377. Temporal and spatial dependence of quantum entanglement
COND-MAT, HEP-TH/PH — By Shih-Yuin Lin on April 28, 2009 at 6:01 pm
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In textbooks, quantum entanglement are often introduced to readers with the simplest case: in an isolated system with two parties or subsystems, if a quantum states can be factorized into a product of the quantum states for each subsystem,
then it is called a separable state, otherwise it is entangled. For a separable state, the quantum state of one party will not be affected by any local measurement on the other, but for an entangled state, it will.
Usually in their examples there is no indication of the positions of the two (or more) subsystems or quantum objects. The quantum objects such as two spins or atoms can be located very far away from each other, e.g. the A-party is at Helsinki Institute of Physics while the B-party is at some planet in Andromeda Galaxy. They can also be located at the same point in space, anyway, with the two parties being different degrees of freedom, provided that it is possible to perform “local” operations on each party without disturbing the other (in other words, those operations on different parties commute).
As a physical property, entanglement is independent of the representation of each party. But entanglement of a quantum system does depend on partition: one can always perform a “global” canonical transformation or a mixing on variables of all parties to make an entangled state in the old variables separable in the new variables. We don?t need to worry about this, however, because when we are talking about quantum entanglement between two objects, we have actually assigned some preferred set of dynamical variables which are going to be measured locally — according to the apparatus in our hands or in our minds. Physical considerations will determine which way of partition or which set of dynamical variables is natural.
Quantum entanglement in open quantum systems
In a more realistic situation, the system we are looking at will be inevitably coupled with the environment, which could be as innocent as a mediating quantum field in free space or in medium at some temperature. Then the dynamics of the system should be described by the reduced quantum state (the reduced density matrix, RDM) for the system, which is obtained by integrating (or averaging) out the degrees of freedom of the environment in the density matrix of the combined system. The RDM of the system is in general a mixed state, which is a probabilistic sum of the density matrices of pure quantum states of the system. It carries both quantum and statistical natures.
What is quantum entanglement for a mixed state? For a bipartite system, a mixed state is said to be entangled if it CANNOT be written as a convex sum of direct products of the density matrices of each party,
(which can be modeled by local hidden-variable theories). Unfortunately this definition of quantum entanglement is not constructive. This makes the construction of well-defined degrees of entanglement for mixed states very hard and rarely successful — So far the well-defined ones exist only for systems with (a) two 2-level atoms, and (b) two Gaussian states!
This is actually a good excuse for us — to get a concrete result on entanglement dynamics, we have to design our thought experiment by employing one of these two categories, both are simple enough. Thus in [1] and [2] we employ the latter: Gaussian states in the Unruh-DeWitt (UD) detector theory.
The UD detector is a point-like object with its internal degree of freedom linearly coupled to a quantum field while its trajectory is put in by hand [1]. It is an analogy for an atom in EM field, rather than a real, macroscopic detector in laboratory. By choosing the internal degree of freedom of the detector as a harmonic oscillator* and the quantum field as a massless scalar field in Minkowski space, the combined system is linear, and we are able to solve the full dynamics of the detector-field system.
*Historically the interacting action of the Unruh-DeWitt detector theory was applied to demonstrate the Unruh effect in time-dependent perturbation theory by Bill Unruh (1976) and Bryce DeWitt (1979), without specifying the structure of the free detector. The present form with harmonic oscillators as the internal degrees of freedom is employed by several authors later for simplicity.
Temporal dependence of entanglement
In open quantum systems, the initial state of interest, e.g., a direct product of the quantum state for the system and the one for the environment, is usually not an eigenstate of the combined system. So the quantum state for the detectors or atoms as well as the one for the field (our environment) will all evolve in time (non-equilibrium phenomena!), which makes quantum entanglement between the detectors or atoms time-varying.
There are all kinds of entanglement dynamics in open quantum systems, some are not quite intuitive due to the somewhat curious definition of quantum entanglement for mixed states. For example, two initially entangled objects immersed in the environment can be completely disentangled in finite time. This is called the “sudden death” of entanglement.
Entanglement dynamics of two 2-level atoms in two independent cavities. C(t) is the “concurrence”, which is a measure of entanglement of these two atoms, and a is a parameter of the initial state. Different values of a gives different behaviors. From Yu, Eberly, PRL 93, 140404 (2004).
In contrast, two initially separated objects in the environment can be entangled after a finite time. This is called the entanglement creation or generation.
Entanglement creation of two spins coupled with a spin chain. From Lai, Hung, Mou, Chen, PRB77, 205419 (2008)
In some cases entanglement can revive after sudden death, or insists until late times and becomes residual entanglement.
Entanglement dynamics of two harmonic oscillators located at the same point in a quantum field. EN is the logarithm negativity, which is a degree of quantum entanglement. From Paz and Roncaglia, PRL100, 220401 (2008).
