+H(M_{2})-H(M_{1}|B)-H(M_{2}|B)\nonumber \\&\quad =\mathcal {J}(B|M_1)+\mathcal {J}(B|M_2), \end{aligned}$$
(11)

for incompatible observables \(M_{1}\) and \(M_{2}\). 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.

The entropic uncertainty relation now reads that \(H(M_{1}|B)+H(M_{2}|B)\ge U_2\), where \(U_2\equiv -\log c_{1}+H(A)-\mathcal {J}(B|M_1)-\mathcal {J}(B|M_2)\). The left hand side (LHS) minus right hand side (RHS) equals to \(\varDelta _2^{M_1 M_2}=H(M_1)+H(M_2)+\log c_1-H(A)\) which is independent of quantum memory B. Actually, \(\varDelta _2\) 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 \(U_2\). 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 \(\mathcal {J}(B|M_i)\) \((i=1, 2)\) is related to the optimal bound of the Holevo–Schumacher–Westmoreland (HSW) channel capacity [45,46,47,48,49], i.e.,

$$\begin{aligned} \mathcal {C_{\text {HSW}}} = \max \limits _{\sum \limits _{j} p_{j} \rho _{j} = \rho } \left\{ H\left( \sum \limits _{j} p_{j} \varepsilon (\rho _{j})\right) - \sum \limits _{j} p_{j} H\left( \varepsilon (\rho _{j})\right) \right\} , \end{aligned}$$
(12)

here \(\varepsilon \) denotes a quantum channel, and \(\{p_{j}, \rho _{j}\}\) is a ensemble decomposition for the density matrix \(\rho \). When we set \(\varepsilon =\text {id}\), the HSW channel capacity degenerates to the Holevo quantity. However, if the particle (quantum memory) is prepared to be entangled with a measuring system \(\rho _{A}\), then the HSW channel capacity \(\mathcal {C_{\text {HSW}}}\) can be generalized to

$$\begin{aligned}&\mathcal {C}_{\text {GHSW}}^{\text {max}} (M_{1}) \nonumber \\&\quad = \max \limits _{\rho _{AB}} \left\{ H\left( \sum \limits _{j} p_{j} \varepsilon (\rho _{B|u_{j}})\right) - \sum \limits _{j} p_{j} H\left( \varepsilon (\rho _{B|u_{j}})\right) \right\} ,\nonumber \\&\mathcal {C}_{\text {GHSW}}^{\text {min}} (M_{1}) \nonumber \\&\quad = \min \limits _{\rho _{AB}} \left\{ H\left( \sum \limits _{j} p_{j} \varepsilon (\rho _{B|u_{j}})\right) - \sum \limits _{j} p_{j} H\left( \varepsilon (\rho _{B|u_{j}})\right) \right\} , \end{aligned}$$
(13)

with measurement \(M_{1}\) perform on measured system A and all \(\rho _{B|u_{j}}\) satisfy the condition \(\sum \limits _{j} p_{j} \rho _{B|u_{j}} = \rho _{B}\). Similarly, we define the generalized HSW (GHSW) quantities for measurement \(M_{2}\), i.e. \(\mathcal {C}_{\text {GHSW}}^{\text {max}} (M_{2})\) and \(\mathcal {C}_{\text {GHSW}}^{\text {min}} (M_{2})\). The maximal value of GHSW is related with the asymptotic rate at which classical information can be transmitted over a quantum channel \(\varepsilon \) per channel use in the presence of quantum memory  [45,46,47,48]. On the other hand, the minimal value of GHSW plays an important role in generalized uncertainty relations, especially the sum form \(\mathcal {C}_{\text {GHSW}}^{\text {min}} (M_{1}, M_{2})\),

$$\begin{aligned}&\mathcal {C}_{\text {GHSW}}^{\text {min}} (M_{1}, M_{2}) \nonumber \\&\quad = \min \limits _{\rho _{AB}} \left\{ H\left( \sum \limits _{j} p_{j} \varepsilon (\rho _{B|u_{j}})\right) - \sum \limits _{j} p_{j} H\left( \varepsilon (\rho _{B|u_{j}})\right) \right. \nonumber \\&\qquad \left. +H\left( \sum \limits _{k} q_{k} \varepsilon (\rho _{B|v_{k}})\right) - \sum \limits _{k} q_{k} H\left( \varepsilon (\rho _{B|v_{k}})\right) \right\} . \end{aligned}$$
(14)

