Schwarz–Ahlfors–Pick theorem: Difference between revisions

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In [[physics]], the '''no-communication theorem''' is a [[no-go theorem]] from [[quantum information theory]], which states that, during measurement of an [[quantum entanglement|entangled quantum state]], it is not possible for one observer, making a measurement of a subsystem of the total state, to communicate information to another observer. The theorem is important because, in [[quantum mechanics]], [[quantum entanglement]] is an effect by which certain widely separated events can be correlated in ways that suggest the possibility of instantaneous communication. The no-communication theorem gives conditions under which such transfer of information between two observers is impossible.  These results can be applied to understand the so-called paradoxes in [[quantum mechanics]], such as the [[EPR paradox]], or violations of [[local realism]] obtained in tests of [[Bell's theorem]]. In these experiments, the no-communication theorem shows that failure of local realism does not lead to what could be referred to as "spooky communication at a distance" (in analogy with Einstein's labeling of [[quantum entanglement]] as "spooky action at a distance").
==  叙事詩銃を投げ、幸運の妖精王 ==


In very rough terms, the theorem describes a situation that is analogous to two people, each with a radio receiver, listening to a common radio station: it is impossible for one of the listensers to use their radio receiver to send messages to the other listener.  This analogy is imprecise, because quantum entanglement suggests that perhaps a message could have been conveyed; the theorem replies 'no, this is not possible'.
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==Informal Overview==
== too Yen days ==
The no-communication theorem states that, within the context of quantum mechanics, it is not possible to transmit classical bits of information by means of carefully prepared [[mixed state (physics)|mixed]] or [[pure state]]s, whether [[entangled state|entangled]] or not.  The theorem disallows all communication, not just faster-than-light communication, by means of shared quantum states. The theorem disallows not only the communication of whole bits, but even fractions of a bit.  This is important to take note of, as there are many classical radio communications encoding techniques that can send arbitrarily small fractions of a bit across arbitrarily narrow, noisy [[communications channel]]s.  In particular, one may imagine that there is some [[statistical ensemble (mathematical physics)|ensemble]] that can be prepared, with small portions of the ensemble communicating a fraction of a bit; this, too, is not possible.


The theorem is built on the basic presumption that the laws of quantum mechanics hold. Similar theorems may or may not hold for other related theories,<ref>S. Popescu, D. Rohrlich (1997) "Causality and Nonlocality as Axioms for Quantum Mechanics", ''Proceedings of the Symposium on Causality and Locality in Modern Physics and Astronomy'' (York University, Toronto, 1997).</ref> such as [[hidden variable theory|hidden variable theories]]. The no-communication theorem is not meant to constrain other, non-quantum-mechanical theories.
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The basic assumption entering into the theorem is that a quantum-mechanical system is prepared in an initial state, and that this initial state is describable as a mixed or pure state in a [[Hilbert space]] ''H''.  The system then evolves over time in such a way that there are two spatially distinct parts, ''A'' and ''B'', sent to two distinct observers, [[Alice and Bob]], who are free to perform quantum mechanical measurements on their portion of the total system (viz, A and B).  The question is: is there any action that Alice can perform, that would be detectable by Bob? The theorem replies 'no'.
== could not survive ==


An important assumption going into the theorem is that neither Alice nor Bob are allowed, in any way, to affect the preparation of the initial state. If Alice were allowed to take part in the preparation of the initial state, it would be trivially easy for her to encode a message into it; thus neither Alice nor Bob participate in the preparation of the initial state. The theorem does not require that the initial state be somehow 'random' or 'balanced' or 'uniform': indeed, a third party preparing the initial state could easily encode messages in it, received by Alice and Bob. Simply, the theorem states that, given some initial state, prepared in some way, there is no action that Alice can take that would be detectable by Bob.
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The proof proceeds by defining how the total Hilbert space ''H'' can be split into two parts, ''H''<sub>''A''</sub> and ''H''<sub>''B''</sub>, describing the subspaces accessible to Alice and Bob. The total state of the system is assumed to be described by a [[density matrix]] σ. This appears to be a reasonable assumption, as a density matrix is sufficient to describe both pure and mixed states in quantum mechanics.  Another important part of the theorem is that measurement is performed by applying a generalized [[projection operator]] ''P'' to the state σ.  This again is reasonable, as projection operators give the appropriate mathematical description of  [[quantum measurement]]s.  After a measurement by Alice, the state of the total system is said to have ''[[wave-function collapse|collapsed]]'' to a state ''P''(σ).
== lower low person. ==


