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|yy| = 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'y'| = 1 focal plane (MOMENTUM MEASUREMENT) YES? NO? (2)

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.
A final remark: The UNITARITY LOOPHOLE is not yet plugged.
The standard argument against signal nonlocality in orthodox
micro-quantum theory is that if
Alice(0)|Alice'(0) = 0 at time t = 0
Then
Alice(t)|Alice'(t) = 0 at time t
Because
|Alice(t) = U(t)|Alice(0)
and U(t) is a UNITARY OPERATOR in qubit Hilbert space
Therefore
U*(t)U(t) = 1
Therefore, sums over y' of
Alice(t)|y'y'|Alice(t)' = 0
Note however that
U*(t)U(t') =/= 1 when t =/= t'
This may be a clue for a loophole to let signal nonlocality creep back
into orthodox micro-quantum theory.
That is, if the focal plane momentum measurement statistical sample
Alice makes correspond to different arrival times t =/= t' then perhaps
local fringes will be seen by Bob with the CCC switched off?
There is the idea that quantum mechanics is an "island in theory space",
due originally to Steven Weinberg. If Schrodinger evolution were not
unitary or linear, one can indeed exploit entanglement toward nonlocal
signaling. However, of course, this goes beyond standard quantum mechanics.

Yes, of course. Henry Stapp's theory of Helmut Schmidt's retro-PK was a
correct example of that in Phys. Rev. letters - exactly.
I have understood the origin of nonlocal signaling in the case of my
thought experiment in terms of the measurement process (von Neumann's
process 1) rather than Schrodinger evolution (von Neumann process 2). I
guess one might be able re-cast measurement noncompleteness as a
non-unitary evolution plus conventional measurement.
Note my Super Cosmos book does NOT ASSUME signal nonlocality in orthodox
micro-quantum theory. I replace orthodox micro-quantum theory with a
post-quantum macro ODLRO theory that explains the "classical limit",
i.e. WHY space-time physics is local physics at the light signal level
and also explains why the early universe has low entropy from
retro-causal signal nonlocality at the cosmic scale in accord with the
empirical Second Law of Thermodynamics. The post-quantum theory reduces
to orthodox micro-quantum theory when non-equilibrium open system
complexity vanishes in the same way that 1915 General Relativity reduces
to 1905 Special Relativity when the curvature vanishes.
On Jul 7, 2006, at 9:40 PM, Jack Sarfatti wrote:
This is a tighter version so you can tell who is saying what for the record.
With best regards,
Srik.