Based on \(\mathcal {C}_{\text {GHSW}}^{\text {min}} (M_{1}, M_{2})\), we obtain the following relation:

$$\begin{aligned} \max \limits _{\rho _{AB}} U_{2} =&\max \limits _{\rho _{AB}} \left\{ -\log c_{1}+H(A)-\mathcal {J}(B|M_1)-\mathcal {J}(B|M_2) \right\} \nonumber \\ =&-\log c_{1}+H(A)-\mathcal {C}_{\text {GHSW}}^{\text {min}} (M_{1}, M_{2})|_{\varepsilon =\text {id}} . \end{aligned}$$
(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

$$\begin{aligned} ds^2=-\bigg (1-\frac{2M}{r}\bigg )dt^2+\bigg (1-\frac{2M}{r}\bigg )^{-1}dr^2+r^2d\varvec{\Omega }^2\nonumber \\ \end{aligned}$$
(16)

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.

Fig. 2
figure 2

Penrose diagram for maximally extended black hole which shows the world-line of Alice, Rob and Anti-Rob. \(i^0\) denotes the spatial infinities, \(i^-\) (\(i^+\)) denotes timelike past (future) infinity. \(J^-\) (\(J^+\)) denotes lightlike past (future) infinity. \(H^{\pm }\) denote the event horizons of the black hole

Full size image

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 \(\left| {0} \right\rangle \) and \(\left| {1} \right\rangle \) are also calculated.

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]:

$$\begin{aligned} \left| {\sigma _{{\omega _i}}} \right\rangle _\text {I}&=c^\dagger _{\text {I},{\omega _i},\sigma }\left| {0} \right\rangle _\text {I},\nonumber \\ \left| {\sigma _{{\omega _i}}} \right\rangle _\text {IV}&=d^\dagger _{\text {IV},{\omega _i},\sigma }\left| {0} \right\rangle _{\text {IV}},\nonumber \\ \left| {p_{{\omega _i}}} \right\rangle _\text {I}&=c^\dagger _{\text {I},{\omega _i},\uparrow } c^\dagger _{\text {I},{\omega _i},\downarrow }\left| {0} \right\rangle _\text {I} =-c^\dagger _{\text {I},{\omega _i},\downarrow }c^\dagger _{\text {I},{\omega _i}, \uparrow }\left| {0} \right\rangle _\text {I},\nonumber \\ \left| {p_{{\omega _i}}} \right\rangle _{\text {IV}}&=d^\dagger _{\text {IV},{\omega _i},\uparrow }d^\dagger _{\text {IV},{\omega _i}, \downarrow }\!\left| {0} \right\rangle _{\text {IV}}=-d^\dagger _{\text {IV},{\omega _i},\downarrow } d^\dagger _{\text {IV},{\omega _i},\uparrow }\!\left| {0} \right\rangle _{\text {IV}}, \end{aligned}$$
(17)

where \(p_{{\omega _i}}\) represents a pair of spin states in the mode with frequency \({\omega _i}\), \(\sigma =\,\uparrow \text {or} \downarrow \), and \(c^\dagger _{\text {I},{\omega _i},\sigma },d^\dagger _{\text {IV},{\omega _i},\sigma }\) are create operators for particle and anti-particle, respectively. Thus, for each mode, a Dirac particle has four basis states: \(\left| {0} \right\rangle ,\left| {\uparrow } \right\rangle ,\left| {\downarrow } \right\rangle ,\left| {p} \right\rangle \).