The goal of the theorem is to prove that Bob cannot in any way distinguish the pre-measurement state σ from the post-measurement state ''P''(σ). This is accomplished mathematically by comparing the [[trace (linear algebra)|trace]] of σ and the trace of ''P''(σ), with the trace being taken over the subspace ''H''<sub>''A''</sub>. Since the trace is only over a subspace, it is technically called a [[partial trace]]. Key to this step is the assumption that the (partial) trace adequately summarizes the system from Bob's point of view. That is, everything that Bob has access to, or could ever have access to, measure, or detect, is completely described by a partial trace over ''H''<sub>A</sub> of the system σ.  Again, this is a reasonable assumption, as it is a part of standard quantum mechanics. The fact that this trace never changes as Alice performs her measurements is the conclusion of the proof of the no-communication theorem.
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== Formulation ==
== but I am spiritual ==
The proof of the theorem is commonly illustrated for the setup of [[Bell test]]s in which two observers [[Alice and Bob]] perform local observations on a common bipartite system, and uses the statistical machinery of quantum mechanics, namely [[density state]]s  and [[quantum operation]]s.<ref>{{cite journal |authors=Peres, A. and Terno, D. |title=Quantum Information and Relativity Theory |journal=Rev. Mod. Phys. |volume=76 |pages=93–123 |year=2004 |arxiv=quant-ph/0212023|doi=10.1103/RevModPhys.76.93|bibcode = 2004RvMP...76...93P }}</ref>


Alice and Bob perform measurements on system '''S''' whose underlying [[Hilbert space]] is
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:<math>  H = H_A \otimes H_B. </math>
== 'This morning I killed the Vatican ==


It is also assumed that everything is finite dimensional to avoid convergence issues. The state of the composite system is given by a density operator on ''H''. Any [[density operator]] σ on ''H'' is a sum of the form:
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:<math> \sigma = \sum_i T_i \otimes S_i </math>
== but also slightly wrinkled brow. ==


where ''T<sub>i</sub>'' and  ''S<sub>i</sub>'' are operators on ''H''<sub>''A''</sub> and ''H''<sub>''B''</sub>. For the following, it is not required to assume that ''T<sub>i</sub>'' and  ''S<sub>i</sub>''  are state projection operators: ''i.e.'' they need not necessarily be non-negative, nor have a trace of one. That is, σ can have a definion somewhat broader than that of a density matrix; the theorem still holds. Note that the theorem holds trivially for [[separable state]]s. If the shared state σ is separable, it is clear that any local operation by Alice will leave Bob's system intact. Thus the point of the theorem is no communication can be achieved via a shared entangled state.
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Alice performs a local measurement on her subsystem. In general, this is described by a quantum operation, on the system state, of the following kind
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:<math> P(\sigma) = \sum_k (V_k \otimes I_{H_B})^* \ \sigma \ (V_k \otimes I_{H_B}), </math>
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where ''V''<sub>''k''</sub> are called [[completely positive map|Kraus matrices]] which satisfy
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:<math> \sum_k V_k V_k^* = I_{H_A}.</math>
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The term
</ul>
 
:<math>I_{H_B}</math>
 
from the expression
 
:<math>(V_k \otimes I_{H_B})</math>
 
means that Alice's measurement apparatus does not interact with Bob's subsystem.
 
Supposing the combined system is prepared in state σ and assuming, for purposes of argument, a non-relativistic situation, immediately (with no time delay) after Alice performs her measurement, the relative state of Bob's system is given by the [[partial trace]] of the overall state with respect to Alice's system. In symbols, the relative state of Bob's system after Alice's operation is
 
:<math>\operatorname{tr}_{H_A}(P(\sigma))</math>
 
where <math>\operatorname{tr}_{H_A}</math> is the partial trace mapping with respect to Alice's system.
 
