Quantum correlation between a qubit and a relativistic boson in an expanding spacetime

We first study the quantum correlations between the qubit $p$ and the boson with momentum $k$. By tracing over the antiboson mode $-k$, we obtain the density operator $\rho_{p,k}$ for the system of qubit and boson

with $\gamma^{2}=|\frac{\beta_{k}}{\alpha_{k}}|^{2}=\frac{\sinh^{2}(\pi\frac{\omega_{-}}{\rho})}{\sinh^{2}(\pi\frac{\omega_{+}}{\rho})}$.

There are many special approaches to the detection of quantum entanglement, among them the Peres-Horodecki positive partial transpose (PPT) criterion Peres1996 ; Horodecki1996 and the afterwards logarithmic negativityL32 are widely used, due to the simplicity of calculations. In this paper, we use the logarithmic negativity as the detection of entanglement, which is defined as $N(\rho)=\log_{2}\|\rho^{T}\|_{1}$, where the trace norm is given by $\|A\|_{1}={\rm tr}\sqrt{A^{{\dagger}}A}$ and $\rho^{T}$ denotes the partial transpose of the bipartite density matrix $\rho$ with respect to its one party. It is worthwhile to point out that logarithmic negativity is only the sufficient condition of entanglement and is not necessarily a guarantee of detecting all the entanglement in any case. However, it is indeed a feasible indicator for the conjecture of universe expansion.

Performing partial transpose on density matrix $\rho_{p,k}$ with respect to the qubit, then the transposed density matrix has the eigenvalues

with $n=0,1,2,\cdots$. Due to $\omega_{-}<\omega_{+}$, thus $\gamma^{2}<1$ and $|\alpha_{k}|^{2}=\frac{1}{1-\gamma^{2}}\neq\infty$, which means that there must be negative eigenvalues and the state $\rho_{p,k}$ is always entangled. The logarithmic negativity $N(\rho_{p,k})$ of the state $\rho_{p,k}$ is

We can also study the total correlation in the state $\rho_{p,k}$ by calculating the mutual information, which is defined as L33

where $S(\rho)=-\rm Tr[\rho\log_{2}(\rho)]$ is the von Neumann entropy of the density matrix $\rho$. From the expression $\rho_{p,k}$ of Eq.(19), we obtain the von Neumann entropies for the joint system and each subsystems as

and

respectively. Due to $\gamma^{2}<1$, we thus have $S(\rho_{k})>S(\rho_{p,k})$ and $I(\rho_{p,k})>S(\rho_{p})$.

Eqs.(20) and (21) suggest that logarithmic negativity and mutual information depend on the cosmological parameters $\epsilon$ and $\rho$, as well as the bosonic mass $m$ and momentum $k$. Assuming that we live in the universe corresponding to the asymptotic future of the spacetime. Then we can find the historical information about the spacetime expansion by inspecting the evolution of logarithmic negativity and mutual information. The historical information shows a promising application in observational cosmology and presents a way to simulate the expansion of universe.

In Fig.1, we show the logarithmic negativity (left panel) and mutual information (right panel) between qubit and boson as functions of the expansion volume $\epsilon$ or the expansion rate $\rho$. It is shown that the change of mutual information $I(\rho_{p,k})$ is very similar to the change of logarithmic negativity $N(\rho_{p,k})$. They reduce monotonically with respect to both the expansion volume $\epsilon$ and the expansion rate $\rho$.

Fig.2 shows the dependency of logarithmic negativity and mutual information on the momentum $k$, where the expansion volume $\epsilon$ and expansion rate $\rho$ are set to be three typical values according to Fig.1. We find that both the logarithmic negativity $N(\rho_{p,k})$ and the mutual information $I(\rho_{p,k})$ increase monotonically with the growth of momentum $k$. For infinite $k$, we have

Note that $\log_{2}\left[1+2\chi\sqrt{1-\chi^{2}}\right]$ and $2S(\rho_{p})$ are respectively the initial values of logarithmic negativity and mutual information in the state of Eq.(17), we thus conclude that, for boson with very large momentum, its logarithmic negativity and mutual information with the qubit are very little affected by the expanding spacetime. If we want to extract historical information about the expanding spacetime, the boson modes with smaller momentum are most preferable.

