No decoherence by entanglement
There are misconceptions that entanglement (e.g. with environment) causes
decoherence, and that decoherence causes classicality. Yet in an entanglement,
barring classical communication, no action taken by one party has any effect on
another party, a fact known as no-signaling theorem. The presented analysis
reveals, it is the measurement, not entanglement, which turns quantum state into
classical event sample, resulting in a loss of correlation terms of density matrix
One can read phrases like: It is now widely accepted that environmental entanglement and the
resulting decoherence processes play a crucial role in the quantum-to-classical transition and the
emergence of “classicality” from quantum mechanics ; … the correlations of a quantum system
with other quantum systems may cause one of its observables to behave in a classical manner .
If one can cause an observable of a remote system “to behave in a classical manner” through
entanglement, it would imply a spooky action at a distance [3, 4, 5], a long-discredited idea. Some
of such misconceptions have been refuted [6, 7], some still persist.
Here I shall pinpoint the source of confusion about effects of entanglement. Consider a qubit
entanglement with an ancilla system (e.g. environment), wherein qubit’s eigenstates ,
are correlated with ancilla’s states ,:
The ancilla states , are normalized but not necessarily orthogonal. The combined state (1) is
pure, and its density matrix features interference terms, highlighted in purple:
The standard approach [2, 8] to demonstrate how entanglement “causes decoherence” is to trace
out ancilla from pure state density matrix (2) [9, 8]:
There are still interference terms, proportional to scalar product , in reduced density matrix
(3). In a limit case, when , are the same, i.e. when , the reduced density matrix (3)
is that of a standalone pure qubit state:
In another limit, when , are orthogonal, , the interference terms in reduced density
matrix (3) disappear, signifying full decoherence.
The expression for reduced density matrix (3) can be derived from measurement performed on
state (1) with some arbitrary operator , acting on qubit only. The expectation value of the
measurement is given by Born rule:
To some authors, the result (5) seems as a proof of decoherence by entanglement, because, in their
view, the application of Born rule in (5) means the measurement performed on qubit only, with
the result showing dependence on ancilla in the form of scalar product . Strangely though
that a question is not asked, if such interpretation complies with no-signaling theorem . Indeed,
if the measurement only involves local qubit, the ancilla system in entangled state (1) can be
chosen arbitrarily. It might not even physically exist, being just a gedanken. The expectation value
of local measurement should not depend on gedanken, or remote system. Such dependence would
mean the remote ancilla can somehow affect the measurement on local qubit. That runs contrary
not just no-signaling theorem, but special relativity as well. Yet the result (5) depends on ancilla.
So, what is the reason for paradox?
The culprit is hiding in plain view. Even though (5) is meant to be the measurement done with
operator on local qubit only, in fact, (5) contains two measurements, with two devices. One
measurement is done with operator , and another measurement is done with identity operator :
The identity operator is POVM operator in its own right. Born rule (6) for entangled state (1)
implicitly includes measurement by identity operator (device ). Generally, any expression in a
form implies measurement, because its output is classical information. The application of
Born rule (6) to (1) means the ancilla is part of a real physical system being measured, not a
gedanken. It is the measurement by device which turns pure state (2) into generally mixed state
(3), not entanglement per se. As is evident from right side of (6), the subsequent measurement by
device is done not on pure state (2), but on output from the first measurement, the mixed state
(3). In the limit case of , the measurement by device completely turns pure state (2)
into classical event sample. It eliminates any uncertainty about subsequent measurement by device
, because orthogonal states , each point  to the respective qubit eigenstate ,.
The event sample from device has to be available to device , i.e. classical information has
to be shared between two devices. It explains why spooky action at a distance, or other claims of
non-locality of quantum theory, is a fantasy, albeit still popular with some authors
[11, 7, 6]. The
Born rule (6) imposes speed limit on two measurements in a form of requirement that the interval
between them is timelike. If interval is spacelike, then classical information between two devices
cannot be shared, and the measurement by device has to be considered as single-device
measurement on standalone qubit state (4), not two-device measurement on entangled state (2).
This condition is equivalent to in (5,6), i.e. to inability of device to distinguish between
different ancilla states, or inability of device to share classical information with device .
