### Quantum Statistical Mechanics forbids any EPR instantaneous information transfer (6/07/04)

#### (Appendix A2 of Proposition d'un test de transmission instantanée d'information par effet EPR )

A2-1 Aims of the present appendix

In the present study (see § 5), our aim is to submit an experimental principle devoted to test the possibility to transfer information with spin measurements thanks to EPR effect. However, such a possibility has been discarded. This appendix reminds the quantum statistical calculations that prove this impossibility so as to better emphasize that this conclusion relies on the assumption of a fundamental indeterminacy of quantum measurements.

First, we will discuss the consequences of the commutability of quantum observables A1 and A2 of EPR correlated parts S1 and S2 of a system S=S1US2. Next, in the framework of quantum statistical mechanics (so as to account for a possible collapse between two measurements) we shall remind the demonstration of the impossibility to modify the probability distribution pk2 of ak2 outcomes of an A2 measurement performed on S2 when performing an A1 measurement on S1 prior to A2 measurement (according to Aristotle absolute chronology). We shall point out that this alleged impossibility relies on the assumption of a fundamental indeterminacy of quantum measurement outcomes.

A2-2 Commutability of quantum operators of EPR correlated parts

A2-2-1 Commutability of quantum operators on disjoint systems

• Let S1 and S2 be two disjoint parts of a system S = S1US2

• Let A1 and A2 be quantum operators respectfully concerning S1 and S2

• Let |jk1> be an Hilbert basis of eigen vectors of quantum operator A1

• Let |yk2> be an Hilbert basis of eigen vectors of quantum operator A2

• Let |j> = S lk1k2 |jk1yk2>            denote the quantum state of S

• Let A1' = A1Ä1 be the extension of operator A1 from S1 to S state space

• and A2' = 1ÄA2 be that of A2. A1' et A2' commute. Indeed,

A1'.A2' |j>  = A1'(1ÄA2) S lk1k2 |jk1 yk2> = (A1Ä1) S lk1k2 |jk1 A2yk2>

= S lk1k2 |A1jk1 A2yk2> = (1ÄA2) S lk1k2 |A1jk1 yk2>

A1'.A2' |j>  = A2'(A1Ä1) S lk1k2 |jk1 yk2> = A2'.A1' |j>

A2-2-2 Spectral Properties of commuting Hermitian operators

Let A and B be two commuting Hermitian operators and let |jk> be an Hilbert basis of eigen vectors of B with eigen values bk

B |jk> = bk |jk>   Þ   AB |jk> = bk A |jk> so that BA |jk> = bk A |jk>

Consequently, A |j k> is also an eigen vector of B with eigen value bk, e.g. A action let |jk> in the eigen subspace Ek of eigen value bk of observable B. Now, let us consider a spectral projector Pl of A (associated to an eigen value al of A) so that Pl |jk> models the collapse of state |jk> subsequent to a measurement outcome al of observable A. It is possible to state that Pl commutes also with B, so that Pl |jk> is also an eigen vector of B with eigen value bk, e.g. the projection Pl |jk> stemming from a measurement A let |jk> in eigen space Ek of bk. Hence, a measurement B of Pl |jk>, subsequent to the collapse of |jk> caused by measurement of observable A, provides the same outcome bk as if the measurement of observable A had not been achieved prior to the measurement of observable B.

A2-2-3 Physical consequences

So, if state |j> of S = S1US2 is in an eigenstate of observable A2 of S2, a measurement A1 of S1 prior to an observation A2 of S2 provides the same outcome as if A2 had been performed first. Hence, an observer of S2 which would know the quantum state of system S=S1US2 (prior to A1 and A2 observations) is deprived of any information about the possible occurrence of a measurement A1 achieved by an observer of S1 prior to a measurement of observable A2 of S2. Now, what if |j> is not in an eigen state of A2 ? Arguing that A1 measurement doesn't influence on A2 outcomes probability distribution because A2A1|j>=A1A2|j> implicitly amounts to assume that A2 acts on the coherent superposition A1|j>, e.g. that |j> hasn't still collapsed subsequently to A1 measurement before A2 measurement has occurred. To overcome this objection, let us perform a quantum statistical calculation to check if a possible collapse achieving A1 measurement before A2 measurement takes place preserves the probability distribution of A2 outcomes whatever the initial state |j> of S=S1US2.

A2-3 Probability distribution of quantum measurement outcomes

If system S = S1US2 is in a known state |j>, the probability distribution pk2 of outcomes ak2 of a measurement A2 of S2 is known. It amounts to pk2 = (Pk2 |jj>)2 = <j|Pk2|j> according to the Born rule, where Pk2 denotes the spectral projector on eigen space of eigen value ak2 of quantum operator A2. Now, let us state the probability distribution pk2' after an ensemble of observations A1 of an ensemble of systems S1 has taken place.

