#### 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).
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