The vacuum corresponding to free falling observer is called Hartle–Hawking vacuum \(|0\rangle _{\text {H}}\), which is analogous to Minkowski vacuum. The vacuum corresponding to proper accelerated observer is called Boulware vacuum \(|0\rangle _{\text {R}}\), which is analogous to the Rindler vacuum. There is another Boulware vacuum \(|0\rangle _{\bar{\text {R}}}\) in region IV. Vacuum is made of different frequency modes \(|0_H\rangle \equiv \bigotimes _i|0_{\omega _i}\rangle _H\) and similarly for first excitation \(|1_H\rangle \equiv \bigotimes _i|1_{\omega _i}\rangle _H\). The relation between different notation is

$$\begin{aligned} \begin{array}{lllll} \left| {0} \right\rangle _\text {R}&{}\leftrightarrow &{}\left| {0} \right\rangle _\text {I},\\ \left| {0} \right\rangle _{\bar{\text {R}}}&{}\leftrightarrow &{}\left| {0} \right\rangle _{\text {IV}},\\ \left| {0} \right\rangle _{\text {A,B}}&{}\leftrightarrow &{}\left| {0} \right\rangle _{\text {H}}. \end{array} \end{aligned}$$
(18)

Just like the case in Rindler spacetime, vacuum and excited state for different observer are related by Bogoliubov transformation [37, 38, 50]

$$\begin{aligned} \left| {0_{\omega _i}} \right\rangle _{\text {H}}&= (\cos q_{\text {d},i})^2 \left| {0_{\omega _i}} \right\rangle _{\text {R}}\left| {0_{\omega _i}} \right\rangle _{\bar{\text {R}}}\nonumber \\&\quad +\sin q_{\text {d},i}\cos q_{\text {d},i} \left( \left| {\uparrow _{\omega _i}} \right\rangle _{{\text {R}}} \left| {\downarrow _{\omega _i}} \right\rangle _{\bar{\text {R}}}+ \left| {\downarrow _{\omega _i}} \right\rangle _{\text {R}}\left| {\uparrow _{\omega _i}} \right\rangle _{\bar{\text {R}}}\right) \nonumber \\&\quad +(\sin q_{\text {d},i})^2\left| {p_{\omega _i}} \right\rangle _{\text {R}}\left| {p_{\omega _i}} \right\rangle _{\bar{\text {R}}}, \end{aligned}$$
(19)

and for one particle state of Hartle–Hawking vacuum

$$\begin{aligned} \left| {\uparrow _{\omega _i}} \right\rangle _\text {H}&= \cos q_{\text {d},i} \left| {\uparrow _{\omega _i}} \right\rangle _{\text {R}}\left| {0_{\omega _i}} \right\rangle _{\bar{\text {R}}}+ \sin q_{\text {d},i}\left| {p_{\omega _i}} \right\rangle _{\text {R}}\left| {\uparrow _{\omega _i}} \right\rangle _{\bar{\text {R}}},\nonumber \\ \left| {\downarrow _{\omega _i}} \right\rangle _\text {H}&= \cos q_{\text {d},i} \left| {\downarrow _{\omega _i}} \right\rangle _{\text {R}}\left| {0_{\omega _i}} \right\rangle _{\bar{\text {R}}}- \sin q_{\text {d},i}\left| {p_{\omega _i}} \right\rangle _{\text {R}}\left| {\downarrow _{\omega _i}} \right\rangle _{\bar{\text {R}}}, \end{aligned}$$
(20)

with

$$\begin{aligned} \tan q_{\text {d},i}=\exp \left( -\frac{\varOmega }{2}\sqrt{1-1/R_0}\;\right) , \end{aligned}$$
(21)

where \(R_0=r_0/R_H=r_0/2M\), \(\varOmega =\omega /T_H=8\pi \omega M\) and \(\omega \) 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. \(R_0-1 \ll 1\) [37].

