# Comments on: ‘Relative facts do not exist. Relational quantum mechanics is
incompatible with quantum mechanics’ by Jay Lawrence, Marcin Markiewicz and
Marek Źukowski.

Aurélien Drezet  Institut Néel, UPR 2940, CNRS-Université Joseph Fourier, 25,
rue des Martyrs, 38000 Grenoble, France

###### Abstract

The Relational Quantum Mechanics (RQM) is an alternative interpretation of
quantum mechanics that was proposed originally by C. Rovelli [11, 12, 8, 3].
RQM can be seen as a logical completion and generalization of the Copenhagen
(orthodox) interpretation but where the arbitrariness of Heisenberg’s quantum
‘shifty-split’ or ‘cut’, which is separating observed and observing
subsystems, is taken more seriously. Unlike, the Copenhagen interpretation the
cut is not confined to the macroscopic domain and the roles of observed and
observing systems are relative and can be inverted. RQM is therefore a more
symmetric and general approach.
Moreover, recently RQM has been criticized and assessed by various authors.
The aim of the present comment (see also [2]) is to give a short reply to the
recent J. Lawrence et al. article [9] that concerns RQM and the role of
quantum contextuality (for previous claims see [1, 10] and see the replies by
Di Biagio and Rovelli [4] and Drezet [5, 6]).

##  1 Introduction

Following the recent analysis [5, 6], I remind that in RQM the main issue
concerns the interpretation of the full wavefunction $|\Psi_{SO}\rangle$
involving observer (O) and observed system (S). In RQM the fundamental object
relatively to (O) is not $|\Psi_{SO}\rangle$ but the reduced density matrix

$\displaystyle\hat{\rho}^{(red.)}_{S|O}=\textrm{Tr}_{O}[\hat{\rho}_{SO}]=\textrm{Tr}_{O}[|\Psi_{SO}\rangle\langle\Psi_{SO}|].$
(1)

As it is well known $\hat{\rho}^{(red.)}_{S|O}$ is independent of the basis
chosen to represent the degrees of freedom for (O). In [5] I showed that it
solves the dilemma discussed in [1, 10] concerning the ‘preferred basis
problem’. Here I show that the same features debunk the claims of [9]
concerning non-contextuality.
[9] starts with a GHZ state [7] for a system S of 3 spins $m=1,2,3$:

$\displaystyle|GHZ\rangle_{S}=\frac{1}{\sqrt{2}}[|+1^{(1)},+1^{(1)},+1^{(1)}\rangle_{S_{1},S_{2},S_{3}}$
$\displaystyle+|-1^{(1)},-1^{(1)},-1^{(1)}\rangle_{S_{1},S_{2},S_{3}}]$ (2)

where $|p^{(1)}\rangle_{S_{m}}\equiv|\textrm{sign}(p)z\rangle_{S_{m}}$ (with
$p=\pm 1$) are spin eigenstates along the $z$ direction. We also have in
different spin bases:

$\displaystyle|GHZ\rangle_{S}=\frac{1}{2}[|+1^{(2)},+1^{(3)},-1^{(3)}\rangle_{S_{1},S_{2},S_{3}}$
$\displaystyle+|+1^{(2)},-1^{(3)},+1^{(3)}\rangle_{S_{1},S_{2},S_{3}}$
$\displaystyle+|-1^{(2)},+1^{(3)},+1^{(3)}\rangle_{S_{1},S_{2},S_{3}}$
$\displaystyle+|-1^{(2)},-1^{(3)},-1^{(3)}\rangle_{S_{1},S_{2},S_{3}}]$ (3)

where we used
$|p^{(2)}\rangle_{S_{m}}=\frac{1}{\sqrt{2}}[|+1^{(1)}\rangle_{S_{m}}\pm|-1^{(3)}\rangle_{S_{m}}]\equiv|\textrm{sign}(p)x\rangle_{S_{m}}$,
and $|p^{(3)}\rangle_{S_{m}}=\frac{1}{\sqrt{2}}[|+1^{(1)}\rangle_{S_{m}}\pm
i|-1^{(3)}\rangle_{S_{m}}]\equiv|\textrm{sign}(p)y\rangle_{S_{m}}$. This
implies

