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No decoherence by entanglement

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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
No decoherence by entanglement
Sergei Viznyuk
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 [1]; … the correlations of a quantum system
with other quantum systems may cause one of its observables to behave in a classical manner [2].
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 ,:
, where
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 [10]. 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 [2] 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 [3]. The following catchy phrase [4] is definitely subject to
misinterpretation: grossly non-local structure... is characteristic of any theory which reproduces exactly quantum
mechanical predictions
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 [12].
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
[13], 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” [2] which select
certain states to be robust” against interaction. What states are preferred will depend on the details
of the interaction [16]. 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 [16] 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:
, where
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 [14], 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 [13]. 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 [17]. 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 [17], 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).
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Full-text available
Bell’s theorem states that some predictions of quantum mechanics cannot be reproduced by a local-realist theory. That conflict is expressed by Bell’s inequality, which is usually derived under the assumption that there are no statistical correlations between the choices of measurement settings and anything else that can causally affect the measurement outcomes. In previous experiments, this “freedom of choice” was addressed by ensuring that selection of measurement settings via conventional “quantum random number generators” was spacelike separated from the entangled particle creation. This, however, left open the possibility that an unknown cause affected both the setting choices and measurement outcomes as recently as mere microseconds before each experimental trial. Here we report on a new experimental test of Bell’s inequality that, for the first time, uses distant astronomical sources as “cosmic setting generators.” In our tests with polarization-entangled photons, measurement settings were chosen using real-time observations of Milky Way stars while simultaneously ensuring locality. Assuming fair sampling for all detected photons, and that each stellar photon’s color was set at emission, we observe statistically significant ≳7.31σ and ≳11.93σ violations of Bell’s inequality with estimated p values of ≲1.8×10−13 and ≲4.0×10−33, respectively, thereby pushing back by ∼600 years the most recent time by which any local-realist influences could have engineered the observed Bell violation.
Major revision of the previous article published on the same website: 'The Role of Decoherence in Quantum Mechanics', in E. N. Zalta (ed.), The Stanford Encyclopedia of Philosophy (Winter 2003 Edition), [under the editorial responsibility of J. Norton] (25pp.).
We show how the correlations of a quantum system with other quantum systems may cause one of its observables to behave in a classical manner. In particular, "reduction of the wave packet," postulated by von Neumann to explain definiteness of an outcome of an individual observation, can be explained when a realistic model of an apparatus is adopted. Instead of an isolated quantum apparatus with a number of states equal to the number of possible distinct outcomes of the measurement, discussed by von Neumann, we consider an apparatus interacting with other physical systems, described here summarily as "environment." The interaction of the quantum apparatus with the environment results in correlations. Correlations impose effective superselection rules which prevent apparatus from appearing in a superposition of states corresponding to different eigenvalues of the privileged pointer observable. It is the propagation of the correlations with the pointer basis states which is ultimately responsible for the choice of the pointer observable. It can be thought of as a process of amplification in which the state of many distinct physical systems becomes correlated with the pointer basis state. Whether these environment systems are regarded as a part of the apparatus setup, or as a part of its environment is irrelevant. What is crucial is the redundancy of the record concerning the pointer observable which is imprinted into the correlations. Eigenspaces of the pointer observable provide a natural basis for the pointer of the quantum apparatus and determine the to-be-measured observable of the quantum system. Decay of those elements of the apparatus-system density matrix, which are off-diagonal in the pointer observable, is caused by the natural evolution of the combined system-apparatus-environment object. For a hypothetical finite environment with N distinct eigenvalues of the apparatus-environment interaction Hamiltonian, off-diagonal terms will decay to become of the order of N-1/2, and will recur only on a Poincaré time scale. No recurrences will be observed in realistic circumstances. As the correlations spread through the environment on a time scale typically much shorter than the recurrence time scale calculated for the environment already correlated with the pointer observable, the measurement becomes effectively irreversible. Relevance of this model of the measurement process for the understanding of the second law of thermodynamics and its relation to Bohr's "irreversible act of amplification" is briefly discussed. The emergence of the pointer observable can be interpreted as a clue about the resolution of the measurement problem in case of no environment. It points towards the possibility that properties of quantum systems have no absolute meaning. Rather, they must be always characterized with respect to other physical systems.
In quantum mechanics it is well known that, if any two states are superposed, they interfere with each other. It is true, we should not deny such interference in principle, but we may assert what follows. When two states different from each other in a great many degrees of freedom are superposed, the interference effect becomes obscure. If they are different in an infinitely many degrees of freedom, they do not interfere at all, and their superposition is nothing but a mere probability function. This assertion enables us to understand how the probability amplitude for a micro-system is converted into a probability function for a measuring apparatus in the cource of measurement.
Correlations of linear polarizations of pairs of photons have been measured with time-varying analyzers. The analyzer in each leg of the apparatus is an acousto-optical switch followed by two linear polarizers. The switches operate at incommensurate frequencies near 50 MHz. Each analyzer amounts to a polarizer which jumps between two orientations in a time short compared with the photon transit time. The results are in good agreement with quantum mechanical predictions but violate Bell's inequalities by 5 standard deviations.
In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) these two quantities cannot have simultaneous reality. Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false. One is thus led to conclude that the description of reality as given by a wave function is not complete.
The quantum-to-classical transition: Bohr's doctrine of classical concepts, emergent classicality, and decoherence
  • M Schlosshauer
  • K Camilleri
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], 2008.