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arXiv:1003.3550 · Physical Review A

Population bound effects on bosonic correlations in noninertial frames

Eduardo Martín-Martínez, Juan León

Published May 6, 2010• 38 citations
arXivHTMLPDF

Abstract

We analyze the effect of bounding the occupation number of bosonic field modes on the correlations among all the different spatial-temporal regions in a setting in which we have a space time with a horizon along with an inertial observer. We show that the entanglement between $A$ (inertial observer) and $R$ (uniformly accelerated observer) depends on the bound $N$, contrary to the fermionic case. Whether or not decoherence increases with $N$ depends on the value of the acceleration $a$. Concerning the bipartition $A\mathrm{R\ifmmode \bar{}\else \={}\fi{}}$ (Alice with an observer in Rindler's region $\mathit{IV}$), we show that no entanglement is created whatever the value of $N$ and $a$. Furthermore, $\mathit{AR}$ entanglement is very quickly lost for finite $N$ and for $N\ensuremath{\rightarrow}\ensuremath{\infty}$. We will study in detail the mutual information conservation law found for bosons and fermions. By means of the boundary effects associated to $N$ finiteness, we will show that for bosons this law stems from classical correlations while for fermions it has a quantum origin. Finally, we will present the strong $N$ dependence of the entanglement in $R\mathrm{R\ifmmode \bar{}\else \={}\fi{}}$ bipartition and compare the fermionic cases with their finite $N$ bosonic analogs. We will also show the anti-intuitive dependence of this entanglement on statistics since more entanglement is created for bosons than for their fermion counterparts.