$\displaystyle{\sigma_{x}}_{S_{1}}{\sigma_{y}}_{S_{2}}{\sigma_{y}}_{S_{3}}|GHZ\rangle_{S}=-|GHZ\rangle_{S}$
(4)

and

$\displaystyle p^{(2)}_{S_{1}}q^{(3)}_{S_{2}}r^{(3)}_{S_{3}}=-1.$ (5)

Similar expressions are obtained by circular permutations:

$\displaystyle p^{(3)}_{S_{1}}q^{(2)}_{S_{2}}r^{(3)}_{S_{3}}=-1,$ (6)

and

$\displaystyle p^{(3)}_{S_{1}}q^{(3)}_{S_{2}}r^{(2)}_{S_{3}}=-1.$ (7)

We can also write:

$\displaystyle|GHZ\rangle_{S}=\frac{1}{2}[|+1^{(2)},+1^{(2)},+1^{(2)}\rangle_{S_{1},S_{2},S_{3}}$
$\displaystyle+|+1^{(2)},-1^{(2)},-1^{(2)}\rangle_{S_{1},S_{2},S_{3}}$
$\displaystyle+|-1^{(2)},+1^{(2)},-1^{(2)}\rangle_{S_{1},S_{2},S_{3}}$
$\displaystyle+|-1^{(2)},-1^{(2)},+1^{(2)}\rangle_{S_{1},S_{2},S_{3}}]$ (8)

implying

$\displaystyle{\sigma_{x}}_{S_{1}}{\sigma_{x}}_{S_{2}}{\sigma_{x}}_{S_{3}}|GHZ\rangle_{S}=+|GHZ\rangle_{S}$
(9)

and thus

$\displaystyle p^{(2)}_{S_{1}}q^{(2)}_{S_{2}}r^{(2)}_{S_{3}}=+1.$ (10)

It is well known that quantum mechanics is highly contextual. If the results
of spin measurements were non-contextual Eqs. 5,6,7 and 10 could be true
together and this, clearly, is not possible: multiplying Eqs. 5,6,7
contradicts Eq. 10. This incompatibility also leads to a well known proof of
quantum nonlocality without inequality [7].

##  2 Debunking a paradox

In [9] the authors use the previous results. First, they consider a single
observer Alice (A) (composed of 3 qubits $A_{1},A_{2},A_{3}$) who measures the
spins of the GHZ system S in the y bases. The GHZ states of Eq. 2 reads in the
y bases:

$\displaystyle|GHZ\rangle_{S}=\sum_{p,q,r}C^{333}_{pqr}|p^{(3)},q^{(3)},r^{(3)}\rangle_{S_{1},S_{2},S_{3}}$
(11)

where the amplitudes $C^{333}_{pqr}$ are non vanishing for each of the 8
combinations $p,q,r$ and where $|C^{333}_{pqr}|^{2}=\frac{1}{8}$. After
entanglement with Alice’s qubits we have

$\displaystyle|GHZ\rangle_{SA}=\sum_{p,q,r}C^{333}_{pqr}|p^{(3)}\rangle_{SA_{1}}|q^{(3)}\rangle_{SA_{2}}|r^{(3)}\rangle_{SA_{3}}$
(12)

with
$|k^{(3)}\rangle_{SA_{m}}:=|k^{(3)}\rangle_{S_{m}}|k^{(3)}\rangle_{A_{m}}$ and
$|k^{(3)}\rangle_{A_{m}}$ is the state of the qubit $A_{m}$, $m=1,2,3$ (for
$k=\pm 1$).
Importantly, for Alice this state shows no correlation. This is clear in RQM
where the reduced density matrix reads
$\hat{\rho}^{(red.)}_{S|A}=\textrm{Tr}_{A}[\hat{\rho}_{SA}]$:

$\displaystyle\hat{\rho}^{(red.)}_{S|A}=\frac{1}{8}\sum_{p,q,r}|p^{(3)}\rangle_{S_{1}}{}_{S_{1}}\langle
p^{(3)}|\otimes|q^{(3)}\rangle_{S_{2}}{}_{S_{2}}\langle q^{(3)}|$
$\displaystyle\otimes|r^{(3)}\rangle_{S_{3}}{}_{S_{3}}\langle r^{(3)}|$ (13)

Due to decoherence i.e., entanglement with the environement (Alice) we have
lost coherence and correlations between spins. In particular we have

$\displaystyle\textrm{Tr}_{S}[{\sigma_{x}}_{S_{1}}{\sigma_{x}}_{S_{2}}{\sigma_{x}}_{S_{3}}\hat{\rho}^{(red.)}_{S|A}]=0,$
$\displaystyle\textrm{Tr}_{S}[{\sigma_{x}}_{S_{1}}{\sigma_{y}}_{S_{2}}{\sigma_{y}}_{S_{3}}\hat{\rho}^{(red.)}_{S|A}]=0,$
$\displaystyle\textrm{Tr}_{S}[{\sigma_{y}}_{S_{1}}{\sigma_{x}}_{S_{2}}{\sigma_{y}}_{S_{3}}\hat{\rho}^{(red.)}_{S|A}]=0,$
$\displaystyle\textrm{Tr}_{S}[{\sigma_{y}}_{S_{1}}{\sigma_{y}}_{S_{2}}{\sigma_{x}}_{S_{3}}\hat{\rho}^{(red.)}_{S|A}]=0,$

which contrast with

$\displaystyle\textrm{Tr}_{S}[{\sigma_{x}}_{S_{1}}{\sigma_{x}}_{S_{2}}{\sigma_{x}}_{S_{3}}\hat{\rho}^{(red.)}_{S}]=+1,$
$\displaystyle\textrm{Tr}_{S}[{\sigma_{x}}_{S_{1}}{\sigma_{y}}_{S_{2}}{\sigma_{y}}_{S_{3}}\hat{\rho}^{(red.)}_{S}]=-1,$
$\displaystyle\textrm{Tr}_{S}[{\sigma_{y}}_{S_{1}}{\sigma_{x}}_{S_{2}}{\sigma_{y}}_{S_{3}}\hat{\rho}^{(red.)}_{S}]=-1,$
$\displaystyle\textrm{Tr}_{S}[{\sigma_{y}}_{S_{1}}{\sigma_{y}}_{S_{2}}{\sigma_{x}}_{S_{3}}\hat{\rho}^{(red.)}_{S}]=-1,$

In RQM Eq. 15 actually describes the correlations available to A before the
interaction occurred, i.e., when the full state of SA is still factorized. It
represents a catalog of knowledge or potentiality in the sense of Heisenberg.
The actualization of measurements in RQM is a debatable issue and we will not
consider this problem here (see e.g.[5]).
In the next step the authors of [9] consider a second observer Bob (B) (also
composed of 3 qubits $B_{1},B_{2},B_{3}$) who measures the spins of the
entangled GHZ system SA in the x bases of the joint system. In analogy with
Eq. 8 we write after entanglement with Bob qubits:

$\displaystyle|GHZ\rangle_{SAB}=\frac{1}{2}\sum_{p,q,r}|p^{(2)}\rangle_{SAB_{1}}|q^{(2)}\rangle_{SAB_{2}}|r^{(2)}\rangle_{SAB_{3}}$
(16)

with
$|k^{(2)}\rangle_{SAB_{m}}:=|k^{(2)}\rangle_{SA_{m}}|k^{(2)}\rangle_{B_{m}}$
and $|k^{(2)}\rangle_{B_{m}}$ is the state of the qubit $B_{m}$, $m=1,2,3$
(for $k=\pm 1$) and where we introduced the entangled ‘x’ states for the SA
system:
$|k^{(2)}\rangle_{SA_{m}}=\frac{1}{\sqrt{2}}[|+1^{(3)}\rangle_{SA_{m}}\pm
i|-1^{(3)}\rangle_{SA_{m}}]\equiv|\textrm{sign}(p)x\rangle_{SA_{m}}$.
Crucially, the numbers $p,q,r=\pm 1$ in Eq. 16 must obey the GHZ-constraint:

$\displaystyle p^{(2)}_{SAB_{1}}q^{(2)}_{SAB_{2}}r^{(2)}_{SAB_{3}}=+1.$ (17)

Once more, in RQM we need to consider the reduced density matrix
$\hat{\rho}^{(red.)}_{SA|B}=\textrm{Tr}_{B}[\hat{\rho}_{SAB}]$:

$\displaystyle\hat{\rho}^{(red.)}_{SA|B}=\frac{1}{4}\sum_{p,q,r}|p^{(2)}\rangle_{SA_{1}}{}_{SA_{1}}\langle
p^{(2)}|\otimes|q^{(2)}\rangle_{SA_{2}}{}_{SA_{2}}\langle q^{(2)}|$
$\displaystyle\otimes|r^{(2)}\rangle_{SA_{3}}{}_{SA_{3}}\langle r^{(2)}|$

with again $pqr=+1$. This defines the information available to B in RQM and
this density matrix shows partial coherence since we have

$\displaystyle\textrm{Tr}_{SA}[{\sigma_{x}}_{SA_{1}}{\sigma_{x}}_{SA_{2}}{\sigma_{x}}_{SA_{3}}\hat{\rho}^{(red.)}_{SA|B}]=+1,$
$\displaystyle\textrm{Tr}_{SA}[{\sigma_{x}}_{SA_{1}}{\sigma_{y}}_{SA_{2}}{\sigma_{y}}_{SA_{3}}\hat{\rho}^{(red.)}_{SA|B}]=0,$
$\displaystyle\textrm{Tr}_{SA}[{\sigma_{y}}_{SA_{1}}{\sigma_{x}}_{SA_{2}}{\sigma_{y}}_{SA_{3}}\hat{\rho}^{(red.)}_{SA|B}]=0,$
$\displaystyle\textrm{Tr}_{SA}[{\sigma_{y}}_{SA_{1}}{\sigma_{y}}_{SA_{2}}{\sigma_{x}}_{SA_{3}}\hat{\rho}^{(red.)}_{SA|B}]=0.$

The first line of Eq. 19 is of course reminiscent of Eq. 17 and shows that
there is a preferred pointer basis defined by the specific measurement
protocol. This is associated with a specific interaction Hamiltonian
$H_{SA,B}$ leading to the state given by Eq. 16.
However, the authors of [9] didn’t correctly analyze the structure of RQM and
the meaning of relative facts. They claim that we can find relations between
facts or information available to Bob and facts or information available to
Alice. This we show below is actually a misunderstanding. More precisely, they
consider that Bob only measures one of the 3 qubits belonging to SA. In the
following we consider the particular case $m=1$ and therefore only the system
$SA_{1}$ will interact with $B_{1}$. This requires to let the two other qubits
of Bob $B_{2}$ and $B_{3}$ in their respective ground states. As the authors
show we get the new state

$\displaystyle|GHZ^{\prime}\rangle_{SAB}=\frac{1}{2}\sum_{p,q,r}|p^{(2)}\rangle_{SAB_{1}}|q^{(3)}\rangle_{SA_{2}}$
$\displaystyle\times|in\rangle_{B_{2}}|r^{(3)}\rangle_{SA_{3}}|in\rangle_{B_{2}}$
(20)

with now the constraint:

$\displaystyle p^{(2)}_{SAB_{1}}q^{(3)}_{SA_{2}}r^{(3)}_{SA_{3}}=-1.$ (21)

Of course, we could develop two similar procedures acting only on the qubit
$B_{2}$ or alternatively the qubit $B_{3}$. We will obtain two different
states $|GHZ^{\prime\prime}\rangle_{SAB}$ and
$|GHZ^{\prime\prime\prime}\rangle_{SAB}$ leading to the relations

$\displaystyle p^{(3)}_{SA_{1}}q^{(2)}_{SAB_{2}}r^{(3)}_{SA_{3}}=-1,$ (22)

and

$\displaystyle p^{(3)}_{SA_{1}}q^{(3)}_{SA_{2}}r^{(2)}_{SAB_{3}}=-1.$ (23)

