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Kitchen experiment on entanglement and teleportation
Sergei Viznyuk
Abstract
I prove teleportation protocol for an arbitrary qubit state can be implemented with
one bit of information transmitted via classical channel, per preparation + measurement
cycle. I show how teleportation protocol can be implemented in a classical setting. I
discuss the contextual meaning of teleportation
Entanglement and teleportation have become popular buzzwords, generating interest in
scientific community, as well as industry, and government, with promise of secure communication
[1], and quantum computing capabilities [2].
Teleportation relies on a shared two-qubit entangled ancilla state between the sender (Alice),
and the receiver (Bob). The subject of teleportation is the third qubit in unknown state, which Alice
wants to teleport to Bob.
As entanglement is deemed a purely quantum feature, the teleportation is assumed to be only
possible via quantum channel. The publicized teleportation schemes [3, 2] involve unitary
transformations, a measurement by Alice in basis on her two accessible qubits, and two bits of
information transmitted by Alice to Bob via classical channel. All that effort is in order for Bob to
reproduce the measurement result sample which would have been obtained by direct
measurements of third qubit.
As an improvement over publicized teleportation schemes, I present teleportation protocol
which only involves one projection transformation, one measurement by Alice, and of
information transmitted by Alice to Bob via classical channel. I show that to the same end result,
teleportation can be realized in a classical setting.
The teleportation protocol is based on Alice and Bob sharing entangled ancilla state of
two qubits: ;
. The subscripts , designate qubits
accessible respectively to Alice and Bob. Alice and Bob are free to choose the shared ancilla state.
A third qubit, in unknown state
(1)
is given to Alice to teleport to Bob. In standard protocol, two unitary transformations (-gate
+ -gate) [2], and two measurements are required for Alice to perform, and of information
to be transmitted by Alice to Bob via classical channel per preparation + measurement cycle
(PMC), in order for Bob to reconstruct the measurement result sample for the third qubit. Alice
performs her measurements in cardinality basis, the results of which require
, to be passed to Bob, per PMC. There is, however, a redundancy in traditional protocol, as
Bob’s measurement is performed in cardinality basis, so Alice’s Hilbert space, one way
or another, gets projected into Bob’s space. In the proposed protocol this redundancy is
eliminated.
To start, Alice and Bob choose the usual, maximally entangled shared ancilla state :
(2)
The proposed teleportation protocol is implemented in PMC steps as follows:
1. Preparation of the standard product state:
(3)
Next, Alice wants to use observation basis of cardinality . She also wants the basis to be
transformed, so the state of Bob’s qubit would look separated and unitarily equivalent to the state
of third qubit.
2. Alice transforms to cardinality observation basis, with basis vectors:
(4)
Transformation of observation basis is performed with projection operator :
(5)
Applying operator (5) to (3) results in:
(6)
In (6), the Alice’s qubits are in orthogonal states and , and Bob’s qubit is in a separated
superposition of state (1), and unitarily equivalent to (1) state .
3. Alice performs the measurement of accessible to her qubits of state (6). With equal probability
Alice finds accessible to her qubits in state , if both qubits are the same, or state , if qubits
are different.
4. If Alice finds her qubits to be identical, i.e. in state, she sends Bob a single bit with value 0,
meaning he has to measure his qubit without any transformation. If Alice finds her qubits in
state, she sends Bob a single bit with value 1, meaning Bob has to apply -gate on his qubit,
swapping and , before taking measurement. The -gate applied to turns it
into target state (1).
Repeating PMC steps 1-4 will let Bob obtain the same measurement result sample, as if he
performed measurement on state (1), which, in minds of many authors, means the state (1) has
been successfully teleported by Alice to Bob. The experimenters on quantum teleportation [4]
reported success when measurement result samples matched with fidelity 0.8.
I shall now show the Bell-type entanglement and associated teleportation protocol can be
implemented in purely classical setting, to achieve the same result as in quantum teleportation
described above.
Bob invites Alice for a dinner and promises to entertain her with teleportation experiment. Bob
prepares two pairs of identical matching gloves, and four black boxes, one per glove, and a mirror.
Bob puts four gloves into four boxes, one glove per box, and closes boxes. Once Alice shows up,
Bob gives boxes to Alice. He asks Alice to randomly shuffle boxes, behind his back, without
looking into them, and then give one box to him. Thus, Bob gets one box, and Alice keeps three
boxes. Bob asks Alice to put one of her three boxes aside. He then says, that he can tell which
glove is in that box, if Alice opens her remaining two boxes and tells Bob only one thing: if the
gloves she sees make a pair or not (i.e. if they are matching left and right gloves). If Alice tells
Bob, she found left and right gloves in her remaining two boxes, Bob opens his box while looking
into mirror image of its content (i.e. using -gate transformation), and records the result, i.e. if he
sees left or right glove in the mirror. The mirror image of the left glove is the right glove, and vice
versa. If Alice tells Bob she sees two identical gloves in her two boxes, then Bob looks straight
into his box and records what he sees. Surely enough, whatever result Bob records matches the
content of the box which Alice puts aside, every time they repeat the experiment.
Similar protocol can be used to teleport a secret binary string of . This protocol is
known as Vernam-Mauborgne one-time pad. In this scenario, Bob generates a random string of
and secretly shares it with Alice, thus establishing a shared ancilla state. In order to teleport
the string , Alice performs binary (XOR) operation between string and her copy of the string
. Then, Alice reads bits of string one by one. If Alice reads value she tells Bob to read
his bit as is. If Alice reads she tells Bob to swap his bit into opposite. Even if Eve eavesdrops on
Alice’s communication to Bob, she would not be able to reconstruct secret string without having
a copy of string . The ancilla string , just like the shared entangled state (2), serves as a
codebook. The actual messages are transmitted via classical channel, but decrypted using shared
codebook.
The thought experiments above prompt some legitimate questions as for the meaning of
quantum teleportation and its potential usefulness, given the same results can be achieved in
classical settings. Similar questions have been asked before [5]. For one thing, in any teleportation
scheme, only of information about target state (1) is obtained by Bob per preparation +
measurement cycle. It requires the same of information (or , if using publicized
teleportation schemes), to be transmitted by Alice to Bob via classical channel. So, it seems nothing
beyond what is transmitted via classical channel gets received by Bob.
References
[1]
C. Elliott, D. Pearson and G. Troxel, "Quantum Cryptography in Practice," arXiv:quant-
ph/0307049, 2003.
[2]
M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge
University Press, 2010.
[3]
C. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres and W. Wootters, "Teleporting an
Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels,"
Physical Review Letters, vol. 70, no. 13, pp. 1895-1899, 1993.
[4]
J. Ren, P. Xu, H. Yong, L. Zhang, S. Liao, J. Yin, W. Liu, W. Cai, M. Yang, L. Li, K. Yang,
X. Han, Y. Yao, J. Li, H. Wu, S. Wan, L. Liu and D. Liu, "Ground-to-satellite quantum
teleportation," arXiv:1707.00934 [quant-ph], 2017.
[5]
O. Cohen, "Classical Teleportation of Classical State," arXiv:quant-ph/0310017, 2003.