DOI
10.1103/physreva.81.052305

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We analyze the effect of bounding the occupation number of bosonic field modes on the correlations among all the different spatial-temporal regions in a setting in which we have a space time with a horizon along with an inertial observer. We show that the entanglement between AAA (inertial observer) and RRR (uniformly accelerated observer) depends on the bound NNN, contrary to the fermionic case. Whether or not decoherence increases with NNN depends on the value of the acceleration aaa. Concerning the bipartition AR\ifmmodeˉ\elseˉ\fiA\mathrm{R\ifmmode \bar{}\else \={}\fi{}}AR\ifmmodeˉ\elseˉ\fi (Alice with an observer in Rindler's region IV\mathit{IV}IV), we show that no entanglement is created whatever the value of NNN and aaa. Furthermore, AR\mathit{AR}AR entanglement is very quickly lost for finite NNN and for N\ensuremath→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞. We will study in detail the mutual information conservation law found for bosons and fermions. By means of the boundary effects associated to NNN finiteness, we will show that for bosons this law stems from classical correlations while for fermions it has a quantum origin. Finally, we will present the strong NNN dependence of the entanglement in RR\ifmmodeˉ\elseˉ\fiR\mathrm{R\ifmmode \bar{}\else \={}\fi{}}RR\ifmmodeˉ\elseˉ\fi bipartition and compare the fermionic cases with their finite NNN bosonic analogs. We will also show the anti-intuitive dependence of this entanglement on statistics since more entanglement is created for bosons than for their fermion counterparts.
We study the reflected entropy and the Markov gap for modes of a free bosonic field shared by inertial observers (Alice, Charlie) and a uniformly accelerated one (Bob), for a bipartite Bell state and the tripartite Werner (W) and Greenberger--Horne--Zeilinger (GHZ) states. The bosonic Bogoliubov transformation spans an infinite-dimensional Fock space with an unbounded squeezing parameter unlike the fermionic case. By identifying a conserved charge, we block-diagonalize the reduced density matrices into exact two-dimensional sectors, yielding closed or semi-analytic forms for all three states. Although bosonic entanglement is known to vanish asymptotically, the Alice--Bob reflected entropy instead saturates at a nonzero floor, retaining the surviving classical correlation, and converges to the value Alice shares with Bob's causally disconnected partner. Crucially, only the inter-wedge reflected entropy diverges, linearly in the squeezing parameter---the sharp distinction from the fermionic case, where it stays bounded---while the Markov gaps saturate. The construction transfers verbatim to a Schwarzschild black hole, where the saturation values become mass-independent constants.
We generalize the study of fermionic and bosonic entanglement in noninertial frames to fields of arbitrary spin and beyond the single-mode approximation. After the general analysis we particularize for two interesting cases: entanglement between an inertial and an accelerated observer for massless fields of spin 1 (electromagnetic) and spin 3/23/23/2 (Rarita-Schwinger). We show that, in the limit of infinite acceleration, no significant differences appear between the different spin fields for the states considered.
Two observers determine the entanglement between two free bosonic modes by each detecting one of the modes and observing the correlations between their measurements. We show that a state which is maximally entangled in an inertial frame becomes less entangled if the observers are relatively accelerated. This phenomenon, which is a consequence of the Unruh effect, shows that entanglement is an observer-dependent quantity in noninertial frames. In the high acceleration limit, our results can be applied to a nonaccelerated observer falling into a black hole while the accelerated one barely escapes. If the observer escapes with infinite acceleration, the state's distillable entanglement vanishes.
We study the entanglement of families of Unruh modes in the Bell states $|{\mathrm{\ensuremath{\Phi}}}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rangle}=1/\sqrt{2}(|00\ensuremath{\rangle}\ifmmode\pm\else\textpm\fi{}|11\ensuremath{\rangle})$ and $|{\mathrm{\ensuremath{\Psi}}}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rangle}=1/\sqrt{2}(|01\ensuremath{\rangle}\ifmmode\pm\else\textpm\fi{}|10\ensuremath{\rangle})$ shared by two accelerated observers and find fundamental differences in the robustness of entanglement against acceleration for these states. States ${\mathrm{\ensuremath{\Psi}}}^{\ifmmode\pm\else\textpm\fi{}}$ are entangled for all finite accelerations, whereas, due to the Unruh effect, states ${\mathrm{\ensuremath{\Phi}}}^{\ifmmode\pm\else\textpm\fi{}}$ lose their entanglement for finite accelerations. This is true for Bell states of two bosonic modes, as well as for Bell states of a bosonic and a fermionic mode. Furthermore, there are also differences in the degradation of entanglement for Bell states of fermionic modes. We reveal the origin of these distinct characteristics of entanglement degradation and discuss the role that is played by particle statistics. Our studies suggest that the behavior of entanglement in accelerated frames strongly depends on the occupation patterns of the constituent states, whose superposition constitutes the entangled state, where especially states ${\mathrm{\ensuremath{\Phi}}}^{\ifmmode\pm\else\textpm\fi{}}$ and ${\mathrm{\ensuremath{\Psi}}}^{\ifmmode\pm\else\textpm\fi{}}$ exhibit distinct characteristics regarding entanglement degradation. Finally, we point out possible implications of hovering over a black hole for these states.
A uniformly accelerated observer perceives the Minkowski vacuum as a thermal bath, so acceleration acts as a noise channel on correlations shared with an inertial partner. For a Dirac field in the standard treatment, the resulting map is a one-sided amplitude-damping channel, and entanglement and quantum discord survive at infinite acceleration. We introduce a symmetric Unruh-like map in which the vacuum and the one-particle state both exchange amplitude with the inaccessible Rindler region; for a Bell state it yields a bit-flip channel whose flip probability is the Fermi-Dirac occupation at the Unruh temperature. Using negativity, mutual information, and all six pairwise quantum discords, we find that the entanglement and discord between the inertial observer and the accessible mode vanish together at infinite acceleration, leaving only one bit of classical correlation. These results show that the fate of quantum correlations in noninertial frames depends on the structure of the Minkowski-Rindler map.
We review a recently introduced unified approach to the analytical quantification of correlations in Gaussian states of bosonic scalar fields by means of Renyi-2 entropy. This allows us to obtain handy formulae for classical, quantum, total correlations, as well as bipartite and multipartite entanglement. We apply our techniques to the study of correlations between two modes of a scalar field as described by observers in different states of motion. When one or both observers are in uniform acceleration, the quantum and classical correlations are degraded differently by the Unruh effect, depending on which mode is detected. Residual quantum correlations, in the form of quantum discord without entanglement, may survive in the limit of an infinitely accelerated observer Rob, provided they are revealed in a measurement performed by the inertial Alice.