For spinless state spanned by \(\{\left| {0} \right\rangle ,\left| {1} \right\rangle \}\), the Hartle–Hawking vacuum \(\left| {0} \right\rangle \) and its first excitation \(\left| {1} \right\rangle \) can be expressed in Rindler basis as [37, 39]

$$\begin{aligned} \left| {0_{\omega _i}} \right\rangle _H&= [1+ \text {exp} (-\varOmega \sqrt{1-1/R_0}]^{-\frac{1}{2}} \left| {0_{\omega _i}} \right\rangle _{I} \left| {0_{\omega _i}} \right\rangle _{IV} \end{aligned}$$
(22)
$$\begin{aligned}&\quad +[1+\text {exp}(\varOmega \sqrt{1-1/R_0}]^{-\frac{1}{2}} \left| {1_{\omega _i}} \right\rangle _{I} \left| {1_{\omega _i}} \right\rangle _{IV}\end{aligned}$$
(23)
$$\begin{aligned} \left| {1_{\omega _i}} \right\rangle _H&= \left| {1_{\omega _i}} \right\rangle _{I} \left| {1_{\omega _i}} \right\rangle _{IV} \end{aligned}$$
(24)

where \(R_0=r_0/R_H=r_0/2M\), \(\varOmega =\omega /T_H=8\pi \omega M\) and \(\omega \) is the mode frequency measured by Bob.

3.2 Incompatible measurements

For spinless field state, normal 2-dimensional Pauli matrices are used. \(\sigma _x = \left| {0} \right\rangle \left\langle {1} \right| +\left\langle {1} \right| \left| {0} \right\rangle , \sigma _y = -i\left| {0} \right\rangle \left\langle {1} \right| +i\left| {1} \right\rangle \left\langle {0} \right| , \sigma _z=\left| {0} \right\rangle \left\langle {0} \right| -\left| {1} \right\rangle \left\langle {1} \right| \).

For Dirac field state, 4-dimensional Pauli matrices are utilized for measurements [51]

$$\begin{aligned} \sigma _x&\equiv \frac{1}{2}\begin{pmatrix} 0 &{} \sqrt{3} &{} 0 &{} 0 \\ \sqrt{3}&{} 0&{} 2&{} 0 \\ 0&{} 2&{} 0&{} \sqrt{3}\\ 0&{} 0&{} \sqrt{3}&{} 0 \end{pmatrix},\end{aligned}$$
(25)
$$\begin{aligned} \sigma _y&\equiv \frac{1}{2}\begin{pmatrix} 0 &{} -i\sqrt{3} &{} 0 &{} 0 \\ i\sqrt{3}&{} 0&{} -2i&{} 0 \\ 0&{} 2i&{} 0&{} -i\sqrt{3}\\ 0&{} 0&{} i\sqrt{3}&{} 0 \end{pmatrix},\end{aligned}$$
(26)
$$\begin{aligned} \sigma _z&\equiv \frac{1}{2}\begin{pmatrix} 3 &{} 0 &{} 0 &{} 0 \\ 0&{} 1&{} 0&{} 0 \\ 0&{} 0&{} -1&{} 0\\ 0&{} 0&{} 0&{} -3 \end{pmatrix}. \end{aligned}$$
(27)

For each pair of them, after calculating their eigenvectors, we can find the incompatible term \(-\log c_1=-\log \max _{i_{1},i_{2}}\mid \langle u^{M_1}_{i_{1}}|u^{M_2}_{i_{2}}\rangle \mid ^{2}\) is always \(\log \frac{8}{3}\). Thus in the following discussion, without loss of generality, we choose \(\sigma _x\) and \(\sigma _y\) only.

Fig. 3
figure 3

Given \(\varOmega =\omega /T_H=10\) or 30, \(U_2^{x y}\) is always better than \(U_1\)

Full size image
Fig. 4
figure 4

The left figure depicts the uncertainty bound \(U_1\) and \(U_2^{x y}\) with respect to \(R_0=r_0/2M\) when \(\varOmega =\omega /T_H=10, 30\). The right figure shows the gap between left side and right side: \(\varDelta _1=H(X|R)+H(Y|R)-U_1\) and \(\varDelta _2^{x y}=H(X|R)+H(Y|R)-U_2^{x y}\). Given \(\varOmega =\omega /T_H=10\) or 30, \(U_2^{x y}\) is always better than \(U_1\). For different \(\varOmega \) and \(R_0\), \(\varDelta _2\) is constant while \(\varDelta _1\) decreases as \(\varOmega \) and \(R_0\) increase