Fig.3 shows the change of logarithmic negativity and mutual information versus the mass $m$ for different expansion volume $\epsilon$ and different expansion rate $\rho$. It is shown that logarithmic negativity $N(\rho_{p,k})$ and mutual information $I(\rho_{p,k})$ firstly decrease to the minimum and then asymptotically recover to their initial values with the increasing of mass $m$. The fact that logarithmic negativity and mutual information asymptotically recover to their initial values can also be demonstrate through the analytical calculations

This result suggests that logarithmic negativity and mutual information between the qubit and the boson with very large or very small mass are less affected by the expanding spacetime. There is an optimal mass that is most prone to extract the information about the expanding spacetime. Note that the optimal mass decreases slightly with the growth of the expansion volume $\epsilon$, and increases slightly with the growth of the expansion rate $\rho$.

In order to study the redistribution of quantum correlations, we need to calculate logarithmic negativity and mutual information between qubit $p$ and the antiboson $-k$. Tracing over the mode $k$ in state of Eq.(18), we obtain the density matrix

The partial transpose of this density matrix with respect to qubit has the eigenvalues

with $D=\chi^{2}+\frac{(n+2)(1-\chi^{2})|\beta_{k}|^{2}}{|\alpha_{k}|^{4}}$ and $n=0,1,2,\cdots.$ Thus the logarithmic negativity between the qubit and antiboson is

This means that spacetime expanding can not generate logarithmic negativity between the qubit and antiboson. However, this does not mean that quantum state between qubit and antiboson is not without quantum correlation. The correlated quantum coherence is nonlocal coherence and is regarded as a kind of quantum correlation. Through simple calculation, we find that the correlated quantum coherence can be generated by the spacetime expanding wsm32 .

The mutual information between the qubit and the antiboson is given by

where $S(\rho_{p})$ is given by Eq.(23), and $S(\rho_{-k})$ and $S(\rho_{p,-k})$ are given by

and

Eq.(27) is positive in general, meaning that spacetime expanding can generate mutual information between the qubit and the antiboson.

In Fig.4, we plot the mutual information between the qubit and the antiboson as a function of the expansion volume $\epsilon$ and the expansion rate $\rho$. It is shown that the mutual information $I(\rho_{p,-k})$ increases monotonically with both $\epsilon$ and $\rho$, but the change is more sensitive to the beginning of $\epsilon$ and $\rho$. This means that smaller $\epsilon$ and $\rho$ are more favorable from the consideration of extracting information.

Fig.5 shows the change of mutual information $I(\rho_{p,-k})$ versus the momentum $k$ or mass $m$, where the expansion volume $\epsilon$ and expansion rate $\rho$ are set as the typical values according to Fig.4. We can see that $I(\rho_{p,-k})$ reduces monotonically with the growth of the momentum $k$ [see Fig.5(a) and 5(b)]. In the limit of infinite momentum, we have $\lim_{k\rightarrow\infty}I(\rho_{p,-k})=0$ (through analytic calculation). This means that the smaller the momentum we pick out, the more the generated mutual information $I(\rho_{p,-k})$ is, which is more favourable for extracting information of spacetime expansion.

From Fig.5(c) and 5(d), we see that the generated mutual information $I(\rho_{p,-k})$ increases from zero to a maximum and then reduces again to zero with the growth of the mass $m$, i.e., satisfying

This means that there are some medium-sized masses which can produce more mutual information between the qubit and the antiboson and thus are more conducive to extract information about the expanding of universe. As displayed by Fig.3, the optimal mass decreases with the growth of the expansion volume $\epsilon$, and increases with the growth of the expansion rate $\rho$. In fact, we can demonstrate (see III.3 below) that the optimal mass displayed here is actually consistent with that in Fig.3.

From the above discussion, we find that the expansion of universe can not produce logarithmic negativity $N(\rho_{p,-k})$, but can produce mutual information $I(\rho_{p,-k})$ between the qubit and the antiboson. However, the production of $I(\rho_{p,-k})$ requires some initial conditions. In Fig.6, we plot the logarithmic negativity $N(\rho_{p,k})$, the mutual information $I(\rho_{p,k})$ and $I(\rho_{p,-k})$ as functions of the initial parameter $\chi$. It is shown that the production of the mutual information $I(\rho_{p,-k})$ needs the trigger of the initial logarithmic negativity $N(\rho_{p,k})$ or initial mutual information $I(\rho_{p,k})$. With no these triggers, the mutual information $I(\rho_{p,-k})$ can not be generated. The more the trigger logarithmic negativity $N(\rho_{p,k})$ or the mutual information $I(\rho_{p,k})$ is, the more the generated mutual information $I(\rho_{p,-k})$ is.