With these mathematical properties the deduction of [9] goes as follows:
i) It is visible from Eq. 21 (and similarly for Eqs. 22,23 by cyclic
permutation) that the number $p^{(2)}_{SAB_{1}}$ characterizes the entangled
system $SAB_{1}$ whereas $q^{(3)}_{SA_{2}}$ and $r^{(3)}_{SA_{3}}$
characterize $SA_{2}$ and $SA_{3}$. Therefore it is tempting to call
$p^{(2)}_{SAB_{1}}$ a relative fact for Bob and $q^{(3)}_{SA_{2}}$ and
$r^{(3)}_{SA_{3}}$ relative facts for Alice. This idea, which is central for
their paper, is clearly summarized by the analysis surrounding their equation
17 in [9]. They call $\mathcal{B}_{m}$ the number $k^{(2)}_{SAB_{m}}$ defined
in our Eqs. 17, 21-23 and similarly they call $\mathcal{A}_{m}$ the number
$k^{(3)}_{SA_{m}}$. Assuming this we go to the next step of their ‘no-go’
deduction.
ii) The 4 relations Eq. 17, 21-23. are clearly incompatible. If we multiply
Eq. 21 by Eqs. 22 and 23 we obtain

$\displaystyle
p^{(2)}_{SAB_{1}}q^{(2)}_{SAB_{2}}r^{(2)}_{SAB_{3}}(p^{(3)}_{SA_{1}}q^{(3)}_{SA_{2}}r^{(3)}_{SA_{3}})^{2}$
$\displaystyle=p^{(2)}_{SAB_{1}}q^{(2)}_{SAB_{2}}r^{(2)}_{SAB_{3}}=-1,$ (24)

which clearly contradicts Eq. 17. Now as they emphasize it clearly: ‘Note
that, most importantly for the sequel, the three [unitary] transformations
[acting on $m=1,2,3$] mutually commute, and thus their order of application is
immaterial.’. In other words: Since the 3 operations leading to
$|GHZ^{\prime}\rangle_{SAB}$,
$|GHZ^{\prime\prime}\rangle_{SAB}$,$|GHZ^{\prime\prime}\rangle_{SAB}$ and thus
Eqs. 21-23 are acting ‘locally’ only on one of the sub systems $SA_{m}$ their
meaning should be non contextual and absolute. This following [9] justifies
why we should apriori compare these states with $|GHZ\rangle_{SAB}$ and Eq.
17. This noncontextual reading is what they believe is contained in RQM.
If we accept this reasoning then RQM contradicts quantum mechanics. Relative
facts for Alice and Bob are contradictory.
However, this deduction is false and we now debunk the contradiction. First,
consider point i): It is clear that $p^{(2)}_{SAB_{1}}$ must be a relative
fact for Bob. However, there is no reason to consider that $q^{(3)}_{SA_{2}}$
and $r^{(3)}_{SA_{3}}$ should be relative facts for Alice. Actually, the
quantum state Eq. 20 contains quantum numbers $p,q,r$ but in RQM the
fundamental description for Bob is the reduced density matrix
$\hat{\rho^{\prime}}^{(red.)}_{SA|B}=\textrm{Tr}_{B}[\hat{\rho^{\prime}}_{SAB}]$
obtained by using
$\hat{\rho^{\prime}}_{SAB}=|GHZ^{\prime}\rangle_{SAB}{}_{SAB}\langle|GHZ^{\prime}|$.
We have:

