Lenny Susskind gives the simplest proof of the no-cloning theorem in his little blackhole book.

Suppose we have a single unitary time evolution operator

U(t) = e^iHt/hbar

ignore issues of time ordering in quantum field theory for now that will apparently not make a significant difference to the conclusion?

Do not here think of |0) as the vacuum with |1) as a single-quantum state in sense of second-quantization a*|0)

Consider a single qubit with a c-bit basis |1) and |0) at time t = 0, at time t these states transform to

|0)' = U(t)|0)

|1)' = U(t)|1)

The general qubit at t = 0 is the fragile coherent superposition

|qubit) = |0)(0|qubit) + |1)(1|qubit)

completeness is

|0)(0| + |1)(1| = 1

Therefore

|qubit)' = U(t)|qubit)

= U(t)|0)(0|qubit) + U(t)|1)(1|qubit) = |0)'(0|qubit) + |1)'(0|qubit)

Note that U*(t)U(t) = U(t)U*(t) = 1 by hypothesis, therefore

(...|qubit) = '(...|qubit)'

i.e. invariance of inner products under the SAME unitary transformation.

Let C(t) be a hypothetical cloning operator. Can it be linear and unitary?

C(t)|0) = |0)|0)

C(t)|1) = |1)|1)

Using only linearity

C(t)|qubit) = C(t)|0)(0|qubit) + C(t)|1)(1|qubit)

= |0)|0)(0|qubit) + |1)|1)(1|qubit) =/= |qubit)|qubit)

Therefore, it cannot be linear and unitary because

|qubit)|qubit) = [|0)(0|qubit) + |1)(1|qubit)][|0)(0|qubit) + |1)(1|qubit)]

= |0)|0)(0|qubit)^2 + |1)|1)(1|qubit)^2 + 2|0)|1)(0|qubit)(1|qubit)

But if C(t) were also unitary, then, for arbitrary inner products

(...|C*(t)C(t)|???) = (...|???) =/= (...|???)^2

Nick Herbert's FLASH theorem is that a C(t) operator for arbitrary |qubit) states permits signal nonlocality.

Lenny Susskind shows that a violation of this no-cloning theory contradicts his theory of black hole complementarity - another story for another time.

Let's now directly look at the theorem forbidding signal nonlocality, i.e. forbidding controlled spooky telepathic paranormal action at a distance using quantum entanglement as a stand-alone Command-Control-Communication (AKA C^3) in which a classical light-cone limited signal is NOT required to decode the message encoded in the spread out entanglement.

Consider a pair-entangled state

|Alice, Bob) = |ab)(ab|Alice, Bob) + |a'b')(a'b'|Alice, Bob)

where a(b) & a'(b') are two possible eigenvalues Alice (Bob) can directly locally measure.

Suppose these are the only two eigenvalues that each is able to measure.

Therefore

|a)(a| + |a')(a'| = 1 local completeness for Alice

|b)(b| + |b')(b'| = 1 " " " Bob

However

|ab)(ab| + |a'b')(a'b'| =/= 1

In fact

|ab)(ab| + |a'b')(a'b'| + |a'b)(a'b| + |ab')(ab'| = 1

Entanglement means incompleteness at the pair-level! If we had pair-completeness the pair state would be a statistically independent product for the measurement of the operators

A = a|a)(a| + a'|a')(a'|

and

B = b|b)(b| + b'|b')(b'|

[A,B] = AB - BA = 0

The forbidding of signal nonlocality, i.e. the enforcement of "signal locality", is that summing ("tracing") over Alice's eigenvalues a & a'

|ab) = |a)|b) etc.

For the irreversible detections of Alice's A quanta, measurement theory demands for what distant Bob will see:

TraceA|Alice,Bob)(Alice,Bob|

= TraceA{[|ab)(ab|Alice, Bob) + |a'b')(a'b'|Alice, Bob)][(Alice, Bob|ab)(ab| + (Alice, Bob|a'b')(a'b'|]}

= TraceA{(|a)(a|)(|b)(b|)|(ab|Alice,Bob)|^2 + (|a')(a'|)(|b')(b'|)|(a'b'|Alice,Bob)|^2

+ (|a')(a|)(|b')(b|)(a'b'|Alice,Bob)(Alice,Bob|ab) + (|a)(a'|)(|b)(b'|)(ab|Alice,Bob)(Alice,Bob|a'b')}

Assume (a|a) = (a'|a') = 1 Normalized states

(a|a') = (a'|a) = 0 Alice's local orthogonality.