Full size image

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 \(r_0\) outside the event horizon. At this stage Alice measures her quantum system with measurement either \(M_1\) or \(M_2\), 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 \(\omega \). Therefore, the states corresponding to mode \(\omega \) 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 \(\{\left| {0} \right\rangle , \left| {1} \right\rangle \}\) can be expressed as

$$\begin{aligned} \left| {\varPsi } \right\rangle _H = \frac{1}{\sqrt{2}}(\left| {0} \right\rangle _A\left| {0} \right\rangle _B+\left| {1} \right\rangle _A\left| {1} \right\rangle _B). \end{aligned}$$
(28)

Figure 3 depicts EUR lower bound for (28) with both \(U_1\) and \(U_2^{x y}\). The figure depicts the uncertainty bound \(U_1\) and \(U_2^{x y}\) with respect to \(R_0=r_0/2M\) when \(\varOmega =\omega /T_H=10, 30\). In the following calculation, \(T_H\) is Hawking temperature and \(\omega \) is the frequency of the mode. The relative distance of Rob to event horizon \(R_0 \le 1.05\) is assumed thus Rindler approximation can be hold. Our calculation for \(U_1\) agrees with bound in [39].

4.2 Dirac field states

4.2.1 A bell-like state

We consider a Bell-like state

$$\begin{aligned} \left| {\varPsi } \right\rangle _{H}=\frac{1}{\sqrt{2}}\left( \left| {0} \right\rangle _A\left| {0} \right\rangle _B+\left| {\uparrow } \right\rangle _A\left| {\downarrow } \right\rangle _B\right) . \end{aligned}$$
(29)

We depict its EUR lower bound for both \(U_1\) and \(U_2^{x y}\) and the difference between \(H(M_{1}|B)+H(M_{2}|B)\) in Fig. 4.

Fig. 5
figure 5

Given \(\varOmega =\omega /T_H=10\) or 30, \(U_2^{x y}\) is always better than \(U_1\). For different \(\varOmega \) and \(R_0\), \(\varDelta _2\) is constant while \(\varDelta _1\) increases as \(\varOmega \) and \(R_0\) increase

Full size image

4.2.2 W state

Consider the case when Alice, Bob and Charlie initially shared a W state from perspective of inertial frame,

$$\begin{aligned} W = \frac{1}{\sqrt{3}} (|00\uparrow \rangle +|0\uparrow 0\rangle +|\uparrow 00\rangle ) \end{aligned}$$
(30)

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 \(U_1\) and \(U_2^{x y}\) and their difference with \(H(M_{1}|B)+H(M_{2}|B)\) are shown in Fig. 5.

4.2.3 Comparison

In all examples we calculated, \(U_2\) is tighter than \(U_1\). When Bob gets closer to the horizon, his uncertainty about Alice’s state gradually increases for both \(U_1\) and \(U_2^{x y}\). In addition, the figures shows that for a particular bound \(U_1\) or \(U_2\), when \(\varOmega = \omega / T_H\) is larger, the uncertainty bound is lower. This is evident since fixing the mode energy \(\omega \), the larger \(\varOmega \) is, the lower Hawking temperature \(T_H\) is, which results in more correlation which can reduce the uncertainty. Besides, there is no surprise that \(\varDelta _2^{x y}=H(M_1)+H(M_2)-(-\log c_1)\) is constant as it is only influenced by the choice of measurements \(M_1, M_2\) and measured system \(\rho _A\), not by the quantum memory \(\rho _B\). We can see from these figures that \(\varDelta _1\) is not always constant but can decrease or increase when \(R_0\) increase. This fact indicates that \(U_2\) is better than \(U_1\) in the sense that, for \(U_2\) 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 \(M_0\) 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 \(M_0\), \(\varDelta _2\) and \(U_2\) for \(M'\).