Mermin-Klyshko Inequality

Consider a non-signaling scenario with a single source and \(n\) measurement devices. The Mermin-Klyshko (MK) inequality bounds the classical correlations in such scenarios but yields a quantum violation that increases exponentially with \(n\) [Mermin1990], [Ardehali1992], [Belinsky1993_1], [Belinsky1993_2]. The inequality is expressed iteratively as [Cabello2002],

\[I_{\text{MK}} := M_n = M_{n-1}(B^n_0 + B^n_1) + M'_{n-1}(B^n_0 - B^n_1) \leq 2^{n-1}\]

where \(B^j_{y_j}\) is the dichotomic observable for the \(j^{th}\) measurement node and \(y_j\in\{0,1\}\). Additionally, the term \(M'_j\) is derived from \(M_j\) simply by exchanging the subscripts \(0 \leftrightarrow 1\) describing the measurement input at each node. The inequality construction is initialized with \(M_1 = B^1_0\) and \(M'_1 = B^1_1\). The inequality \(I_{\text{MK}}\) contains \(2^{2 \lfloor n/2 \rfloor}\) correlator terms resulting in an exponential computational complexity in the number of quantum circuits to execute.

The classical bound for the form of the inequality \(I_{\text{MK}}\) is \(2^{n-1}\) while the quantum bound is \(2^{3(n-1)/2}\) [Cabello2002]. A remarkable feature of the Mermin-Klyshko inequality is that the separation between the quantum and classical bounds grows exponentialy with \(n\).

qnetvo.mermin_klyshko_cost_fn(ansatz, **qnode_kwargs)

Constructs an ansatz-specific cost function based upon the Mermin-Klyshko (MK) inequality.

Parameters:

• ansatz (NetworkAnsatz) – The network ansatz for which to apply the MK inequality.

• qnode_kwargs – Keyword arguments passed through to the qnode constructors.

Returns:

A cost function, cost(*network_settings), that evaluates \(-I_{\text{MK}}\) for the supplied network settings.

Return type:

Function

qnetvo.mermin_klyshko_inputs_scalars(n)

Helper function for handling the algebra of combining correlator terms in the Mermin-Klyshko (MK) inequality.

This function supports mermin_klyshko_cost_fn().

Parameters:

n (Int) – The number of measurement nodes in the scenario.

Returns:

The first element of the returned tuple is a list of measurement inputs \(y_j\in\{0,1\}\) for the MK correlator terms. The second element of the returned tuple is a list of scalar multipliers for each correlator term.

Return type:

Tuple(List[Int], List[Int])

qnetvo.mermin_klyshko_classical_bound(n)

The classical bound for the Mermin-Klyshko inequality is \(2^{n-1}\).

Parameters:

n (Int) – The number of measurement nodes.

Returns:

The classical bound.

Return type:

Float

qnetvo.mermin_klyshko_quantum_bound(n)

The quantum bound for the Mermin-Klyshko inequality is \(2^{3(n-1)/2}\).

Parameters:

n (Int) – The number of measurement nodes.

Returns:

The quantum bound.

Return type:

Float

References

[Mermin1990]

Mermin, N. David. “Extreme quantum entanglement in a superposition of macroscopically distinct states.” Physical Review Letters 65.15 (1990): 1838.

[Ardehali1992]

Ardehali, Mohammad. “Bell inequalities with a magnitude of violation that grows exponentially with the number of particles.” Physical Review A 46.9 (1992): 5375.

[Belinsky1993_1]

Belinsky, A. V., and D. N. Klyshko. “A modified N-particle Bell theorem, the corresponding optical experiment and its classical model.” Physics Letters A 176.6 (1993): 415-420.

[Belinsky1993_2]

Belinski, A. V., and David Nikolaevich Klyshko. “Interference of light and Bell’s theorem.” Physics-Uspekhi 36.8 (1993): 653.

[Cabello2002] (1,2)

Cabello, Adan. “Bell’s inequality for n spin-s particles.” Physical Review A 65.6 (2002): 062105.