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arXiv:2002.07316 · The European Physical Journal C

Correlation redistribution by causal horizons

Leonardo Pipolo de Gioia, M. C. de Oliveira

Published Jun 28, 2019• 1 citation
arXivHTMLPDF

Abstract

Abstract The Minkowski vacuum $$|0\rangle _M$$ | 0 ⟩ M , which for an inertial observer is devoid of particles, is perceived as a thermal bath by Rindler observers living in a single Rindler wedge (Unruh in Phys Rev D 14:870, 1976) as a result of the discrepancy in the definition of positive frequency between the two classes of observers and a strong entanglement between degrees of freedom in the left and right Rindler wedges. We revisit the problem of quantification of the correlations in a two-mode state of a free neutral scalar field which is observed by an inertial observer Alice and left/right Rindler observers Rob/AntiRob, a problem that pertains to the field of relativistic quantum information and has been previously studied in Martin-Martinez et al. (Phys Rev D 82:064006, 2010) and Datta (Phys Rev A 80:052304, 2009). We focus on the analysis of informational quantities like the locally accessible and locally inaccessible information (Koashi and Winter in Phys Rev A 69:022309, 2004; Fanchini et al. in Phys Rev A 84:012313, 2011; Fanchini et al. in New J Phys 14:013027, 2012) and a closely associated entanglement measure, the entanglement of formation. We conclude that, with respect to the correlation structure probed by inertial observers alone, the introduction of a Rindler observer gives rise to a correlation redistribution which can be quantified by the entanglement of formation. Given the informational meaning of the derived correlations, we discuss on the capacity of a quantum channel to communicate classical information between accelerated parties.

