Quantum Discord and Quantum Entanglement in the Background of an Asymptotically Flat Static Black Holes
The quantum discord and tripartite entanglement are discussed in the presence of an asymptotically flat static black holes. The total correlation, quantum discord and classical correlation exhibit decreasing behavior with increasing Hawking temperature. It is shown that the classical correlation is less than the quantum discord in the full range of Hawking temperature. The tripartite entanglements for Greenberger-Horne-Zeilinger and W-states also exhibit decreasing behavior with increasing Hawking temperature. At the infinite limit of Hawking temperature the tripartite entanglements for Greenberger-Horne-Zeilinger and W-states reduce to and of the corresponding values in the flat space limit, respectively.
Recently much attention is paid to the quantum information theories in the relativistic frameworkperes1 (); peres2 (); peres3 (); inertial1 (); inertial2 (); inertial3 (); inertial4 (); noninertial1 (); noninertial2 (); noninertial3 (); noninertial4 (); noninertial5 (); noninertial6 (); noninertial7 (); noninertial8 (); noninertial9 (); noninertial10 (); noninertial11 (); noninertial12 (). The most remarkable fact in the inertial frames is the fact that entanglement of given multipartite quantum state is conserved although the entanglement between some degrees of freedom can be transfered to othersinertial1 (); inertial2 (); inertial3 (); inertial4 (). In non-inertial frames, however, the entanglement is in general degraded, which implies that the quantum correlation between rest and accelerating observers is reduced more and more with increasing the accelerationnoninertial1 (). The main reason for the reduction of the quantum correlation is that the accelerating observer located in one Rindler wedge loses an information arising from the other Rindler wedge due to the causally disconnected nature between the wedges. This means that some quantum information is leaked into other causally disconnected Rindler space, which makes the reduction of the quantum correlation. In fact, this is a main scenario of the well-known Unruh effectunruh1 (); unruh2 (). Recently, this Unruh-type decoherence effect beyond the single-mode approximation is discussed in the context of the quantum information theoriesbeyond ().
More recently, the quantum entanglement in the black hole background is examinedblack1 (); black2 (). Especially, in Ref.black1 () the Hawking temperature-dependence of the bipartite entanglement is studied in the arbitrary spherically symmetric and asymptotically flat black hole background. The purpose of this paper is to explore the quantum discordzurek02 (); vedral01 () and the tripartite entanglement in the same black hole background.
Ii spacetime background
Throughout this paper we use . The metric we consider in this paper is
where the functions , , and satisfy , , and . Thus, the line element (1) includes the various black holes such as Schwarzschild and Reissner-Nordström black holes. The Hawking temperature in this metric is , where is a surface gravity defined as .
As shown in Ref. black1 () one can consider three-different vacuum states , , and in this background. First two vacuum states are the Fock vacua inside and outside horizon, respectively, and the last one is the Kruskal vacuum outside the event horizon. The interrelation between these vacua is
where and are -particle states constructed from and by operating the corresponding creation operators times, and is a frequency of the scalar field. Applying the creation operator of the Kruskal spacetime in Eq. (2) and using the Bogoliubov coefficients, one can construct , whose expression is
Iii Quantum Discord
Quantum discordzurek02 (); vedral01 () is a measure for quantumness of given bipartite quantum state. Usually these two parties consist of system and corresponding apparatus. In this paper, however, we will call these parties as Alice and Bob. We will examine in this section how the quantum discord is changed in the presence of the black hole (1).
We assume that initially Alice and Bob share a entangled state
in the asymptotic region. After sharing, Bob moves to the near-horizon region with his own particle detector while Alice stays in the asymptotic region. Since, then, the Bob’s detector registers only thermally excited particles due to the Hawking effect, Bob’s state can be represented by tensor product of the and states. However, since the inside region of the black hole is causally disconnected from Alice and Bob, we have to take a partial trace over state. Then, the state between Alice and Bob becomes a mixed state whose density matrix becomes
It is worthwhile noting and .
Now, we discuss the quantum discord. We assume that Alice performs a projective measurement with a complete set of the measurement operators . The usual mutual information between Alice and Bob is
where denotes the von Neumann entropy . In our paper, all logarithms are taken to base . The classical analogue of Eq. (7) is , where denotes the Shannon entropy. In classical information theories different representation of the mutual information is , where is the conditional entropy of given . The quantum analogue of this representationzurek02 () is
where denotes a complete set of the measurement operators prepared by the party and is a von Neumann entropy of the party after the party has a measurement outcome . Of course, is a probability for getting outcome in the quantum measurement. Usual quantum mechanical postulatespostulate () imply
where . Unlike , therefore, is dependent on the complete set of the measurement operators. The quantum discord is defined as
where minimum is taken over all possible complete set of the measurement operators
Now, we would like to compute the quantum discord in the black hole background. From Eq. (5) it is easy to show that is a completely mixed state and
Also it is easy to show
Now, we introduce the complete set of the projective measurement operators with
In Eq. (13) denotes the Pauli matrix and . Then, it is straightforward to show and
Since it is impossible to compute the eigenvalues of the , we should compute numerically. One can perform this numerical calculation with parametrizing , and . Then, it is possible to show that the eigenvalues of are independent of .