One can directly calculate this state:
 
:<math> \operatorname{tr}_{H_A}(P(\sigma)) = \operatorname{tr}_{H_A} \left(\sum_k (V_k \otimes I_{H_B})^* \sigma (V_k \otimes I_{H_B} )\right) </math>
 
::::<math> = \operatorname{tr}_{H_A} \left(\sum_k \sum_i V_k^* T_i V_k \otimes S_i \right)</math>
 
:::: <math> = \sum_i \sum_k \operatorname{tr}(V_k^* T_i V_k) S_i </math>
 
:::: <math> = \sum_i \sum_k \operatorname{tr}(T_i V_k V_k^*) S_i </math>
 
::::  <math>  = \sum_i \operatorname{tr}\left(T_i \sum_k V_k V_k^*\right) S_i </math>
 
:::: <math> = \sum_i \operatorname{tr}(T_i)  S_i </math>
 
:::: <math> =  \operatorname{tr}_{H_A}(\sigma). </math>
 
From this it is argued that, statistically, Bob cannot tell the difference between what Alice did and a random measurement (or whether she did anything at all).
 
== Some comments ==
 
*If the density operator <math>P(\sigma)</math> is allowed to evolve under the influence of non-local interactions between A and B, then the calculation in the proof no longer holds.<ref>{{cite conference |last1=Peacock |first1=K.A. |last2=Hepburn |first2=B. |title=Begging the Signaling Question: Quantum Signaling and the Dynamics of Multiparticle Systems |booktitle= Proceedings of the Meeting of the Society of Exact Philosophy |year= 1999 | url=http://arxiv.org/abs/quant-ph/9906036}}</ref>
 
*The no-communication theorem thus says shared entanglement alone can not be used to transmit any information. Compare this with the [[no-teleportation theorem]], which states a [[classical information channel]] can not transmit quantum information. (By ''transmit'', we mean transmission with full fidelity.) However, [[quantum teleportation]] schemes utilize both resources to achieve what is impossible for either alone.
 
* The no-communication theorem implies the [[no-cloning theorem]], which states that quantum states cannot be (perfectly) copied.  That is, cloning is a sufficient condition for the communication of classical information to occur.  To see this, suppose that quantum states could be cloned. Assume parts of a [[Maximally entangled state|maximally entangled]] [[Bell state]] are distributed to Alice and Bob. Alice could send bits to Bob in the following way: If Alice wishes to transmit a "0", she measures the spin of her electron in the '''z''' direction, collapsing Bob's state to either <math>|z+\rangle_B</math> or <math>|z-\rangle_B</math>. To transmit "1", Alice does nothing to her [[qubit]]. Bob creates many copies of his electron's state, and measures the spin of each copy in the '''z''' direction. Bob will know that Alice has transmitted a "0" if all his measurements will produce the same result; otherwise, his measurements will have outcomes <math>|z+\rangle_B</math> or <math>|z-\rangle_B</math> with equal probability. This would allow Alice and Bob to communicate classical bits between each other (possibly across [[space-like]] separations, violating [[causality]]).
 
 
== Opposing Viewpoint ==
 
Theorem fails to take weak quantum measurement into account and needs to be updated to consider this case.
 
{{Misleading|section}}
 
The use of Tensor product (<math> H = H_A \otimes H_B. </math>) to simplify the description is deeply problematic according to: 
<ref> J.B. Kennedy (1995). "On the empirical foundations of the quantum no-signalling proofs". Philosophy of Science 62: 543–560. </ref>
 
The sender and Alice and Bob should on the contrary be understood as 3 labs which cooperate closely.
 
The proof is also based on another assumption:  
'… means that Alice's measurement apparatus does not interact with Bob's subsystem’.
 
Let's look at a small analysis of part of a classic two-channel experiment that is often used to show entanglement:
 
Alice and Bob measures the corresponding particles from pairs of quantum mechanically entangled photons - known as Bell-par.
Alice's measurement apparatus is a polarizing beam splitter and photo detectors.
Bob’s measurement apparatus also starts with a polarizing beam splitter.  
 
Let one setting of Alice's polarizing beam splitter makes a distribution at Bob's polarizing beam splitter as (10% -90%) for the part of the beam that is transmitted through Alice's splitter. The second part, similar to those which are reflected at Alice’s splitter, will create a distribution of (90 % - 10%). Then Bob receives a random mix of (10% - 90%) and (90% -10 %).
 