TraceA ... = (a|...|a) + (a'|...|a')

Therefore

TraceA|Alice,Bob)(Alice,Bob| = (|b)(b|)|(ab|Alice,Bob)|^2 + |b')(b'||(a'b'|Alice,Bob)|^2

without any local fringe interference terms ~ |b)(b'| and |b')(b|

So this is the proof of no stand-along LOCAL FRINGES in entangled systems - in a generic pair state somewhat simplified to dichomatic eigenvalues without loss of generality.

Now what can go wrong with this proof? More than one thing, but note if we used ODLRO macro-quantum coherent Glauber states then

(a|a') --> (z|z') ~ e^-|z-z''|^2 =/= 0 generalize "a" to a continuous complex variable z

i.e. macro-quantum ODLRO signal nonlocality.

Another approach would be a anholonomy in which the evolution of the different Alice states is path dependent so that at the local Alice (sender) measurements

|a)' = U(a)|a) & |a')' = U(a')|a')

where

U(a)*U(a') =/= 1

Therefore in effect even though (a|a') = 0 initially on a fuzzy spacelike hypersurface of finite thickness ~ resolving time

'(a|a')' =/= 0 on the evolved fuzzy local spacelike hyperspace where Alice's quanta are irreversibly detected.

i.e. an effective non-unitary evolution making Alices local states non-orthogonal even if they started out orthogonal - i.e. emergent signal nonlocality in a more general post-quantum theory.

On Jul 18, 2006, at 2:42 PM, Jack Sarfatti wrote:

Good news. :-)

I hope to be able to focus in on this more intensely soon. Meantime I have seen no one able to refute your argument or Cramer's argument - nothing from Nick Herbert on this for example. You seem to have found a new loophole that bypasses the standard no-cloning based on unitarity and linearity - a new incompleteness in orthodox QM. I am not sure yet myself - but that is how it appears at the moment.

On Jul 18, 2006, at 1:58 PM, Srikanth R wrote:

Hi, Dr. Jack,

Just back from out of station...

Thanks a lot for your comments. As you point out, an actual experimental test of nonlocal signaling using the modified Dopfer experiment would help at this point.

I have been talking to an experimentalist colleague of mine about testing the idea of noncomplete measurement at the level of unentangled photons (since we entanglement based experiments are not yet available here). He seems to be quite enthusiastic, so I hope to have interesting results to report soon!!

With best regards,

Srik.

On Mon, 10 Jul 2006, Jack Sarfatti wrote:

On Jul 10, 2006, at 6:45 AM, Srikanth R wrote:

On Sat, 8 Jul 2006, Jack Sarfatti wrote:

OK here is my morning-after assessment of the situation.

Quantum Reality is complex. ;-)

Just as Classical Reality is real... ;-)

Exactly! Seriously, Roger Penrose has some really interesting insights on the physical meaning of complex numbers in "The Road to Reality". Born probability rule breaks "complex holomorphic structure" - curious clue.

Woke up this morning with a lucid dream of a technicolor 3D tour of Dante's Inferno underneath The Vatican - seemed very real. :-)

Clarifications on "counter-factual definiteness" and the equation for the complete set of Alice's photon states |y> that washes out any stand-alone local fringes on Bob's side if the CCC is switched off in the usual orthodox quantum theory.

Previously I wrote:

The real idea here is counter-factual definiteness that what might happen even if it doesn't would be definite if it were to happen.

Now what happens is that we need to wait for a large enough statistical sample or "Born ensemble" of photon pairs to register on each side to see what is happening. This is like Lenny Susskind's populated "peppered" cosmic landscape in eternal chaotic inflation on the much larger scale in which our universe has a small enough cosmic dark energy allowing us to come into being and becoming in the sense of the Weak Anthropic Principle AKA WAP.

