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The origin of unitary dynamics
Sergei Viznyuk
I argue the unitary evolution of quantum state is imposed by implicit
extraction of classical information. I show the generating self-adjoint
operator of time-driven unitary transformation is entropy, not Hamiltonian
The description of physical phenomena in terms of measurement outcomes, i.e., objective
facts, vs. inferred possibilities, provides the principal distinction between the realm of classical
physics, and the realm of quantum mechanics (QM) [1]. While former operates with definite values
of observables, [which can only arise as a result of measurement], the latter derives the
corresponding measurement probabilities, for a known device configuration (measurement
eigenbasis)
1
. The concept of a device is pivotal for both realms. The device is both, the source of
classical information contained in measurement events, and the reference frame (measurement
eigenbasis), in which events are obtained.
Typically, there are two devices to consider: preparation device, and measuring device. The
output of preparation device serves as input to measuring device. The output of a device is, by
definition
2
, one of its eigenstates. As eigenbasis of measuring device may not coincide with
eigenbasis of preparation device, the projection of preparation device’ output onto eigenspace
of measuring device is a superposition
3
of measuring device’ eigenvectors :
, where is a set of all possible mutually exclusive (orthogonal) measurement events.
When we express as superposition of eigenvectors of measuring device with defined
amplitudes , it means we already possess
4
some [classical] information about . Generally,
however, such information can only be extracted by the measurement, as an event sample [2]. This
information is only enough to determine event probabilities , in limit, where is
the measurement sample size. It means, no measurement can completely determine input to
measuring device [3, 4]. The description, based on extracted classical information, cannot be
complete. It is the core problem which prompted development of QM during first half of 20th
century. While QM was not able to resolve this (measurement) problem, prompting further
incompleteness claims [5], it succeeded in predicting dependence of event probabilities ,
and associated expectation values, on parameter-driven change of measurement basis.
1
The adjective known always implies classical information. As pertaining to device configuration, the very notion of
it being known, necessitates the classicality of [measuring] device [1]
2
A device is completely defined by its eigenbasis, i.e., by a set of distinct states device can be in
3
The quantum state thus refers to the description of preparation device’ output in terms of measuring device’
eigenvectors [14]. The phenomenon of quantum superposition [of measuring device’ eigenvectors] arises from
inability of measuring device to distinguish eigenvectors of preparation device
4
The amplitudes are known exactly if experimenter is in control of both, preparation, and measuring devices,
meaning the experimenter determines the output of preparation device, and the measurement eigenbasis
If an event probability in old measurement basis , in new basis
. It shows that if transformation is applied to measuring device,
the expectation values are the same as if adjoint transformation is applied to preparation device,
and v.v. A transformation
5
of quantum state is, therefore, indistinguishable from transformation of
measurement basis.
The knowledge of basis transformation is crucial for QM analysis. Besides linearity, there is
no restriction on transformation considered above. The restriction appears once a third party (an
observer) is introduced, with a premise that event probability is invariant with respect to
transformation of observer basis relative to both, measuring, and preparation device:
The independence of an event [probability] on observer [basis] is a defining property of objective
fact. For objectivity to hold, the change of observer basis has to be unitary or anti-unitary
operation
6
. The objectivity imposes restriction on how device transformation , considered in
previous paragraph, alters with a change of observation basis. Without objectivity restriction we
have: , i.e., alters by
similarity transformation. The objectivity restricts similarity to unitarity , if the change
of device’ basis is relative to a third party (observer). The presence of third party is a crucial
element of unitary dynamics which we shall investigate further below. The unitarity is imposed
not by the mere presence of observer [6], but by the extraction of classical information, implied by
the presence of observer. The classical information is carried by the event sample, in a form, e.g.,
of event probabilities .
The generator of a unitary transformation is a self-adjoint operator , i.e., .
If basis transformation is parameter-driven, i.e., , then:
In differential form, becomes:
, where
. One would readily recognize as Schrödinger equation [7], for
an arbitrary parameter . If is a distance ; and
, then becomes
The Schrödinger equation in a form serves as definition of momentum operator
. Any
parameter-driven unitary transformation, where generator is a differentiable function of the
parameter, can be written in a form of Schrödinger equation. In case is time, it was postulated [7]
by Schrödinger, that is Hamiltonian. As of now, it still remains one of QM postulates [4, 8].
