Quantum bounds on Bell inequalities

Károly F. Pál and Tamás Vértesi

Here we give details of our results on maximum quantum values of tight bipartite Bell inequalities with two-outcome measurements. All such inequalities with up to five measurement settings known to date (13/02/2009) have been considered. The main results have been published in arXiv:0810.1615. The details, including the measurement operators and the state vectors can be downloaded from here case by case, or all files together as a compressed tar archive from here.

Bell inequalities

Let Alice and Bob be two experimenters, and let each of them has a physical object. The objects may have interacted in the past, they may even have been prepared together with some procedure that may have included stochastic elements. Alice performs one of the , and Bob performs one of the  local two-otcome measurements allowed to choose from on her and his object, respectively. For each measurement setting let the measurament value be 1 for one outcome and 0 for the other one.  Let us introduce the following notations:

•  Probability of Alice getting measurement value one when performing measurement  (equal to the expectation value for the corresponding measurement).
•  Probability of Bob getting measurement value one when performing measurement .
•  Probability of both Alice and Bob getting measurement value one when performing measurement  and, respectively (equal to the expectation value for the product of the corresponding measurements).

Let us call a linear combination of these probabilities a Bell expression:

.

Two Bell expressions are equivalent if one can be derived from the other one by permuting the measurement settings, changing the assignment of measurement values to measurement outcomes and/or swapping the roles of Alice and Bob.

By making two very reasonably looking assumptions, one can derive bounds on the value a Bell expression may take. One assumption is the so called counterfactual definiteness, when we suppose that had someone done a different measurement on a physical system than one have actually performed, one would still have got some definite result (in this case either 1 of 0). By this we also implicitly suppose the freedom of choice, namely that one actually could have chosen to do a different measurement had one decided to do so. The other assumption is locality,  which means that the actual measurement value of one party can not depend on what the other party does with his/her object at a distant place, specifically which  measurent he/she performs on it.

The expression derived from these conditions is called a Bell inequaliy. A Bell inequalitiy is necessarily obeyed if the objects behave according to classical physics, or at least if they may be described by a local realistic model.  The set of probabilities  allowed by the condition above constitute a polytope in the -dimensional space, an object with corners and faces. The corners are given by cases with deterministic setting, when the outcome of each measurement is always the same, that is each is either 0 or 1. A Bell inequality expresses the fact that the whole polytope is on one side of a hyperplane. A tight Bell inequality corresponds to a face of the polytope.

The importance of the Bell inequalities lies in the fact that quantum mechanics may violate them.  The shape of the region in the space of probabilities allowed by quantum mechanics is less known than the one allowed by classical physics. The maximum quantum values of Bell expressions (or the maximum violations of Bell inequalities) carries information about this. The maximum value may  depend on the dimensionality of the Hilbert spaces describing the systems to be measured. If this is the case, the Bell expression is called a dimension witness, because if one experiences a value larger than allowed with systems of a certain dimensionality, than one can be sure that the Hilbert spaces of the actual systems has a higher dimensionality than  that. The notion was introduced by Brunner et al. (Phys.Rev.Lett. 100 (2008) 210503; arXiv:0802.0760). Concrete examples with two-outcome measurements were given first by Vértesi and Pál (Phys.Rev A77 (2008) 042106; arXiv:0712.4225, Phys.Rev A77 (2008) 042105; arXiv:0712.4320).

A general procedure based on semidefinite programming ­­– among other things – makes it possible to derive a hierarchy of upper limits for the maximum quantum values of Bell expressions (Navascués et al., Phys.Rev.Lett. 98 (2007) 010401; arXiv:quant‑ph0607119, New J.Phys 10 (2008) 073013; arXiv:0803.4290). Although the computational requirement grows quite fast as we go higher levels in this hierarchy, our experience is that even for the largest cases we considered here the upper limit has already reached the exact value at a well affordable level for the majority of the inequalities. For a short description of the procedure specified to the present case, and the explanation of the (partial) levels  in the hierarchy we considered see arXiv:0810.1615.