Spatial dependence of entanglement
Moreover, the properties of the environment will enter the entanglement dynamics of the system through the coupling. This is why the Unruh-DeWitt detectors are called ?detectors?: Their responses reveal the characteristics of the environment. For localized quantum objects, one of the consequences from coupling with the environment is the dependence of entanglement dynamics on the positions of them.
For two localized detectors 
and 
coupled with a common massless quantum field in free space, the back-reaction of the detector to the field will propagate out in light speed (because the field is massless) then drive the other detector , whose response will again back-act to the field and propagate back to drive the detector . The additional response of the detector to these echoes will then propagate out, and so forth. These generate the mutual influences between the detectors, which depend on how the information propagates in the environment very much. They cannot be faster than light, so they are explicitly causal.
Two detectors L and R are at rest and separated at a distance d in Minkowski space. The coupling between the detectors and the field is switched on at t=0.
Besides, in quantum fields there exist non-vanishing correlations between vacuum fluctuations of the fields at spacelike separated events. While these correlations are non-local in some sense, they still cannot be used for sending signal to violate causality.
Of course, in more complicated situations with more complicated environment such as spin chains or cavities with strange boundaries, or with more nontrivial motions of the detectors such as those in acceleration or sitting around a black hole [2], or with nonlinear interactions in the detectors and/or the environment, the response of the detectors or atoms will be more complicated. But as far as we know even in the simplest models the full dynamics were hardly well-understood by physicists. Thus in [1] we start with the simplest case we can imagine: We consider a model with two Unruh-DeWitt detectors, both at rest and separated at a distance d, coupled with a common massless scalar field at zero temperature in Minkowski coordinate.
We find that the dynamics of entanglement in our model depend on the spatial separation between the detectors in a non-trivial way. Both the two factors mentioned above, (1) phase difference of the quantum noise experienced locally by each detector, and (2) interference of the retarded mutual influences, can lead to the spatial dependence.
For an initially entangled pair of detectors, when one gets inside the light cone of the other (started from the other detector at the initial moment of switching on the interaction), certain interference pattern in d develops. At distances where the interference is constructive the disentanglement times are longer than those at other distances. This behavior is more distinct when the mutual influences are negligible; It is mainly coming from the factor (1).
Evolution of quantum entanglement of an initially entangled pair of detectors sitting at rest in vacuum state of a massless scalar field in Minkowski coordinate. EN is the logarithm negativity, which is a degree of quantum entanglement. For fixed d, as t increases, EN is falling so the detectors are disentangling. EN,rel is roughly the relative value of the EN at finite d to the EN at spatial infinity on the same time slice. You can see there are some interference patterns in d inside the light cone and in t outside the light cone.
On the other hand, for an initially separable state, the mutual influences (factor (2)) can generate entanglement from nothing if the spatial separation is sufficiently small.
We noticed that the entanglement is created deeply in the light cone, meaning that before the entanglement is generated the two detectors have had conversations back and forth for several rounds to synchronize their steps and build up the coherence.
Some authors suggested that entanglement can be generated by the environment even when one atom is still outside the light cone of the other. But in our simple model, we did not see any evidence of entanglement creation outside the light cone, though for the initially entangled states, the phase difference of quantum noise of the environment does produce interesting entanglement dynamics between the detectors located far apart: At some moments the larger the separation the weaker the entanglement, but at other moments, the stronger the entanglement. Such a behavior is again caused by factor (1).
Outlook
As pointed out by Erwin Schr?dinger, quantum entanglement is ?not ONE but THE characteristic trait of quantum mechanics?, so it is THE resource in quantum information science (including, e.g., quantum imaging, quantum teleportation, quantum communication, and so on) and plays the most important role in understanding the foundation of quantum physics. Nevertheless, quantum entanglement is more delicate than correlations and coherences in open quantum systems: It can give you surprises even in the simplest models. Today we are still in a status of gaining more and more experiences and intuitions on entanglement dynamics in different non-equilibrium systems. We hope we can handle and manipulate it in the near future.
References
[1] Shih-Yuin Lin and Bei Lok Hu, Phys. Rev. D 79, 085020 (2009) [arXiv: 0812.4391].
[2] Shih-Yuin Lin, Chung-Hsien Chou, and Bei Lok Hu, Phys. Rev. D 78, 125025 (2008) [arXiv:0803.3995].
[3] E. Calzetta, B.-L. Hu, Non-equilibrium quantum field theory.
[4] F. Benatti, Irreversible quantum dynamics.
26 Comments
2 electrons are entengled, but one is destroyed by a singularity. Now, what happens?
I have even better question: electron and positron pair is created on the horizon (in the entangled state), positron enters the black hole, while electron escapes to infinity. Will electron feel the fact that position has reached singularity?