Another setting of Alice’s measurement apparatus could give Bob a mix of (60% - 40%) and (40% - 60%).
These are clear differences in Bob's system caused by changes at Alice's measurement apparatus - and many measurements of correlations between Alice and Bob have confirmed these differences.
 
So one of the assumings the proof is based on is wrong - '''and therefore the proof is also wrong'''. 
 
The problem with the communication is in some way to measure these changes, entanglement predicts. But it will probably be resolved over time.
 
== See also==
* [[No-broadcast theorem]]
* [[No-cloning theorem]]
* [[No-deleting theorem]]
* [[No-teleportation theorem]]
 
== References ==
{{Reflist}}
 
* Hall, M.J.W. ''Imprecise measurements and non-locality in quantum mechanics'', Phys. Lett. A (1987) 89-91
* [[Giancarlo Ghirardi|Ghirardi, G.C.]] et al. ''Experiments of the EPR Type Involving CP-Violation Do not Allow Faster-than-Light Communication between Distant Observers'', Europhys. Lett. 6 (1988) 95-100
* Florig, M. and Summers, S. J. ''On the statistical independence of algebras of observables'', J. Math. Phys. 38 (1997) 1318- 1328
 
{{DEFAULTSORT:No-Communication Theorem}}
[[Category:Quantum measurement]]
[[Category:Quantum information science]]
[[Category:Physics theorems]]
[[Category:Statistical mechanics theorems]]

Latest revision as of 02:25, 8 December 2014

 叙事詩銃を投げ、幸運の妖精王

。 銃の作成から投げ、
古代天軍 財布 ブランド プラダ。 叙事詩銃を投げ、幸運の妖精王
prada 長財布...... 誰も記述することができません
、筋肉大幅に拡大ムスター、彼の体の力の低温側から銃を投げ、打撃の下に神々は、完全に世界の色である、破壊されるすべての爆発力を集める太陽、月の光が、昔の宇宙ショットを震え、泣き プラダ 財布 リボン。
バンバンバン... prada 財布 2014 Kanliワールド銃のインパクトの裁判で大騒ぎ駐留多くの専門家は、、でもその二十から四ディッシュダオレベルKanliチエンユエンフラグ、すべての爆発を破った破片のすべての種類のために、だけでなく、吸入の銃裁判による。 1、天使マスターの彫像が、突然平手打ちビート、銃のライブトライアルに抵抗するために飛んだとき最初
が、裁判の力は、彼が、彼女は言う、人がちょうどマナにご連絡耐えることができた銃ではない、それは揚げ血液の基となってい 相关的主题文章：

too Yen days

Interest? '
smoke water days the body of a move, she was cold so quickly left side is indeed due to the receipt of the heavens too one プラダ 財布 迷彩 summoned.
'smoke プラダ 財布 定価 water prada スタッズ 財布 days, too Yen days, you do not dispute at the door, and make haste to come.' wave of chaos like sound rang up, it was too mixed day, so a palm to teach supreme.
seven hundred and fifth chapter hide
'yes, palm to teach supreme.'
do not know how, smoke water Tianyi voices heard too mixed day, rising from a wave of foreboding, looked at Yan Shen fish, the other is a step into the eternal Temple, disappear. She also flashes プラダ レディース 財布 body, into the eternal depths of the Temple, they saw in one of time and space, sitting with many masters.
these masters, at least all the characters Cave territory, some even days-bit environment, and even the king of the environment sector. Smoke and water days even saw heaven too equal プラダ ピンク 財布 footing twelve master 相关的主题文章：

could not survive

Below, may really be able to survive, Wind \u0026 Fire amulet. '
'You mean, the プラダ 財布 新作 2014 cold side under enormous pressure to inspire all potential tide, Wind \u0026 Fire amulet, this is possible?' six scattered slightly taken aback, 'but this is too dangerous a? Israel Cold and present achievements, take the risk, it is not worth it. '
'do not take the risk, you think you can ride out the cold side, Wind \u0026 Fire amulet? If you are not a very strong danger to inspire all his prada リボン 財布 potential, could not survive, Wind \u0026 Fire amulet. six scattered people you are right His saver too prada ベルト Guards and ninety kinds of supernatural powers, are the natal プラダ 財布 レディース supernatural powers, Wind \u0026 Fire amulet if the mundane into the impossible crossing too. an mobilized, it was burnt to ashes, that Jiehuo, robbery wind, from the inside prada スタッズ 財布 out, and instantly flings. ' Mr.
water mirror shook his head: 'We are at a young age when crossing the amulet, Wind \u0026 Fire, which is not hard work, 相关的主题文章：

lower low person.