The -dependence of is plotted in Fig. 1. As this figure exhibits, the minimum is occurred at . Therefore, the quantum discord is obtained from by letting . If we assume that the total correlation is a mutual information , it is possible to compute the classical correlation by
In Fig. 2 we plot the Hawking temperature-dependence of the total correlation, quantum discord, and classical correlation. As Fig. 2 shows, all correlations exhibit a decreasing behavior with increasing . In the limit all correlations approach to the values in the absence of the black hole. In the opposite limit, i.e. , , , and approach to , , and , respectively. The remarkable fact is that the classical correlation is less than the quantum discord in the full range of Hawking temperature. Similar behavior was derived when the classical correlation and quantum discord sharing of Dirac field are discussed in the non-inertial framenoninertial5 (). In next section we will discuss on the tripartite entanglement in the presence of the black hole (1).
Iv Tripartite Entanglement Degradation
The most well-known measure for the tripartite entanglement is a three-tangleckw (). Since, however, the three-tangle is not defined in the qudit system, we cannot use it because of Eq. (2) and Eq. (3). Thus, instead of the three-tangle, we will use -tangleou07 () in this paper as a measure of the tripartite entanglement.
iv.1 Greenberger-Horne-Zeilinger state
Let Alice, Bob, and Charlie share the Greenberger-Horne-Zeilinger (GHZ) state
where . Since the Charlie’s -state is a qudit state, it is impossible to compute the genuine tripartite entanglement measure called the three-tangleckw (). As we commented before, therefore, we choose the -tangleou07 () as a tripartite measure defined as
due to more tractable computation. In Eq. (18) , , and are defined by
where and with being a partial transposition over -state and . It is easy to show in the absence of the black hole background.
Now, let us compute the one-tangle . From Eq. (IV.1) it is easy to show that is a diagonal. Therefore, the eigenvalues of can be computable easily. Since is a sum of square root of the eigenvalues, one can derive , whose final expression is
where is a polylogarithm function defined as
Using a property of the polylogarithm function one can show that approaches to when . From a symmetry of the GHZ state it is also easy to show .
Now, let us compute the last one-tangle . Since becomes
where and are
the off-diagonal part makes it difficult to compute the eigenvalues of . However, one can make block-diagonal by ordering the basis as . Thus, one can compute the eigenvalues of analytically, which are . Here, are eigenvalues of each block given by
where . Therefore, reduces to
Finally, one can show that all two tangles , , and are identically zero.
The one-tangles and -tangle are plotted in Fig. 3 as a function of Hawking temperature. As this figure shows, the -tangle decreases with increasing the Hawking temperature, and eventually reduces to at . At the -tangle exactly coincides with that in the absence of the black hole. Thus, as expected, the tripartite entanglement is degraded when Charlie moves to the near-horizon region from the asymptotic region with his own particle detector.
iv.2 W state
Let Alice, Bob, and Charlie share the W-statedur00-1 ()
in the asymptotic flat region. It is easy to show that the -tangle for the W-state is in the flat space limit.
By following similar calculation to the case of GHZ state, one can compute the Hawking temperature-dependence of in the presence of the black hole background. We do not want to repeat the computational procedure again in this paper. Instead, we present Fig. 4, which shows one-tangles, two-tangles, and as a function of Hawking temperature. In Fig. 4 (a) we plot the one- and two-tangles. All tangles exhibit decreasing behavior with increasing Hawking temperature except , which is independent of . At and approaches to while has a vanishing limit. The remarkable fact is that the two-tangles and becomes abruptly zero in the region . This reminds us of the concurrence, one of the bipartite entanglement measure. In Fig. 4 (b) we plot as a function of . At in the flat space is recovered. However, it exhibits a decreasing behavior with increasing , and eventually reduces to at limit.
In this paper we discussed the quantum discord and tripartite entanglement in the presence of the asymptotically flat static black holes. Both the quantum discord and the tripartite entanglement exhibit decreasing behavior with increasing Hawking temperature. This implies that the presence of the black holes reduces the quantum correlation when one party moves from asymptotic to near-horizon regions with his (or her) own particle detector.
Although we have not commented here, the tripartite entanglement of Alice, Bob, and Charlie’s in state (or AntiCharlie) does not completely vanish. Probably, this fact implies that some quantum information processes can be performed partially across the black hole horizon. To confirm this it seems to be important to compute the teleportation fidelity by making use of the tripartite teleportation schemetele-1 (); tele-2 (). If the tripartite teleportation is possible, even if incompletely, across the horizon, what this means in the context of causality? The answer may be important in the context of quantum gravity. We would like to explore this issue in the future.
Acknowledgement: This work was supported by the Kyungnam University Research Fund, 2012.
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