I guess the real question is whether there is a singularity or it is smoothed out by quantum effects, isn’t it?
No, I wasn’t thinking about that. I asked that because I am really clueless about the answer, since a singularity is not a detector, but a destroyer of states. It’s like erasing part of wave function before its detection. I am not sure, but if you try to build a double slit experiment, where instead of a normal mask, one would use a singluarity like mask, with 2 holes. If you think in terms of path integral formulation of the problem, one of the possible outcomes is to collide with the singularity wall. So, I am unsure if the possibility of destruction would cripple all the probability wave. You see, there is no interaction with the singularity, but the destruction of the state, at least, as I see it.
Perhaps you’d get a classical pattern on the screen, perhaps.
A quick answer is “I don’t know”, because physical theories break down at singularity
“A destroyer of states” looks like something which breaks unitarity, so that the probability is not conserved. Is this what you mean?
Sure, exactly!
I was about to write that but this is so weird that I gave up. Gerard t’Hoof proposed a version of quantum mechanics with loss of states.
http://arxiv.org/abs/gr-qc/9903084
I also recomment the articles that cited it.
It’s not the usual QM, but something different, that is supposed to work at and below plank scale, although it is supposed to yield QM at low energies. It would be nice if more people worked on that.
I see. Thanks a lot!
‘t Hooft is more or less trying to construct a quantum mechanics (or a toy model of it) as an emergence theory from a more fundamental structure (which is unknown!). His ideas are original and very illuminating, but I have to say I have difficulty to get all his points. (so embarrassed…, for example, in this paper (gr-qc/9903084) I just don’t understand what his “dissipation” means.
)
Your original question is very interesting, though I am really clueless, too. Let us keep that question in mind and watch the development of the emergence QM.
But I rather avoid these issues by thinking that a real Black Hole is a mimicker of the one of GR, in the classical limit, like a fuzzy sphere.
No. As a theorist, when I am doing calculation in my laptop I do know every bit of information in each UD detector and each mode of the field, no matter where they are in the universe. However, in reality, if I am allowed to look at one UD detector only, I can’t realize two detectors were entangled or not until I received the classical information from the other detector, such as the outcome of the local measurement there, then compared with mine.
Imagine that you measure the spin of the electron outside the black hole in z-direction. Then the outcome is either up or down. This is all information you can obtain if you never know the information of the positron behind the event horizon.
Suppose you have an ensemble of such electron-positron pairs. You perform a series of similar measurements, then you will find a 50-50 distribution of up and down, no matter what happens to the positron. Then you can re-construct the reduced density matrix (2×2) of the electron spin(“quantum tomography”), but you will never be able to re-construct the RDM of the electron-positron pair (4×4), so you just can’t talk about the entanglement between those two particles.
Well, if information is not lost in the end, I guess, I’ll be able to construct 4×4 RDM for electron-position pairs after BH is completely evaporated, so that the information inside it gets revealed.
But if it is never revealed, then I see: if I measure the z component of spin of the electron and find it to be 1/2, there is no way for me to confirm that the z component of spin of the positron is -1/2.
Well, I am not sure if you can speak of the inside of a black hole in terms of information recover, maybe the border of the inside, the horizon. This is where the information of the black hole is stored according to any holography theory I am aware of. You can see this because the horizon is a non essential sigularity, and thus it can be removed by a suitable coordinate choice, and so no state is destroyed. On the other hand, the singularity is problematic since it is an essecial one…
Daniel,
I am talking about the situation when BH is completely evaporated, so horizon is absent. If the information in BH is not lost, it should be then recovered from what is left after complete BH evaporation.
Cheers,
Dmitry.
But I was thinking about the particle of the virtual pair particle/anti-particle that went inside of the black hole. If the horizon disapears, the inside will go away with it, so you won’t know whatever was the state of that particle. But doesnt the particle that enters the black hole subtract its mass? The emited one will be part of the hawking radiation. I mean, this is the traditional particle explanation for evaporation of a blackhole.
For example:
http://nrumiano.free.fr/Estars/bh_thermo.html
You can see that this interpretation leads to the conclusion of why information is lost: the black hole just emits radiation acording to the random pair it absorbs.
This explanation is contained somewhere in Stephen Hawking’s book “A Brief History of Time”…
As you may notice, currently the folklore is that no information will be lost in black hole evaporation:
http://ptonline.aip.org/journa.....57_1.shtml
If this belief is true, then the information must be encoded in the black hole radiation, which will not be completely thermal. (Hawking’s result is a thermal radiation because the spacetime geometry is a classical background in his calculation.)
Of course, the debate will not end until people can handle the back-reaction of quantum fields to the spacetime, or even better, a quantum gravity tested by experiments or observations.
