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The basics of information transmission
Sergei Viznyuk
There is a glaring contradiction, duly noted in [1], between ubiquity of transmission of classical
information in everyday life, and multitude of no-go theorems, such as no-cloning [2, 3, 4], no-
teleportation [5], no-broadcasting [6], no-communication [7], no-signaling [8], which disallow
information, contained in arbitrary state, from being copied. The unfeasibility could be ascertained
multiple ways:
• Suppose there is a linear operator able to clone an arbitrary normalized state over target
: . Since state is arbitrary, operator is also able to clone orthogonal to
normalized state : , as well as normalized state :
==+. From linearity of :
+ + +. Taking
scalar product with of both sides of last equation, I obtain which is only
possible if or . Thus, operator is not able to clone an arbitrary state. In particular,
it is not able to copy state , where ,. Also, contrary to statement “a
cloning device can only clone states which are orthogonal” on page 532 [1], it is easy to
see that it is not able to clone orthogonal to state
• Cloning of arbitrary state would enable measurement, i.e., extraction of classical
information, to be performed on replicas without destroying the original state. Such
proposition is impossible in quantum theory
Could it be that cloning is possible for particular states, such as eigenstates of measuring device?
It is widely assumed (see, e.g., exercise 12.1 in [1]), that such states can be easily cloned via
quantum channel. As I show below, that is also not the case.
• If and are eigenstates of measuring device, they are unitarity equivalent to superposed
normalized states , and
. Namely, there exist
unitary transformation , such that and . If we assume there exists linear
operator able to copy over target : , then
. Thus, based on our assumption, there exists linear
operator able to copy state over target : =,
which has been proven impossible above.
This result may seem controversial, as eigenstates of measuring device are classical, in a sense,
they are not destroyed by measurement, and therefore, can be cloned. However, no-go theorems
and ascertainment in previous paragraph only apply to cloning via quantum channel, i.e., via
unitary operation, not via classical channel. It has been shown [9], no information is transmitted
via quantum channel. Hence, no cloning is possible via quantum channel only. Teleportation or
cloning always involves transmission of information via classical channel. Then, what exactly is
classical channel? That is the question I answer in this paper.
Consider a generic setup with Alice sending a message to Bob. The available operations are
unitary transformation, which preserves both quantum and classical information, and
measurement, which converts quantum information into classical. There must be a shared state to
perform these operations on. Consider a basic shared state of two entangled qubits, one accessible
to Alice and another accessible to Bob:
Here are Alice’s entangled qubit states, and are Bob’s qubit states. For and
in to be orthogonal, I consider normalized, but not necessarily orthogonal, and
normalized and orthogonal:
The amount of information extracted from shared state with measurement performed by Alice is
given by Von Newmann’s entropy
, where is shared state
with Alice’s part traced out:
, with , being the length of Bloch vector [10, 11]. Product
tells how reliably can Alice’s device distinguish and states. If , then ,
are perfectly distinguishable, i.e., orthogonal, in Alice’s measurement basis. In this case,
. If , Alice’s device cannot perfectly distinguish , states, as in case,
e.g., when and are horizontal and vertical polarizations of a photon, with non-ideal PBS
(polarizing beam splitter) used to separate them (Figure 1 in [10]). In this case .
In particular, if , then . Thus, the amount of information extracted by Alice
depends on orthogonality of entangled Alice’s states.
If Bob’s measurement basis is , then expectation value of Bob’s measurement
operator is:
. It does
not depend on Alice’s measurement, as stipulated by no-communication and no-signaling
theorems [7, 8]. However, if Bob chooses to do his measurement in basis, wherein
and , then from we would have:
To simplify , consider
, and
. With that:
, where and are real and imaginary parts of .