A2-3-1 Effect of measurement A1 on density operator r of S=S1US2

The statistical quantum physics model (resting on density operator) of the decoherence process stemming from measurements A1 of systems S1 confirms the impossibility (for observer of S2) to know if this decoherence has happened whatever the measurements A2 of systems S2 subsequent to measurements A1. Indeed,

• let |j k1> be a basis of eigen vectors of observable A1,
• let |y k2> be a basis of eigen vectors of observable A2,

Let us consider S to be in the coherent state

|y>   =       S        lk1k2  |jk1 yk2>
k1k2

As far as no measurement has been performed, system S stays in a shear state |y> and operator r writes 

r  =     |y><y|      =       S      lk1k2 ll1l2* | jk1 yk2><jl1 yl2 |
k1 l1 k2 l2

Let us perform measurement A1. Interference effects coming from S1 quantum state components superposition disappear and r transforms to

r'  =    S   |jk'1><jk'1| Ä <jk'1| r |jk'1>
k'1

Now <jk'1|jk1> = dk'1k1     and <jl1|jk'1> = dl1k'1      so that

r'  =   S  |jk'1 >< jk'1Ä   S lk'1k2 lk'1l2* < jk'1| jk'1 yk2>< jk'1 yl2 | jk'1>
k'1                             k2 l2

e.g.  r'    =    S      lk1k2 lk1l2* |jk1 yk2> < jk1 yl2|
k1 k2 l2

So, any measurement A1 on S1 causes to disappear interference terms of the type lk1... ll1...* | jk1 y... >< jl1 y...| coming from the coherent superposition of S1 quantum states components. However, the observer standing on side 2 fails to know if system S is in quantum state r or in quantum state r'. Indeed, we will see shortly that the change from coherent state r of S to partially incoherent state r' doesn't change anything to the probability distribution pk2 governing the statistics of ak2 measurement outcomes of observable A2 of S2.

A2-3-2 Probabilities pk2 of ak2 outcomes when A1 has been measured

Let us calculate probabilities pk2 of outcomes ak2, when S is in the mixed state r'.
To perform this calculation, the partial trace of S2 amounts to 

r'(2) =  S   <ji1| r'| ji1>   =     S     lk1k2 lk1l2* < ji1| jk1 yk2> < jk1 yl2| ji1>    e.g.
i1                             i1 k1 k2 l2

r'(2) =    S   lk1k2lk1l2* |yk2 > < yl2|                      (because <ji1|jk1> = di1k1)
k1 k2 l2

Probabilities pk2 to get outcomes ak2 on S2 writes

pk2 = tr (r'(2) |yk2> <yk2|) =  S  lk1k2 lk1k2*   =    S  | lk1k2 |2
k1                                 k1

If interference terms lk1k2 ll1k2* satisfying k1 ¹ l1 would pop up in this sum pk2 would be the square of a sum, e.g.

pk2   =    S  lk1k2 ll1k2   =    |S  lk1k2|2
k1 l1                              k1

However, we will state that the absence of any measurement A1 before measurement A2 doesn't preserve interference effects (that would enable observer 2 to know that no prior measurement A1 has been performed by observer 1)

A2-3-3 Probabilities pk2 if A1 measurement has not been performed

If A1 has not been performed, the partial trace r(2) of r on S2 writes

r(2) =   S   < ji1| r | ji1>    =    S      lk1k2 ll1l2* < ji1| jk1yk2> < jl1 yl2 | ji1>,  e.g.
i1                            i1 k1 l1 k2 l2

r(2) =   S    lk1k2 lk1l2* | yk2> < yl2| = r'(2)    (same operator as in A2-3-2)
k1 k2 l2

Interference cross terms between jk1 and jl1 components of S1 quantum state have disappeared. Only terms such that k1 = l1 don't vanish.

pk2 =  tr ( r(2) |yk2> <yk2| )

pk2 =   S  lk1k2 lk1k2*  =   S   | lk1k2|2
k1                                 k1

So, according to quantum statistics formalism, statistically, an ensemble of observations A1 of systems S1 prior to an ensemble of observations A2 of systems S2 isn't up to disturb the statistical distribution of probabilities pk2 of outcomes ak2 of observable A2 of systems S2.

However, in paragraph 3-1, we recall that if an individual system S undergoes an individual measurement A1 of its part S1, its state |y> transforms to |y1> and the probabilities of future measurements A2 on S2 are not the same as if A1 had not been performed.

The above stated impossibility for observer 1 to send information to observer 2 thanks to quantum state changes caused by measurements A1 on identical systems S=S1US2 in identical quantum states (where S1 ans S2 are EPR correlated) relies on an assumed fundamental quantum indeterminacy. Indeed, this interpretation implies that observer 1 isn't up, for instance, to increase systems S1 probability to reach a given state |y1> subsequently to measurements A on systems S1. As far as the probability to get any quantum state |y1>, subsequently to measurements A1 on S1, satisfies the statistics dictated by the Born rule, A2 measurements on systems S2 can't inform observer 2 of observer 1 action on systems S1 prior to A2 measurements.