By the way, one should not be too serious on “pair creation at the horizon”: that explanation contained in Hawking’s “A Brief History of Time” is not based on any mathematical formulation, according to Bill Unruh. It’s just a hand-waving story to make readers feel easier
Dear Shih-Yuin,
thanks again for the post! Here is the first question of mine as inoculum for the discussion: what is the physical meaning of
(and how is it defined for your model of free field linearly interacting with detector)?
Cheers,
Dmitry.
It’s my honor!
The “concurrence” C(t) is first introduced for two-qubit(spin or 2-level atom) system by Hill and Wootters [PRL78, 5022(1997); Wootters, PRL80, 2245(1998)]. It is a measure of entanglement for two-qubit system, with the value going from 0 to 1. Separable states always give C=0, and entangled states have C>0. If a quantum state has C=1, e.g. (|01> – |10>)/\sqrt{2}, we say it is maximally entangled. In quantum teleportation it can be related to the fidelity of teleportation.
C(t) cannot be applied to the systems with continuous variables or infinite-level atoms such as our Uuruh-DeWitt detectors, though it has been generalized to higher (but finite) dimensional systems. In our model we are using “logarithmic negativity” E_N [Vidal and Werner, PRA65, 032314(2002)] and a quantity \Sigma [see our papers] as indicators of entanglement. E_N is the most popular well-defined degree of entanglement for Gaussian states in 2-harmonic-oscillator systems. Just like the concurrence, separable states always have E_N = 0, and if one state has a larger value of E_N than the other, we say the entanglement of that state is the stronger.
By the way, the last plot in section “Temporal dependence of entanglement” is from Paz and Roncaglia, PRL100, 220401 (2008).
Thanks for the explanations! I’ll add the caption to the plot.
Dear Shih-Yuin,
can one actually measure a quantum phase of the given mode k the field
is in? (we can certainly measure 
for the given mode k – the spectrum of the field).
Cheers,
Dmitry.
Good question, I don’t know the answer! Let me think about it and search the literature.
Got one:
Noh, Fougeres, and Mandel, PRL67, 1426 (1991).
Digesting…
I also went through the literature a bit: this one seems relevant –
Shapiro, Shepard, PRA43, 3795 (1991), http://prola.aps.org/abstract/PRA/v43/i7/p3795_1
Basically, what they say is that it is possible to construct probability operator measure for Susskind-Glogover operator
, which is exactly the quantum phase for a given mode.
Your question is so interesting and nontrivial that people even make a book to answer it:
Barnett and Vaccaro (eds) “The Quantum Phase Operator: A Review” (Chapman & Hall, 2007).
(This is a collection of reprints. I got the table of content only, but that’s good enough.)
Canonical phase operators for a given field mode are not made well-behaved until 1988, when Pegg and Barnett successfully constructed a unitary phase operator, which is conjugate to the particle number operator. Unfortunately, due to the lack of the suitable coupling in nature (all one can do in optics is counting the photon number), DIRECT measurement of canonical phase has never been realized in laboratory. So far experimentalists can at best estimate the quantum phase and its variance using heterodyne detection or adaptive homodyne measurement. (For a simple introduction on adaptive measurement, see Jacob, quant-ph/0605015; a tutorial review, see Pegg and Barnett, J. Mod. Opt. 44, 225 (1997).)
There are some operational definitions of quantum phase,
such as those constructed by Noh, Fougeres, and Mandel. However, those phases are dependent on specific experimental arrangement, and behave differently from the canonical phase.
Thank you for raising this question, I learned so much in seeking an answer
Dear Shih-Yuin,
you are most welcome
I have learned quite a lot of stuff from your post, too.
The reason I asked that question was the following. What if one reformulates everything you did with Bei-Lok Hu in the holomorphic representation (i.e., explicitly use
‘s instead of 
)?
This would, for example, allow one to count how many exchanges of particles exactly you need to build up coherence. Then, the question is what is coherence (of, say, two modes
and 
) in the holomorphic representation language. I guess it should be correlation of phases of those modes.
What do you think?
Cheers,
Dmitry.
Dear Dmitry,
Good point. Campo and Parentani have a series of nice work on quantum coherence of the inflaton field (e.g. arXiv:0805.0548 and the references therein). They study how the quantum-ness of a given pair of field modes
and 
depends on the particle numbers and phase correlations. The tricky part is that the covariance matrix of the two modes 
and 
is not in a standard form of those used in quantum information for two parametric oscillators, so even the criterion of quantum-ness or classicality is not obvious in their system, and they have to look at it in many different perspectives. Anyway in their results one can see that quantum coherence of the modes does have certain dependence on the relative phase of the modes.
I think the challenge is, still, how to observe or estimate the quantum phase.
Cheers,
Shih-Yuin
Dear Shih-Yuin,
Ok, so the question is basically how to construct a kind of holomorphic representation for parametric oscillators. Let me think about that for a bit…
Cheers,
Dmitry.
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