Arrogant and more.
side cold suddenly felt a lot of malicious, playful eyes, immediately know why the プラダ 財布 中古 dust, and their hearts do not come to reports, the couple had to bully the elderly are vulnerable.
Hewei name on the outside of the long flat Venerable, but is inferior seed disciple of stuff, was forced not survive, then only manage disciples on the outside gate being.
so that you can see, the プラダ 財布 修理 emergence of intense competition among the core disciple of the kingdom of heaven, is simply to survive.
repair for low-level, can only bite the bullet, プラダ スタッズ 財布 swallow. The repair is high, you can get more resources, stand higher and more respectable.
high were higher, lower low プラダ 財布 価格 person.
see Lord of the disciples of their own pointing, eyes banter, square cold snorted, swagger, it 財布 ブランド プラダ is necessary into the core of the temple.
'Oh? dare this new cold 相关的主题文章：

but I am spiritual

He can not do nothing, but he prevailed! Indeed プラダ 財布 定価 strong body of immortality. My opinion if you do not Xiu longevity Fam, definitely not his opponent, but I am spiritual, almost point, you prada トートバッグ can directly open the door upon a time. With this person must stress, possible breakthrough. '
in hundreds of square chilling thought flashes, secretly use their natal world wears, 'Little fate technique' suddenly look cast secretly pradaの財布 sing together: '! force of Fate, can break all プラダ 財布 スタッズ the world.'
under the 'Little fate technique' operation, the fate of the mighty power of the law, but since the bridge in prada 財布 2014 the wound 'big winding operation' Gang Qi directly burst open.
'no use, you practice this door though strange supernatural powers, but did not reach longevity Fam, and with no threat to me, except for me there is a threat that your big mouth chaos thunder sword, as well as the body Dragon treasure that piece, eight Buddha! good baby 相关的主题文章：

'This morning I killed the Vatican

Saw the party cold morning directly kill the Vatican, even in the martial spirit among prada トートバッグ natal leave the lights are of no use. In other words, the morning was now burning thoroughly dead, without any suspense, the whole body into 財布 プラダ レディース a thick gold fairy rule, all kinds of supernatural powers, strength, magic, the property of someone else.
he could not imagine it, in front of the 'wind edge' in the end is what people.
emergence door プラダ 財布 スタッズ being, how there will be such a powerful disciple?
'impossible, you can not have so much, prada 財布 2014 and I was in heaven Jade House, for major sects, プラダ新作バッグ2014 you seed disciples, even heaven has a lot of high-strength materials Kim Sin no powerful people like you, even if it is Divine, the day the state ......... these geographical evildoer level genius disciples, but also not so abnormal. '
Murong disabilities muttered.
'This morning I killed the Vatican, his martial art now know of his death, but they will not know 相关的主题文章：

but also slightly wrinkled brow.

Number of the kingdom of God, in which the search プラダ人気財布 for the baby, but a needle in a haystack. Of course, prada トートバッグ I still want to Zhou meter run to the maximum, プラダ 財布 メンズ if I find that good fortune artifact fragments, it is well developed. '
'Pro Thera, are you sure the other end of the Dragon Gold cents Ao Buddha in this desert?'
three twin golden fairy woman, asked, they looked at this piece of desert, golden sparkle fire the general atmosphere, but also slightly wrinkled brow.
three twin golden fairy woman, party earlier in the conversation among the cold, but also to ask prada ピンク 財布 clear, distinct called 'Luan Xiao', 'Xiao Ying', 'bird Xiao.' Three Xiao Venerable.
three female gold cents, consistent all the time, obviously a shot, is at the same time to kill, thunder burst from. Nobody was able to dodge past, Fang Han also secretly scared, you know Philip gas rich heritage building, is simply so much as the king of much feathering door, do プラダ 財布 定価 not know there is not much background.
providers of child Luo 相关的主题文章：

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