No communication theorem

The no-communication theorem is a sentence from quantum information theory that states that measurements on a quantum mechanical subsystem cannot be used to transfer information to another subsystem. This is true even if the system is in an entangled state . With the help of Bell's inequality, violations of local realism could be shown. The no-communication theorem is now important to understand that these non-local correlations can still not be used to communicate faster than light. Based on Einstein's objection of the “spooky action at a distance” ( EPR paradox ), the no-communication theorem could roughly be summarized as “there is no spooky long-distance communication ”. The causality is consequently not violated by this type of measurement.

Explanation using the example of entangled photons

As an example, we consider a photon pair (or electron pair) as a system that is prepared in an entangled state - more precisely in the singlet state . The two particles are now sent to two spatially distant observers, Alice and Bob. Alice rotates her detector by the angle α and Bob by the angle β (to the z-axis). The probabilities of the measurements of | ↑⟩ and | ↓⟩ can be calculated using quantum mechanics to give the following results: ${\displaystyle |\psi \rangle ={\tfrac {1}{\sqrt {2}}}\left(|{\mathord {\uparrow }}{\mathord {\downarrow }}\rangle -|{\mathord {\downarrow }}{\mathord {\uparrow }}\rangle \right)}$

	Alice measures ↑	Alice measures ↓
Bob measures ↑	${\displaystyle p_{\uparrow \uparrow }={\frac {1}{2}}\sin ^{2}\left({\frac {\alpha -\beta }{2}}\right)}$	${\displaystyle p_{\downarrow \uparrow }={\frac {1}{2}}\cos ^{2}\left({\frac {\alpha -\beta }{2}}\right)}$
Bob measures ↓	${\displaystyle p_{\uparrow \downarrow }={\frac {1}{2}}\cos ^{2}\left({\frac {\alpha -\beta }{2}}\right)}$	${\displaystyle p_{\downarrow \downarrow }={\frac {1}{2}}\sin ^{2}\left({\frac {\alpha -\beta }{2}}\right)}$

Regarding the above table, it should be noted that the probabilities only depend on the difference angle α - β, because there is no _specific spatial direction, and that p _↑↑ + p _{↓ ↑} + p _{↑ ↓} + p _↓↓ = 1 for the sum of all probabilities . These probabilities are confirmed very well in the experiment, even if the detector angles α and β are selected while the particles are flying apart. If the particles are far enough apart, no signal moving at the speed of light or slower can communicate information (regarding detector angle or measurement result) to the other subsystem (or back to the source) in time. It can thus be shown that there are correlations between the measurement results for Alice and Bob, which cannot be explained by a common past of the photon pair. For details, see Bell's inequality .

If we consider the probability for Alice to measure | ↑>, we get:

${\displaystyle p_{\uparrow }^{A}=p_{\uparrow \uparrow }+p_{\uparrow \downarrow }={\frac {1}{2}}\sin ^{2}\left({\frac {\alpha -\beta }{2}}\right)+{\frac {1}{2}}\cos ^{2}\left({\frac {\alpha -\beta }{2}}\right)={\frac {1}{2}}}$

For almost all angles Alice's measurement result is (stochastically) dependent on Bob's measurement result, i.e. This means that the measurement results are correlated :

${\displaystyle p_{\uparrow }^{A}\cdot p_{\uparrow }^{B}\neq p_{\uparrow \uparrow }}$

Nevertheless, the probability is independent of β (i.e. not a function of β), i.e. i.e. Alice does not notice anything when Bob turns his detector. Since neither Alice nor Bob can choose whether to measure | ↑⟩ or | ↓⟩, the results for both appear completely random (each with probability ½) and they will only notice the correlations when they meet again after the experiment to compare their records. Neither of the two can consequently use any measurement on their subsystem for communication with the other subsystem. This is the essence of the no-communication theorem. ${\displaystyle p_{\uparrow }^{A}}$

General evidence

A more general proof considers a quantum mechanical state ψ over a Hilbert space H made up of two subsystems , each accessible to Alice and Bob. Furthermore, let Ω ^(A) and Ω ^{(B) be} observables that only act on H _A and H _B and have corresponding eigenvalues ​​λ _n : ${\displaystyle H=H_{A}\otimes H_{B}}$

${\displaystyle \Omega ^{(A)}|n^{(A)}\rangle =\lambda _{n}^{(A)}|n^{(A)}\rangle }$and analogously for Ω ^(B)

Because Alice's observable Ω ^(A) does not affect Bob's subsystem, the effect on the entire system can be described as , where I ^{(B) is} the identity matrix (or the identical mapping). A measurement of Ω _A now causes the entire system to collapse into an eigenstate of Ω _A. If one calculates the probabilities of obtaining the measured values ​​for Bob using Born's rule for this eigenstate , then these are the same as for the original state ψ. This means that Bob cannot tell any difference from the statistics of his measured values ​​whether Alice measured or not. Details of this calculation can be found in the article Quantum Entanglement . ${\displaystyle \Omega _{A}=\Omega ^{(A)}\otimes I^{(B)}}$ ${\displaystyle \lambda _{n}^{(B)}}$