Consider a double-slit experiment with a beam of electrons. A charged particle, passing
through the slit, would surely be detectable by sufficiently sensitive detector (device ) placed next
to the slit. The slit itself can act as a detector, if wired appropriately. In the absence of measurement
at the slits one would see interference pattern at the screen (device ) behind the slits. The
interference signifies inability of device to tell which slit the electron passed through. The
interference is preserved even if the slits are wired to detect electrons, but the detected information
The continuing controversy about non-locality of quantum theory has a lot to do with J.S. Bell making confusing
statements on the subject, and an unfortunate EPR article . The following catchy phrase  is definitely subject to
misinterpretation: grossly non-local structure... is characteristic… of any theory which reproduces exactly quantum
is not registered anywhere in classical form
. As soon as device is turned on to register electrons
passing through the slits, the interference pattern at device will degrade, in accordance with
reduced density matrix (3). The more accurate is the detection of electrons at the slits, the more
orthogonal are states ,, the smaller is the product , and the smaller are interference
terms in (2,3,5,6).
The fact that measurement produces classical event sample, substantiates Bohr’s postulate that
the measuring device has to be classical: [the] necessity of discriminating in each experimental
arrangement between those parts of the physical system considered which are to be treated as
measuring instruments and those which constitute the objects under investigation may indeed be
said to form a principal distinction between classical and quantum-mechanical description of
physical phenomena .
Consequently, the measurement itself can be defined as extraction of classical information.
Thus, the classicality emerges as a result of measurement, in a form of classical event sample
, not as a result of decoherence. As evident from (6), the decoherence itself is an aftermath of
measurement. From this prospective, classicality and decoherence go hand in hand. No unitary
process, described by, e.g. Schrödinger equation, can turn quantum state into classical, no matter
how much interactions one plugs in
, only measurement does.
The fact that entanglement itself does not produce decoherence or any other measurable effect
has been attested in multitude of experiments on violation of Bell’s inequalities, including those
where measurements on two entangled entities are separated by spacelike interval [14, 15]. The
strongest violation happens when no classical information is exchanged between two measuring
devices, which corresponds to case in (5,6), i.e. when measurement by one device does
not predetermine to any degree the measurement by another device
The entanglement has to be considered a logical construct, indicating correlation, not
causation, between measurements by different devices. The causation emerges when one
measurement predetermines the results of another measurement. As I have shown, that happens
when classical information is shared between measurement devices.
The claims that entanglement involves interaction can also be disproven
. The measurement
result (6) does not depend on ancilla if measurement is done in qubit eigenbasis, i.e. when
. Such measurement would not detect interference. Why would interaction
depend on how we choose to measure the qubit? The proponents of interaction explain this fact as
follows: the ancilla (e.g. environment) enforces “effective superselection rules”  which select
certain states to be “robust” against interaction. What states are preferred will depend on the details
of the interaction . Apparently, in case (1), ancilla chooses , to be such robust states.
However, by changing the measurement basis from , to ,, the experimenter can make
I refute the statement  that “the pattern of detections at the screen cannot distinguish mere entanglement with
some other systems from the actual use of those systems for detection at the slits”. It does distinguish, because,
according to (6), without measurement of ancilla, there is no decoherence, no degradation of interference.
The proof is trivial: the criterion for a pure quantum state is invariant with respect to unitary transformations.
Tracing out part of density matrix to illustrate decoherence is equivalent to performing measurement.
One must acknowledge the numerous experiments on violation of Bell’s inequalities are just variations of double-
slit experiment, albeit done at much greater expense and effort.
On high level the proof is trivial: the interaction by definition implies causation, which, as I have shown, only
emerges as a result of measurement.
ancilla to “prefer” ,. The measurement in , basis, just like the measurement in ,
basis would not detect interference:
There are infinitely many such measurement bases, which are “robust” against effects of
entanglement. Since the bases which are “robust” depend on the will of experimenter, the
assumption that they are selected by interaction between qubit and ancilla is false. The very notion
of interaction as pertaining to physical reality independent of experimenter has to be abandoned.
One has to recognize that quantum mechanical expressions describe measurement setups, and
predicted measurement outcomes, not abstractions like “quantum state”, “entanglement”,
“superposition”, “wave function”, etc. E.g., the expression (1) describes configuration for two-
device measurement. The expression describes single-device measurement,
where , and
are device eigenstates. As proved by Alain
Aspect’s famous experiment , the quantum state is not just unknown, it does not even exist
prior to measurement.