$\displaystyle\hat{\rho^{\prime}}^{(red.)}_{SA|B}=\frac{1}{2}|+1^{(2)}\rangle_{SA_{1}}{}_{SA_{1}}\langle+1^{(2)}|$
$\displaystyle\otimes|\Phi\rangle_{SA_{2},SA_{3}}{}_{SA_{2},SA_{3}}\langle\Phi|$
$\displaystyle+\frac{1}{2}|-1^{(2)}\rangle_{SA_{1}}{}_{SA_{1}}\langle-1^{(2)}|$
$\displaystyle\otimes|\Psi\rangle_{SA_{2},SA_{3}}{}_{SA_{2},SA_{3}}\langle\Psi|$
(25)

where

$\displaystyle|\Phi\rangle_{SA_{2},SA_{3}}=\frac{1}{\sqrt{2}}[|+1^{(3)}\rangle_{SA_{2}}|-1^{(2)}\rangle_{SA_{3}}$
$\displaystyle+|-1^{(3)}\rangle_{SA_{2}}|+1^{(2)}\rangle_{SA_{3}}],$
$\displaystyle|\Psi\rangle_{SA_{2},SA_{3}}=\frac{1}{\sqrt{2}}[|+1^{(3)}\rangle_{SA_{2}}|+1^{(2)}\rangle_{SA_{3}}$
$\displaystyle+|-1^{(3)}\rangle_{SA_{2}}|-1^{(2)}\rangle_{SA_{3}}].$ (26)

are two Bell states. The fact that EPR-like Bell states are present show that
there is a coherence preserved in the description of the system SA by Bob. In
particular we deduce

$\displaystyle\textrm{Tr}_{SA}[{\sigma_{x}}_{SA_{1}}{\sigma_{x}}_{SA_{2}}{\sigma_{x}}_{SA_{3}}\hat{\rho^{\prime}}^{(red.)}_{SA|B}]=+1,$
$\displaystyle\textrm{Tr}_{SA}[{\sigma_{x}}_{SA_{1}}{\sigma_{y}}_{SA_{2}}{\sigma_{y}}_{SA_{3}}\hat{\rho^{\prime}}^{(red.)}_{SA|B}]=-1,$
$\displaystyle\textrm{Tr}_{SA}[{\sigma_{y}}_{SA_{1}}{\sigma_{x}}_{SA_{2}}{\sigma_{y}}_{SA_{3}}\hat{\rho^{\prime}}^{(red.)}_{SA|B}]=0,$
$\displaystyle\textrm{Tr}_{SA}[{\sigma_{y}}_{SA_{1}}{\sigma_{y}}_{SA_{2}}{\sigma_{x}}_{SA_{3}}\hat{\rho^{\prime}}^{(red.)}_{SA|B}]=0.$

The second line is of course reminiscent of Eq. 21 but the meaning is here
different from what is claimed in [9]. Indeed, this a relation for Bob
measurements not for Alice. The idea to have a mixture of information for Bob
and Alice in the same Eq. 21 is thus unjustified and is not a part of RQM but
instead of the reading of RQM made by the authors of [9]. The first line is
also very interesting: It shows coherence associated with the
$|\Phi\rangle_{SA_{2},SA_{3}}$ and $|\Psi\rangle_{SA_{2},SA_{3}}$ Bell states.
Moreover this result can be compared with Eq. 19 for
$\hat{\rho}^{(red.)}_{SA|B}$. The difference clearly stresses that the
measurement procedures are more invasive that claimed in [9]. In RQM like in
the Copenhagen interpretation the experimental context and the choice of the
interaction Hamiltonian is key. This allows us us to answer to point ii)
concerning non-contextuality. Indeed, the non contextuality supposed by the
authors of [9] is not a part of the RQM. For RQM (like in the orthodox
interpretation) the different contexts are not compatible and we have no right
to compare Eqs. 17 and Eqs. 21-23 as claimed. This would otherwise contradict
the central axioms of RQM and therefore the demonstration discussed in [9] is
a non sequitur fallacious reasoning.
I conclude by adding that the RQM is an interesting interpretation. The
formalism is perfectly agreeing with standard quantum mechanics and recovers
the orthodox interpretation at the limit were observers are essentially
macroscopic but without the problems concerning the definition of agents in
Qbism. Several issues concerning interactions can still be debated but we
should not focus on wrong problems. Often objections are done without
carefully considering the philosophy of RQM but by adding some prejudices or
preconceptions concerning the interpretation (i.e., [9, 1, 10]). I hope this
note will contribute to clarify a bit this issue.

## References

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