5
The “evolution of quantum state” is a popular, albeit misleading, term
6
From , , where is real and is identity operator. Since is self-adjoint,
No theory is complete if it’s based on postulates, i.e., if it cannot explain itself. Below, I show
that if is time, then
is entropy operator.
Generally, measurements are done in preparation + measurement cycles (PMC). The PMCs
produce statistical ensemble needed for determination of probabilities and expectation values.
Consider the preparation and measuring devices have the same eigenspace, i.e., in .
It means the total measurement probability is . The PMC output is one of eigenvectors
of measuring device, where is the dimension of measurement basis. The full output
of measuring device is given by
[3, 9], or, in Fock representation, by
, where is the number of occurrences of event ;
. The event sample
contains classical information which has to persist in some encoded form. The encoded
information constitutes the final result of the measurement, not the transient output sample
7
. The
number of symbols needed to encode output information is given by Boltzmann’s entropy [2]:
, where
is the statistical weight of the sample
8
. As has been shown elsewhere
[10], , i.e., for a finite-size sample, the number of symbols needed to encode
measurement output is less than the size of the sample expressed in the same units. In units of
, the extracted information is represented by symbol of cardinality , i.e., .
Each of possible values of this symbol, corresponds to a quantum state with certain phase
relationship between amplitudes . Such -states constitute eigenbasis in Fock space of
event sample. The -states are eigenvectors of entropy operator , with the same eigenvalue
. The entropy operator, in respective units, is then:
, where is the identity operator, acting in Fock space spanned by -states. Different units of
entropy correspond to different encoding symbols, e.g., bits or nats. If encoded in bits, -state
is represented
9
by qubits :
7
The eigenvectors are not classical outcomes; rather, they can be thought of as radiation modes
8
The Boltzmann’s entropy is the correct measure in this case, as opposed to von Neumann’s [15] entropy
. Von Neumann’s entropy cannot be defined from measurement events, because it requires
knowledge of density matrix , unattainable from the measurement [3]. Diagonal elements of can be
evaluated as
. However, without off-diagonal elements, would represent a mixture, not the pure
quantum state . Therefore, von Neumann’s entropy is not the eigenvalue of Fock state
9
The encoded information can only be represented by integer number of symbols. The value can be non-
integer, depending on cardinality of encoding alphabet, which introduces some noise into encoded information.
Therefore, the encoded information may not unambiguously represent the event sample. It adds to be fact that event
sample itself does not unambiguously represent the input to measuring device
, where qubits are in definite bit states. From , .
The fact that -states are bit product states, and not entangled qubit states, allows for derivation
of parameter-driven unitary transformation between -states. Different parameter values
correspond to different observation bases. The parameter-driven transformation of observation
basis is limited to Fock space of event sample by the objectivity condition .
The transformation of observation basis, considered so far, may seem a continuous function of
the parameters, especially if written in a form of Schrödinger equation . However, the
parameter values can only arise as a result of measurement, even if implied measurement. The
continues values of the parameters would indicate a possibility of continuous measurement, with
infinitely small intervals. That is an improbable proposition, from multiple perspectives, e.g., from
uncertainty relations. The discreteness of -states forces parameters to only take discrete values.
Any self-adjoint operator is expressed [4] in terms of spacetime -vector as
, where ; ; ; are Pauli matrices. Operator
generates qubit’s unitary transformation [4]:
Any two qubits differ by unitary transformation , i.e., by real-valued classical parameters
: . The -vector is the relative spacetime position
of the qubit. The qubits, making up -states, give rise to spacetime as an entity. The change
of observation basis may only transform these qubits between bit states. It forces -vector
to only take discrete values, i.e., spacetime is quantized.