The expectation value of Bob’s measurement operator in basis is:
As transpired, Alice can alter expectation value of Bob’s measurement by tuning
(modulating) her own measuring device (transmitter), i.e., by controlling product. This
mechanism, involving modulation of transmitter, and measurement of the reduced shared state by
receiving device, constitutes classical communication channel. The actual results of any
measurement are not transmitted. It underscores the absoluteness of classical information [10], in
that information can only be extracted once, and cannot be re-extracted again from any state or
“channel”. To illustrate the point, consider a shared entangled state of and [qubits, qdits],
accessible respectively, to Alice and Bob. It’s not too difficult to see that reduced states
and are unitarily equivalent, i.e., , such that ; . In
case of , it is obvious from diagonal form of and :
The shared state [of entangled qubits] is a crucial element of any communication. Such state
could be the output state of higher cardinality measurement operator, encoded in
entangled qubits [12]. Entangled qubits make up elements of objective reality, which transmitter
and receiver are parts of. The transmitter must be in correlated (entangled) relationship with the
receiver for communication to work, thuswise objective reality to be observable [13].
Alice’s or Bob’s measurement reduces shared state to unitarily equivalent or . Yet,
if Alice’s and Bob’s devices are spacetime separated, their measurement operators are not unitarily
equivalent. It is obvious from spacetime parameterization of qubit operator:
. A change in and/or would result in a change to
eigenvalues of , and therefore, non-equivalent operator. The
relationship between Hermitian operators , having shared state, is intertwist ,
where is a linear transformation. For qubit operators, such transformation only exists if [10]:
Three non-trivial situations satisfy :
1. ;
. In this case is unitary; , are unitarily equivalent, but not
necessarily commuting
2. ;
; . In this case
is non-unitary; , are commuting, but not unitarily equivalent
3. ;
;
. is non-unitary; , are not unitarily equivalent, and
not necessarily commuting
The outcomes of measurement by Alice and Bob on shared state implicitly incorporate speed limit
in relationship between observables: or
;
whenever
. The speed limit is effectuated by Hermiticity of [qubit] measurement operators.
References
[1]
M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge University
Press, 2010.
[2]
J. Parker, "The concept of transition in quantum mechanics," Foundations of Physics, vol. 1, no. 1,
p. 23–33, 1970.
[3]
W. Zurek and W. Wootters, "A single quantum cannot be cloned," Nature, vol. 299, no. 5886, p.
802–803, 1982.
[4]
D. Dieks, "Communication by EPR devices," Physics Letters A, vol. 92, no. 6, p. 271–272, 1982.
[5]
R. Werner, "Optimal Cloning of Pure States," arXiv:quant-ph/9804001, 04 1998.
[6]
H. Barnum, C. M. Caves, C. A. Fuchs, R. Jozsa and B. Schumacher, "Noncommuting Mixed States
Cannot Be Broadcast," Physical Review Letters, vol. 76, no. 15, p. 2818–2821, 1995.
[7]
A. Peres and D. Terno, "Quantum Information and Relativity Theory," arXiv:quant-ph/0212023,
2003.
[8]
T. De Angelis, F. De Martini, E. Nagali and F. Sciarrino, "Experimental test of the no signaling
theorem," arXiv:0705.1898 [quant-ph], 2007.
[9]
S. Viznyuk, "Kitchen experiment on entanglement and teleportation," 2020. [Online]. Available:
https://www.academia.edu/42219989/Kitchen_experiment_on_entanglement_and_teleportation.
[10]
S. Viznyuk, "Where unfathomable begins," Academia.edu, 2021. [Online]. Available:
https://www.academia.edu/45647801/Where_unfathomable_begins.
[11]
S. Haroche and J. Raymond, Exploring the Quantum, Oxford University Press, 2006.
[12]
S. Viznyuk, "Ontogenesis, Part 1," 11 2023. [Online]. Available:
https://www.academia.edu/109685658/Ontogenesis_Part_1.
[13]
S. Viznyuk, "Dimensionality of observable space," Academia.edu, 2021. [Online]. Available:
https://www.academia.edu/66663981/Dimensionality_of_observable_space.