The classical state, represented by an event sample, is encoded via a set of symbols from some
alphabet . Each symbol equates to a classical parameter value. E.g., the spatial coordinate of
an object represents a [continuous] alphabet. The value of coordinate represents a symbol,
produced as a result of measurement. No quantum object can have coordinate defined until the
measurement is done, in coordinate eigenbasis. This dispels frequently heard claims that quantum
particle can be at two different locations at the same time. It cannot, because the coordinate is not
defined until the measurement. It is not just unknown; it is not defined even as a parameter. A
presence of classical parameter in quantum mechanical expression, even implicit hidden variable,
would indicate a carried-out measurement, just like the presence of scalar products , .
The macroscopic objects are usually associated with classical behavior, i.e. with absence of
interference. In view of things said, it can be explained as follows. Commonly accessible
observables correspond to operators, which only act within small subset of object’s degrees of
freedom . For simplicity, I consolidate those into Hilbert space formed by eigenstates ,.
Every vector from this Hilbert space is correlated with a vector from Hilbert space of the rest of
object’s degrees of freedom, e.g. is correlated with , is correlated with , as in (1). Due
to the large number of degrees of freedom, macroscopically different states , correspond to
vectors ,, which differ in many of the degrees of freedom , i.e. , are orthogonal
pointer states, . This leads to vanishing interference terms in (5, 6).
To conclude, I have elucidated the origins of decoherence, classicality, and causation. I have
argued that quantum mechanical expressions have to be watched for implicit measurements, which
signify emerging classicality, such as the measurement on ancilla present in Born rule (6).
M. Schlosshauer and K. Camilleri, "The quantum-to-classical transition: Bohr’s doctrine of
classical concepts, emergent classicality, and decoherence," arXiv:0804.1609 [quant-ph],
W. H. Zurek, "Environment-induced superselection rules," Phys. Rev. D, vol. 26, no. 8, pp.
A. Einstein, N. Rosen and B. Podolsky, "Can Quantum-Mechanical Description of Physical
Reality Be Considered Complete," Phys. Rev., vol. 47, p. 777, 1935.
J. S. Bell, "On the Einstein Podolsky Rosen Paradox," Physics, vol. 1, no. 3, pp. 195-200,
A. Einstein, "Letter to Max Born," 1947. [Online]. Available:
R. Griﬃths, "Nonlocality Claims are Inconsistent with Hilbert Space Quantum
Mechanics," arXiv:1901.07050 [quant-ph], 2019.
S. Boughn, "There Is No Action at a Distance in Quantum Mechanics, Spooky or
Otherwise," arXiv:1806.07925 [quant-ph], 2018.
M. Schlosshauer, "Quantum Decoherence," arXiv:1911.06282 [quant-ph], 2019.
L. Landau, "The damping problem in wave mechanics," Z. Phys., vol. 45, no. 63, pp. 430-
T. De Angelis, F. De Martini, E. Nagali and F. Sciarrino, "Experimental test of the no
signaling theorem," arXiv:0705.1898 [quant-ph], 2007.
N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani and S. Wehner, "Bell nonlocality,"
arXiv:1303.2849 [quant-ph], 2014.
N. Bohr, "Quantum Mechanics and Physical Reality," Phys.Rev., vol. 48, pp. 696-702,
S. Viznyuk, "From QM to KM," 2020. [Online]. Available:
A. Aspect, J. Dalibard and G. Roger, "Experimental Test of Bell’s Inequalities Using
Time-Varying Analyzers," Phys. Rev. Lett., vol. 49, pp. 1804-1807, 1982.
J. Handsteiner, A. Friedman, D. Rauch, J. Gallicchio, B. Liu, H. Hosp, J. Kofler, D.
Bricher, M. Fink, C. Leung, A. Mark, H. Nguyen, I. Sanders, F. Steinlechner, R. Ursin, S.
Wengerows, A. Guth, D. Kaiser, T. Scheidl and A. Zeilinger, "Cosmic Bell Test:
Measurement Settings from Milky Way Stars," Phys.Rev.Lett., vol. 118, no. 6, p. 060401,
G. Bacciagaluppi, "The Role of Decoherence in Quantum Mechanics," The Stanford
Encyclopedia of Philosophy (Summer 2020 Edition).
H. Wakita, "Measurement in Quantum Mechanics," Progress of Theoretical Physics, vol.
23, no. 1, pp. 32-40, 1960.
M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge
University Press, 2010.