Any two -states differ by a tensor product of individual qubits’ unitary transformations:
Here , where subscript refers to parameters of, and to operators
acting on, qubit ; are space coordinates of individual qubits. The unitary transformation
between given -states is by no means unique. We are looking for the one which has time-driven
part separated, as in . With , the tensor product expands into
Using entropy operator , I write as
The above shows the generating self-adjoint operator for time-driven unitary transformation is
entropy operator . The underlying reason is: the Fock state is the eigenstate of
entropy, not of Hamiltonian. Unlike entropy , the [information] energy is not
determined solely from event sample . It requires additional information, in a form of event
probabilities , as evident from formula in [2]:
In , the knowledge of probabilities is inseparable from the knowledge of quantum state
proper. The concept of knowledge is based on entropy as measure of missing information. The
entropy is the amount of unknown. The maximum entropy state has zero known. Thus, the amount
of known, i.e., knowledge, equals difference between max entropy
,
and entropy of event sample [11, 3, 2]:
If probabilities of events are same
10
,
, the second term under sum in
disappears. The energy then equals knowledge . The time-driven part of becomes
Transformation leads to traditional Schrödinger equation with Hamiltonian
.
The eigenvalues of are , as expected, for bound states with discrete spectrum .
From , the spacetime quantization interval of -states is by factor smaller than
quantization interval of standalone qubits. The use of differential Schrödinger equation , or any
differentiation by classical parameters, may only be justified if is sufficiently large, as in case
of large event sample .
-state represents classical information, obtained from the measurement, and encoded as
spacetime configuration of qubits in definite bit states. -states do not have phase. A phase
would mean the presence of unencoded information. The phase is lost when measurement turns
amplitude into classical probability . However, the internal phase relationship
between components of quantum state is not lost upon measurement. Consider a quantum state
consisting of two components: . The event probability is
11
:
It shows, the phase difference between two components of quantum state affects event
probabilities, and, therefore, is converted into classical information by the measurement.
10
The equal probabilities are expected in canonical scenarios, as e.g., in case of double-slit experiment, where
probabilities of a particle passing through either slit are equal; or, if, e.g., is a distribution of photons among
radiation modes of the same energy
11
Naturally, and would not be normalized to 1
I have shown the unitary dynamics is conditioned on extraction of classical information,
implied by the presence of third party (observer). This is contrary to orthodox QM interpretations,
which postulate unitary dynamics, and consequently, the Schrödinger equation, as an inherent
property of quantum state. The conclusions of this paper, to a significant extent, are foretold by
Bohr and Heisenberg [1, 6, 12, 13]. I have shown the generating self-adjoint operator of time-
driven unitary transformation, in general, is entropy, not Hamiltonian. Perhaps the most intriguing
finding is the emergence of spacetime as encoding structure for classical information.
References
[1]
N. Bohr, "Can Quantum Mechanical Description of Physical Reality be Considered Complete?,"
Phys.Rev., vol. 48, pp. 696-702, 1935.
[2]
S. Viznyuk, "From QM to KM," 2020. [Online]. Available:
https://www.academia.edu/41619476/From_QM_to_KM.
[3]
S. Viznyuk, "The measurement and the state evaluation," 2020. [Online]. Available:
https://www.academia.edu/42855863/The_measurement_and_the_state_evaluation.
[4]
M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge University
Press, 2010.
[5]
A. Einstein, N. Rosen and B. Podolsky, "Can Quantum-Mechanical Description of Physical Reality
Be Considered Complete?," Phys. Rev., vol. 47, p. 777, 1935.
[6]
W. Heisenberg, Physics and Philosophy, New York: Harper & Row Publishers, Inc, 1962.
[7]
E. Schrödinger, "An Undulatory Theory of the Mechanics of Atoms and Molecules," Phys. Rev.,
vol. 28, no. 6, p. 1049–1070, 1926.
[8]
M. Kober, "Copenhagen Interpretation of Quantum Theory and the Measurement Problem,"
arXiv:0905.0408 [physics.hist-ph], 2009.
[9]
R. Werner, "Optimal Cloning of Pure States," arXiv:quant-ph/9804001, 04 1998.
[10]
S. Viznyuk, "Shannon's entropy revisited," arXiv:1504.01407 [cs.IT], 2015.
[11]
C. Adami, "The Physics of Information," arXiv:quant-ph/0405005, 2003.
[12]
N. Bohr, "The Quantum Postulate and the Recent Development of Atomic Theory," Nature, pp.
580-590, 14 April 1928.
[13]
N. Bohr, "Causality and Complementarity," Phylosophy of Science, vol. 4, no. 3, pp. 289-298, 1937.
[14]
S. Viznyuk, "No decoherence by entanglement," 2020. [Online]. Available:
https://www.academia.edu/43260697/No_decoherence_by_entanglement.
[15]
J. Von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press,
1955.