In fact, the trigger phenomenon of mutual information is obviously. We can view the boson and antiboson as a bipartite system. If there is no initial mutual information between the qubit and the boson, then also no initial mutual information between the qubit and the bipartite system. As no interaction exists between the qubit and the bipartite system in the expanding of spacetime, the mutual information between them always remains zero. Thus no mutual information exists between the qubit and the antiboson. What’s interesting is that the more the trigger mutual information, the more the generated mutual information $I(\rho_{p,-k})$, and this trigger phenomenon does not take place in the logarithmic negativity.

From Figs.1-5, we find that the changes of $I(\rho_{p,k})$ and $I(\rho_{p,-k})$ are always opposite: Whenever the mutual information between the qubit and the boson $I(\rho_{p,k})$ degrades, then the mutual information between the qubit and the antiboson $I(\rho_{p,-k})$ increases. After careful inspection of Eqs.(21) and (27), we find the monogamy relationship

where $2S(\rho_{p})$ is the initial mutual information for the state of Eq.(17). This implies that the mutual information is re-distributable: The initial mutual information is redistributed into $\rho_{p,k}$ and $\rho_{p,-k}$, but the total mutual information is conserved in the expansion of universe. This is evidenced by the numerical simulations, i.e., the value of $I(\rho_{p,k})$ in Fig.3(b)(d) plus the value of $I(\rho_{p,-k})$ in Fig.5(c)(d), for the same $\epsilon$ and $\rho$, is equal to 2. This conservation also demonstrates that the optimal mass appeared in Fig.3(b)(d) and Fig.5(c)(d) coincide.

Ball $et$ $al$ firstly studied the entanglement entropy of bosonic field in an expanding spacetime L28 . Then, the comparison between fermionic and bosonic entanglement entropy for free field modes was studied L8 ; L29 . It is shown that the more information about cosmological parameters can be extracted by using fermionic entanglement entropy. Next, Li $et$ $al$ studied decoherence and disentanglement of scalar fields for Unruh-Wald qubit detector model in an expanded spacetime L31 . The results show that the expansion of the spacetime in its history destroys the entanglement between the qubits. Based on these work, Moradi $et$ $al$ investigated spin-particles entanglement between two modes of Dirac field in Robertson-Walker spacetime wsm33 . The analysis highlight the role of polarization of particles and compare the polarization results with those obtainable without accounting for it. Mohammadzadeh $et$ $al$ studied entropy production due to Lorentz invariance violation (LIV) in an expanding spacetime wsm34 , and Liu $et$ $al$ used the optimal estimation of parameters for scalar fields to extract information in expanding universe exhibiting LIV L30 . The extracted information about the past existence of LIV might be useful in recovering the quantum structure of gravity. Really, the occurrence of a peak in the behavior of entanglement entropy for a specific momentum can provide information about the expansion parameters. Therefore, the information about the LIV parameter is codified in this peak.

In III.1, we use the logarithmic negativity and mutual information between the qubit and the boson to detect the law of universe expansion, because they contain the information about universe expansion. In fact, we find from Eq.(24) that the entanglement entropy (i.e. von Neumann entropy) $S(\rho_{k})$ of the boson also contains the information about universe expansion, thus it also can be used to detect the law of universe expansion. As the whole system (qubit, boson and antiboson) is always a pure state in the universe expansion, $S(\rho_{k})$ describes actually the quantum correlation of the boson with the qubit and the antiboson. In Fig.7, we plot the entanglement entropy $S(\rho_{k})$ as a function of the expansion volume $\epsilon$ and the expansion rate $\rho$, where the parameters are chosen as the same as in Fig.4. It is shown that the evolution of $S(\rho_{k})$ is also similar to that of $I(\rho_{p,-k})$ in Fig.4. $S(\rho_{k})$ increases monotonically with both $\epsilon$ and $\rho$, and the change is more sensitive to the beginning of $\epsilon$ and $\rho$, meaning that smaller $\epsilon$ and $\rho$ are more favorable for the extraction of information.