Determination of the maximum quantum values

To determine the maximum quantum value for the type of Bell expressions we considered here, we can confine ourselves to pure states for the system incorporating the subsystems of Alice and Bob, and projective measurements. Then

, , and ,

Where  and  are the observables corresponding to the allowed measurements of Alice and Bob, respectively, acting in  and , the Hilbert spaces of the respective subsystems.  and  are the unit operators in  and , respectively, and  is the state vector of the composite system. The measurement operators are projection operators having eigenvalues of 1 and 0. We will call the dimensionality of a projection operator the dimensionality of the subspace it projects to, that is the number of eigenvalues 1 (equal to the trace of its matrix).

We have calculated the maximum quantum values by determining the measurent operators and the state vector via numerical optimization. The details of the procedure and the way of parametrizing the operators for the optimization are given in arXiv:0810.1615.

The dimensionalities of the component Hilbert spaces and the dimensionalities of the measurement operators we have done calculations with are the following:

·        All combinations of  1, 0 and 2-dimensional projectors in 2-dimensional real and complex spaces. The 0 and the 2-dimensional projectors are the zero and the unity operators in those spaces, corresponding to degenerate measurements with the same outcome every time for any system, measurements need not be performed at all.

·        2-dimensional projectors in 4-dimensional real and complex component spaces. Such a calculation also covers the 3-dimensional cases with 1 and 2-dimensional projectors, and the 2-dimensional cases with projectors of any dimensionality (for explanation see arXiv:0810.1615).

·        3-dimensional projectors in 6-dimensional real and complex component spaces. Such a calculation also covers the 5-dimensional cases with 2 and 3-dimensional projectors, the 4-dimensional cases with 1, 2 and 3-dimensional projectors, and all smaller dimensional cases with projectors of any dimensionalities.

·        4-dimensional projectors in real 8-dimensional component spaces.  Such a calculation also covers the 7-dimensional cases with 3 and 4-dimensional projectors, the 6-dimensional cases with 2, 3 and 4-dimensional projectors, the 5-dimensional cases with 1, 2, 3 and 4-dimensional projectors, and all smaller dimensional cases with projectors of any dimensionalities.

Although the higher-dimensional calculations cover the lower dimensional ones, and the complex calculations cover the real ones of the same dimensionalities, it is still worth doing the smaller calculations as well. They have much less parameters, therefore they are much faster, global optimum is more reliably found, and the numerical accuracy of the results are very often better. Whenever the upper limit for the maximum value is reached by the result obtained in lower dimensions, which is the case for most inequalities we considered, the higher dimensional calculation need not be done at all.

The content of the files

Each file contains results for one Bell inequality. The the name of the file is derived from the notation of the inequality. We follows the notations of arXiv:0810.1615. The original sources of the inequalities are also cited there. 129 inequalities with 4 (2-valued) measurement settings for each party, denoted as 4422_ Jx’ (corresponds to , where x=1,…,129, x’ denotes 3 digits with the value of x completed with leading zeros, if necessary) were introduced in arXiv:0810.1615. For purely technical reasons the inequalities given here are different from, but equivalent with  the ones given in the paper.

In the files we give:

•  and  (same as first two characters of file name).
• The classical bound .
• The maximum quantum value found (with different component Hilbert spaces).
• The state vector in Schmidt basis to get the maximum value above.
• The matrices of the measurement operators of Alice and Bob in Schmidt basis to get the maximum value above.  The dimensionalities of the subspaces the operators project to are also given (equal to the traces of the matrices).
• The largest quantum value found repeated, with the type of the smallest component Hilbert spaces to achieve that value.
• A series of upper limits calculated at increasing (partial) levels, until either the upper limit reaches the value found numerically (in this case the best result found numerically corresponds to the exact quantum limit), or until we could not afford to go to a higher level. For the explanation of the levels considered here see arXiv:0810.1615.

The maximum quantum value and the corresponding state vector and operators are given in component spaces:

• Real qubits (2-dimensional spaces) without degenerate measurement (all measurement operators are 1-dimensional projectors).
• Real qubits allowing degenerate measurements, if larger value is achieved this way.
• Complex qubits without degenerate measurements, if larger value is achieved than with real qubits without degenerate measurements. If this value is smaller than the one acheved with real qubits allowing degenerate measurements, this fact is noted.
• Complex qubits allowing degenerate measurements, if a larger value is achieved this way than previously.
• The best solution found, if it has been achieved in higher than two-dimensional spaces.