An even more general proof considers not only pure quantum mechanical states, but even ensembles of states that can be represented with a density operator . Every local operation (including measurement) that Alice can perform on her subsystem can therefore be represented algebraically as follows: ${\displaystyle {\hat {\rho }}}$

${\displaystyle P({\hat {\rho }})=\sum _{k}(A_{k}\otimes I^{(B)})^{*}\ {\hat {\rho }}\ (A_{k}\otimes I^{(B)})}$

With

${\displaystyle \sum _{k}A_{k}\,A_{k}^{*}=I^{(A)}}$

Now one would like to show that Bob cannot statistically differentiate whether Alice performed the operation or not. Because all possible measurement results and probabilities for Bob's upper servables can be calculated by means of trace formation , one has to show this . It is a partial track because it is only summed over the subspace H _A. For details of this rather technical calculation, reference is made to the source, where the influence of a relativistic view is also discussed. ${\displaystyle P({\hat {\rho }})}$ ${\displaystyle {\hat {\rho }}}$ ${\displaystyle \operatorname {Spur} _{H_{A}}(P({\hat {\rho }}))=\operatorname {Spur} _{H_{A}}({\hat {\rho }})}$ ${\displaystyle \operatorname {Spur} _{H_{A}}}$

Remarks

• In quantum field theory , too, the no-communication theorem can be shown on the assumption that Alice and Bob are spatially separated.
• If the no-cloning theorem did not apply, faster than light communication would be possible. This should be shown using the example of the singlet state above : If Alice wants to send a "0", she carries out a measurement. If it wants to send a "1", it does not measure. When it measures, the superposition disappears and the wave function collapses into its own state, either or . Bob then makes many copies of this condition and measures himself. If he always gets the same result (always ↑ or always ↓), he had the already collapsed wave function and knows that Alice wanted to send a "0". If he measures ↑ or ↓ each with a probability of ½, then Alice wanted to send a “1”. ${\displaystyle |\psi \rangle ={\tfrac {1}{\sqrt {2}}}\left(|{\mathord {\uparrow }}{\mathord {\downarrow }}\rangle -|{\mathord {\downarrow }}{\mathord {\uparrow }}\rangle \right)}$ ${\displaystyle |{\mathord {\uparrow }}{\mathord {\downarrow }}\rangle }$ ${\displaystyle |{\mathord {\downarrow }}{\mathord {\uparrow }}\rangle }$
• If Born's rule did not apply, faster than light communication would be possible.
• If nonlocal interaction between A and B is also allowed, then the above proof is not valid. However, this can be remedied if certain commutator relations are assumed. ${\displaystyle P({\hat {\rho }})}$

Individual evidence

1. ↑ MJW Hall: Imprecise measurements and non-locality in quantum mechanics . In: Phys. Lett. A . 125, No. 2-3, 1988, pp. 89-91. doi: 10.1016 / 0375-9601 (87) 90127-7 .
2. ↑ Giancarlo Ghirardi et al .: Experiments of the EPR Type Involving CP-Violation Do not Allow Faster-than-Light Communication between Distant Observers . In: Europhys. Lett. . 6, No. 2, 1988, pp. 95-100.
3. ↑ M. Florig, SJ Summers: On the statistical independence of algebras of observables . In: J. Math. Phys. . 38, No. 3, 1997, pp. 1318-1328. doi : 10.1063 / 1.531812 .
4. ^ John S. Bell: Bertlmann's socks and the nature of reality . In: Le Journal de Physique Colloques . 1981, pp. C2-41-C2-62. doi: 10.1051 / jphyscol: 1981202 . Equation (4).
5. ↑ ^{a b} A. Peres, D. Terno: Quantum Information and Relativity Theory . In: Rev. Mod. Phys. . 76, 2004, pp. 93-123. arxiv : quant-ph / 0212023 . doi : 10.1103 / RevModPhys.76.93 .
6. ↑ Phillippe H. Eberhard, Ronald R. Ross: Quantum field theory cannot provide faster than light communication . In: Foundations of Physics Letters . 2, No. 2, 1989, pp. 127-149. doi: 10.1007 / BF00696109 .
7. ↑ Wojciech Hubert Zurek: Environment-assisted invariance, entanglement, and probabilities in quantum physics . In: Physical Review Letters . 90, No. 12, 2003, p. 120404. arxiv : quant-ph / 0211037 . doi: 10.1103 / PhysRevLett.90.120404 .
8. ^ KA Peacock, B. Hepburn: Begging the Signaling Question: Quantum Signaling and the Dynamics of Multiparticle Systems . In: Proceedings of the Meeting of the Society of Exact Philosophy . 1999. arxiv : quant-ph / 9906036 .