Fig.8 shows the change of entanglement entropy $S(\rho_{k})$ versus the momentum $k$ or mass $m$, where the parameters are chosen as the same as in Fig.5. We see that Fig.8 is also very similar to Fig.5. $S(\rho_{k})$ reduces monotonically with the growth of the momentum $k$ [Fig.8(a) and 8(b)]. In the limit of infinite momentum, we have $\lim_{k\rightarrow\infty}S(\rho_{k})=0$ (through analytic calculation). This means that the smaller the momentum we pick out, the more the generated entanglement entropy $S(\rho_{k})$ is, which is more favourable for extracting information of spacetime expansion.

From Fig.8(c) and 8(d), we see that the generated entanglement entropy $S(\rho_{k})$ increases from zero to a maximum and then reduces again to zero with the growth of the mass $m$, i.e., satisfying

This means that there are some medium-sized masses which can produce more entanglement entropy for the boson and thus are more conducive to extract information about the universe expansion. Also, the optimal mass decreases slightly with the growth of the expansion volume $\epsilon$, and increases slightly with the growth of the expansion rate $\rho$. However, this optimal mass is conceptually not the same thing as that in Fig.5(c)(d) for $I(\rho_{p,-k})$.

The ability to recover information about the cosmological parameters which are given access the generated mutual information between the qubit and the antiboson as well as entanglement entropy, were analysed in the limit of small mass $m\ll 2\rho\sqrt{\epsilon}$. In this limit, we find an approximate relation between the generated mutual information and the mass $m$

We can see that the generated mutual information is a monotonically increasing function of the expansion volume $\epsilon$, and is a monotonically decreasing function of the momentum $k$. We can also see that the generated mutual information is a increasing function of the mass $m$ for smaller $m$. However, for bigger mass $m$, the generated mutual information is a decreasing function of the mass $m$. In this same limit, the expansion rate $\rho$ can be related to the variation of the generated mutual information

where $E=\sqrt{m^{2}+k^{2}}$ is the energy of the $k$-momentum mode. Unfortunately, for ${m\rightarrow 0}$, the generated mutual information is almost zero due to the conformal symmetry of the theory. Therefore, the generated mutual information of massive states in our simple toy universe encodes the complete information about the cosmological parameters.

In the hopes of stimulating further research into the relation between the generated mutual information and universe expansion, we conclude with some discussion of further simulation avenues in this area. We discuss the possibility of the cosmic neutrino background that possesses quantum entanglement and mutual information from the early universe and surviving all the way to the present because of its weak coupling to matter. We can also discuss the importance of analogue experimental models providing a laboratory-accessible testbed for cosmological models. Finally, we can use quantum entanglement and mutual information as a tool for theoretical and experimental cosmology.

Perhaps, it may be applied in the modeling of universe expansion with superconducting circuit Zehua2017 . We theoretically propose a setup to realize Jackiw-Teitelboim expansion universe as solitons of sine-Gordon. Note that our configuration can be similar to the superconducting coplanar waveguide. However, each capacitor in our configuration is parallel with an identical superconducting quantum interference devices. It is the added superconducting quantum interference device providing a nonlinear potential to the Lagrangian of fluxes in the superconducting coplanar waveguide, which plays a very crucial role in the simulation of expansion universe. Each superconducting quantum interference device consists of two identical parallel Josephson junctions. Besides, the geometric size of superconducting quantum interference devices loop is assumed to be small enough such that its selfinductance is negligible compared to its kinetic inductance. In this case, each superconducting quantum interference device can be referred to as an effective Josephson junction with a junction capacitance and a tunable Josephson energy. We can realize the Jackiw-Teitelboim expansion universe as solitons of sine-Gordon equation in our setup. Thus, it allows us to understand the mechanics of solitons and especially the duality between Jackiw-Teitelboim gravity and sine-Gordon soliton from the perspective of experiment. The analogous dynamic behavior of quantum fields under the expanding universe in principle could be implemented in the setup. By appropriately adjusting the external magnetic flux through the superconducting quantum interference device loop, we can simulate the behavior of the conformal scale factor. Therefore, the analogous dynamic behavior of quantum fields under an expanding universe in principle could be implemented in setup. The in-mode and out-mode are two particular solutions in the past and future, respectively. Physically, these two particular solutions are a description of two different vacuum and excited states in the two asymptotic regions. We consider the boson (B) that is affected by the simulated spacetime expanding and the qubit (A) entangled with boson in shielding environment.