DOI
10.1140/epjc/s10052-022-10090-w

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Abstract The Minkowski vacuum ∣0⟩M|0\rangle _M∣0⟩M​ | 0 ⟩ M , which for an inertial observer is devoid of particles, is perceived as a thermal bath by Rindler observers living in a single Rindler wedge (Unruh in Phys Rev D 14:870, 1976) as a result of the discrepancy in the definition of positive frequency between the two classes of observers and a strong entanglement between degrees of freedom in the left and right Rindler wedges. We revisit the problem of quantification of the correlations in a two-mode state of a free neutral scalar field which is observed by an inertial observer Alice and left/right Rindler observers Rob/AntiRob, a problem that pertains to the field of relativistic quantum information and has been previously studied in Martin-Martinez et al. (Phys Rev D 82:064006, 2010) and Datta (Phys Rev A 80:052304, 2009). We focus on the analysis of informational quantities like the locally accessible and locally inaccessible information (Koashi and Winter in Phys Rev A 69:022309, 2004; Fanchini et al. in Phys Rev A 84:012313, 2011; Fanchini et al. in New J Phys 14:013027, 2012) and a closely associated entanglement measure, the entanglement of formation. We conclude that, with respect to the correlation structure probed by inertial observers alone, the introduction of a Rindler observer gives rise to a correlation redistribution which can be quantified by the entanglement of formation. Given the informational meaning of the derived correlations, we discuss on the capacity of a quantum channel to communicate classical information between accelerated parties.
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.
In this Ph.D. thesis, I investigate the communication abilities of non-inertial observers and the precision to which they can measure parametrized states. I introduce relativistic quantum field theory with field quantisation, and the definition and transformations of mode functions in Minkowski, Schwarzschild and Rindler spaces. I introduce information theory by discussing the nature of information, defining the entropic information measures, and highlighting the differences between classical and quantum information. I review the field of relativistic quantum information. We investigate the communication abilities of an inertial observer to a relativistic observer hovering above a Schwarzschild black hole, using the Rindler approximation. We compare both classical communication and quantum entanglement generation of the state merging protocol, for both the single and dual rail encodings. We find that while classical communication remains finite right up to the horizon, the quantum entanglement generation tends to zero. We investigate the observers' abilities to precisely measure the parameter of a state that is communicated between Alice and Rob. This parameter was encoded to either the amplitudes of a single excitation state or the phase of a NOON state. With NOON states the dual rail encoding provided greater precision, which is different to the results for the other situations. The precision was maximum for a particular number of excitations in the NOON state. We calculated the bipartite communication for Alice-Rob and Alice-AntiRob beyond the single mode approximation. Rob and AntiRob are causally disconnected counter-accelerating observers. We found that Alice must choose in advance with whom, Rob or AntiRob she wants to create entanglement using a particular setup. She could communicate classically to both.
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.
Emission of photons by atoms can occur into modes which extend into a region causally disconnected with the emitter.For example, a uniformly accelerated ground-state atom emits a photon into the Unruh-Minkowski mode which is exponentially larger in the causally disconnected region.This makes an impression that photon emission is acausal.Here we show that conventional quantum optical analysis yields that a detector atom will not detect the emitted photon in the region noncausally connected with the emitter.However, joint excitation probability of atoms in the causally disconnected regions can be correlated due to entanglement of Minkowski vacuum and be much larger than the product of independent excitation probabilities.Moreover, atoms uniformly accelerated in the same Rindler wedge cannot become simultaneously excited without changing the state of the field, that is, the Unruh-Minkowski photon emitted by one atom cannot be absorbed by the other atom.We discuss examples demonstrating interesting features of Minkowski vacuum entanglement.
Quantum effects in non-inertial frames and curved space-time have been studied for decades. One of the most intriguing discoveries is the existence of virtual particles in the quantum vacuum. One example in which such particles can be observed is through the Unruh effect, which predicts that a uniformly accelerated observer would see a thermal bath of (virtual) particles in the quantum vacuum. Initial doubts on the existence of these virtual particles were allayed by the Unruh De-Witt (UDW) detector, which promoted these virtual particles to real excitations of the detector. Ever since, semiclassical approaches to relativistic quantum field theory (QFT) in flat and curved space-time have been predominantly analysed through a two-level Unruh De-Witt detector utilizing perturbation theory. It is noted, however, that the change of relativistic frame preserves Gaussian properties (e.g. Gaussian distribution of the Wigner function) of the system. In this thesis, we specifically explore the effects of relativistic (non-inertial) reference frames on QFT by considering relativistic QFT in flat-space time. These systems are analysed from the perspective of Gaussian quantum information, via homodyne detection and UDW detectors with a harmonic oscillator degree of freedom.Our project starts by reviewing and developing bipartite/multi-partite entanglement measures (specifically, entanglement of formation), in the hope that we can utilize these measures to understand the entanglement properties of the quantum vacuum. In general, these measure have infinite degrees of freedom and are incomputable. In this project the multi-partite entanglement measure is applied to the Gaussian regime, which reduces the number of degrees of freedom to a finite one, hence making it a computable measure.In our next project, we attempt to understand how relativistic reference frames affect the communication between different observers. To obtain an intuition of its effect, a passive (time-delay) signal sent from a uniformly accelerated observer to a stationary observer was analysed. When we naively implemented the self-homodyne detection scheme, the signal appeared noisy. This effect was referred to as apparent decoherence, and its effect could be traced back to our naive assumption that all of the signal’s information would be stored in the time-delayed mode; the vacuum entanglement that pre-existed before the (time-delay) signal was created must be accounted for. We follow by analysing the effects that emerge due to communication between different relativistic reference frames. We then develop a communication technique referred to as ideal-homodyne which is a homodyne detection scheme robust to these effects. We used this detection scheme to analyse the correlation/entanglement properties of an accelerated signal (created via accelerated mirror, squeezer and phase shifter) sent to an inertial observer.In our last project, we utilize the knowledge that was gained through the prior projects to analyse (subcycle) electro-optic sampling. In the literature, there were doubts as to whether electro-optic sampling were truly detecting virtual particles in the quantum vacuum, or particle excitations that were created as a by-product of the detection mechanism. We were able to pin-point some similarities between electro-optic sampling and subcycle probe via the UDW detector. The regime in which the electro-optic sampling directly maps virtual particles from the vacuum into real excitations of the probe field was identified by establishing an equivalence to the UDW detector in certain parameter regimes.
We investigate the redistribution of continuous-variable Gaussian quantum discord under the Unruh effect for both one and two uniformly accelerated observers. The discord shared between the inertial mode and the accessible accelerated mode, as well as that between the two accessible accelerated modes, decays monotonically with acceleration and vanishes in the limit a→∞a\to\inftya→∞, while the Unruh effect generates discord in all causally disconnected Rindler mode pairs. The initial quantum correlation is therefore redistributed across Rindler partitions rather than destroyed. We further characterize how the squeezing parameter sss and field frequency ωωω control the discord in different mode-pair classes. For causally connected pairs, sss and ωωω exert comparable, interchangeable effects; for cross-region pairs, the squeezing parameter dominates; and for intra-observer cross-region pairs, the field frequency is more influential. In the two-observer case, the AIIA_{II}AII​-BIIB_{II}BII​ pair exhibits non-monotonic behavior with a peak whose position shifts to larger aaa as ωωω increases, and both parameters contribute only weakly except near an optimal ωs\fracω{s}sω​ ratio.