Therefore, the equation of completeness

Integral |y)(y| = 1

of everything Alice might have done in all the multiple branches or parallel classical worlds add up to what Bob actually sees locally without the CCC. That's the basic implicit subliminal ontological-epistemological Ansatz in the orthodox thinking I think?

Now Alice the "sender" has two choices to measure in the image plane or in the focal plane in the picture. Call the two variables y and y' respectively.

The issue is

Sum|y)(y| = 1 image plane (POSITION MEASUREMENT) YES? NO? (1)

YES! Let us denote the outcomes on the image plane y and -y (following the figure). Outcome y corresponds to the measurement given by the sum of annihliation operators for modes |-ps> and |-pd>. That is:

Y = a_{-ps} + a_{-qs}

= |vac><-ps| + |vac><-qs| = |vac>(<-ps| + <-qs|),

assuming single photon modes. Written as a projector, Y^{daggar}Y, it has the form:

(|-ps> + |-pd>)(<-ps| + <-pd|)

Restricted to the subspace spanned by {|-ps>,|-qs>}, it is simply the projector |-ps><-ps|. Likewise, restricted to this same subspace, the measurement corresponding to Alice's detection at image plane point -y is |-qs><-qs|. Clearly |-ps><-ps| + |-qs><-qs| = 1 in this subspace, giving completeness.

Sum|y'>

THIS IS THE KEY POINT RIGHT HERE!

NO! Restricting to horizontal modes, only one possible measurement outcome is: at m (cf. Figure). In this case, Alice's measurement operator is given by the sum of annihilation operators for modes |-ps> and |-qs>, which converge to m. Expressed as projector, this is:

P = (|-ps> + |-qs>)(<-ps| + <-qs|)

Within this subspace, this is the only possible outcome for Alice's focal plane measurement. Noncompleteness is the statement that P not = 1.

OK - THIS IS THE CRUCIAL IDEA TO THINK ABOUT. Of course, doing an actual experiment here would help! :-)

We note that Alice's measurement here is not the incomplete (as against what I have called "noncomplete") measurement

|-ps><-ps| + |-qs><-qs|,

which would have precluded two-photon interference even with CCC.

These sums add up all the actual places Alice's photons land in the statistical sample that act as nonlocal entanglement random noise on what Bob sees LOCALLY when the CCC (Coincidence Counting Circuit) is switched off.

Specifically (2) at the focal plane. Standard theory says (2) is still true only the domain of y' at the focal plane has been squeezed compared to the domain of y at the image plane.

Also at issue here is

<-ps|-pd> = 0 YES? NO? image plane (3)

YES! It is so irrespective of measurement plane. That is, the 4-stream model assumes that |-ps>, |-qs>, |-pd> and |-qd> are orthogonal modes. This assumption is necessary if we assume, quite apart from the signal nonlocality question, that the two-photon correlations are tight (with CCC).

<-ps|-qs> = 0 YES? NO? focal plane (4)

YES! Same arguments apply as with (3).

Note that in the case of Alice's image plane ensemble of y measurements, the nonlocally entangled pair state has already collapsed to

(y,x|A,B) = (y|-ps)(-ps|A)(x|ps)(ps|B) + (y|-pd)(-pd|A)(x|pu)(pu|B)

+ (y|-qs)(-qs|A)(x|qs)(qs|B) + (y|-qd)(-qd|A)(x|qu)(qu|B)

->

(y|-ps)(-ps|A)(x|ps)(ps|B) + (y|-pd)(-pd|A)(x|pu)(pu|B)

from Alice's filters

-> (y|-ps)(-ps|A)(x|ps)(ps|B)

from Bob's filters

This final pair state is completely disentangled with the Bohm random phase factor e^i@ so that we have a random statistical mixture of (x|ps) and (x|qs) on Bob's screen with no local fringes at all!

Indeed.

The issue then is what happens when Alice freely chooses to do a focal plane measurement of the y' variable?

