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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.