The two-sided Copenhagen "collapse" of the nonlocally entangled photon pair state is now

(y',x|A,B) = (y'|-ps)(-ps|A)(x|ps)(ps|B) + (y'|-pd)(-pd|A)(x|pu)(pu|B)

+ (y'|-qs)(-qs|A)(x|qs)(qs|B) + (y'|-qd)(-qd|A)(x|qu)(qu|B)

->

(y',x|A,B)' = (y'|-ps)(-ps|A)(x|ps)(ps|B) + (y'|-qs)(-qs|A)(x|qs)(qs|B)

After both Alice's and Bob's filters operate - this is the entangled state left over.

The key to what Bob sees locally is then the integral of (-ps|y')(y'|-qs>) over the statistical ensemble of Alice's photons collected in the extended focal region of Alice's lens. If

(-ps|-qs) = 0 ORTHOGONALITY

and if

Sum |y')(y'| = 1 COMPLETENESS

Then Bob still sees NO STAND-ALONE LOCAL FRINGES when the CCC is switched off.

An important point to address. It is avoid this scenario that I introduced the "direction filter". To be precise, noncompleteness exists even otherwise, but may be submerged under the integrated signal coming from fringe patterns corresponding to Alice's different focal plane measurement outcomes.

If you are correct here you will get a Nobel Prize for sure. :-) See Martin Gardner's remark about me in this context in "Magic and Paraphysics" - late 1970's. I need to think more about this before I take a stand, but you are arguing well. :-)

The direction filter ensures that insofar as Alice's focal plane measurement is considered, Bob will observe only the horizontal modes, i.e., those coincident with her detection at m (cf. Figure), which will leave Bob's twin photon in a definite momentum state. This can be construed as a horizontally moving plane wavefront, which, impinging on his double-slit diaphragm, will interfere to produce a *fixed* stand-alone fringe pattern. Other potential wavefronts that could have washed out the fringes in the sense you point out are filtered out by the direction filter because they are not horizontally moving. That is, if Alice detects photons elsewhere than m on the focal plane, Bob registers no corresponding photons.

If you switch on the CCC one will see Bob's fringes emerge after the fact TOO LATE for any retro-causal (BACK FROM THE FUTURE) or faster-than-light SIGNAL NONLOCALITY.

This is what most physicists will say will happen contradicting what Cramer, Woodward, Srikanth think might happen.

Indeed. As far as I understand, the modified Dopfer experiment's success depends precisely on measurement non-completeness at the focal plane being verified to exist.

One might at first suspect that non-completeness violates probability conservation in an undesirable way, but I find that it can be easily interpreted as a modification of the probability to produce entanglement in the nonlinear crystal.

If the mainstream is correct here then the no perfect cloning theorem of orthodox quantum theory is correct and if orthodox quantum theory is complete in Bohr's sense as THE FINAL SOLUTION FOR ALL PHYSICAL REALITY then Lenny Susskind's black hole complementarity is also correct. From that it follows that:

1) Signal nonlocality is impossible.

2) Remote viewing is impossible.

3) We can never directly see beyond the horizons (event or particle) to the parallel universes on the cosmic landscape.

This is a creative tension in Lenny Susskind's theory because as David Gross pointed out in Nature it makes Lenny's theory untestable in Popper's sense. That makes everyone uneasy.

On the other hand from AAAS USD Russell Targ's comments on Ingo Swann in the CIA SRI tests remote viewing is ALLEGEDLY a fact. We also heard from Roger Nelson the Global Consciousness data and from other people. So the debate will be on how good the evidence is?

Is it junk science? Is it pathological science? Or is it good science? No double standards here. The same rules need to be applied not only to Hafnium isomer triggers but also to string theory and to loop quantum gravity theory. No one is above the Rule of Law.

Does the claimed remote viewing data demonstrate superluminality?

No, but it is clearly precognitive. Russell Targ gives an example with a CIA test of Ingo Swann in which Ingo correctly identified a location of a Chinese nuclear test and its failed outcome "uranium burn" fully FOUR DAYS before it happened. This is consistent with other information told to me by CIA Chief of Station Harold Chipman in the mid-1980's.

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