Nonlocality is one of the rather rare topics that is interesting in fundamental physics while at the same time already having the potential for applications. As such, it opens up at least five interesting areas of research.
First, it is fair to say that quantum nonlocality and its implications are at this point not yet fully understood. For example, while entanglement and nonlocality are strongly linked, the exact nature of the link is not clear. Not all entangled states can produce nonlocal correlations and maximally entangled states are not always optimal when it comes to exhibiting nonlocality. Partially entangled states have, among other things, a higher resistance to certain types of noise and seem stronger when measurement independence is relaxed. Furthermore, we are still discovering implications of exhibiting different amounts of nonlocality, as shown by our work on k-producibility. Investigating such properties is an interesting area of current research.
Second, in order for us to harvest the inherent power of nonlocality that we have already discovered and are still discovering in the first area, especially in privacy related tasks, it is necessary to develop practical protocols that accomplish these tasks. Most nonlocality protocols, except for entanglement verification, are only starting to move from theoretical possibility towards potential practicality. They often require unrealistic resources, like a large amount of quantum systems, all at spacelike separation, perfect, potentially even multipartite, quantum states with very little or even zero noise tolerance and/or perfect measurement apparatuses1. Note that this is not to say that the protocols assume that these are the under-lying resources, they just do not work unless everything is perfect or very nearly perfect. Due to the complex nature of these privacy tasks, it is to be expected that the corresponding protocols have not yet fully matured. This may be one of the most interesting current areas of research since device independence removes many problems that standard privacy protocols may encounter. On a smaller scale, we hope that our work on measurement dependence can be of use for protocols for randomness generation. The (`, h) assumption was originally strongly inspired by the so-called Santha-Vazirani assumption given by
1 2 −
≤P(xn|λ, x1. . . xn−1)≤ 1
2 −
∀n
1Some may function with currently achievable levels of noise; unfortunately their figure of merit is often dependent on it. A device independent quantum key distribution protocol may not be that useful if it produces only one bit of key per hour.
and the min-entropy assumption given byHmin(X~|Λ)≥H and implying P(~x|λ)≤2−H ∀~x, λ.
These assumptions underlie many of the current randomness amplification and ex-traction protocols. Whether or not the (`, h) measurement dependent local polytope will prove useful remains an open question for future work.
One of the biggest issues when it comes to implementing any of the already-existing protocols is that the requirements to work truly device independently are very strict. There can be no post-selection, all possible communication must be prevented by e.g. working in space-like separation and so forth. This is why the more interesting field may be that of semi-device independence, where some care-ful additional assumptions are made. Practical semi-device independent protocols need to fulfill two conditions: on the one hand, they need to be easier to implement than their device independent counterpart, on the other hand, whichever additional assumption is made needs to be justifiable in an implementation.
The third area goes hand in hand with the second: even the nicest protocol is useless if it is not implemented. Many quantum physicists are already working on pushing our practical limits; after all, a fully functional quantum computer would be very useful for problem solving in today’s world. Using this work and developing it further in order to implement nonlocality-based protocols would be a big achieve-ment. In order to succeed, good, frequent communication between theorists devising protocols and experimentalists familiar with the current practical limitations is re-quired.
Fourth, nonlocality challenges our basic intuitions, leading to the area of fun-damental physics. Fortunately, many do not take this challenge lightly. It is, after all, not clear whether our local view of the world has to be abandoned; it is per-fectly reasonable to question other underlying assumptions that are made or even simply our interpretation of quantum mechanics. When it comes to interpretations, quantum physicists are already split among traditional quantum mechanics (such as considered within this thesis) Bohmian mechanics, many-world interpretations and QBism2. While explorations on this interpretational level are definitely fascinat-ing and useful in providfascinat-ing new perspectives, there is still room for more targeted approaches. Our work on measurement dependence in Chapter 3 falls into this cate-gory, showing that relaxations in the form of (`, h) measurement dependence do not make the weirdness of nonlocality go away for` >0 orh ≤ 13. The case of `= 0 and h ∈ [13,1[ is still an open problem and worth investigating. It would be even more interesting if new ways to restore the principle of locality could be found, whether by doing away with the concept of probability in physics or by restructuring our current vision of spacetime.
Finally, any work in fundamental theoretical physics has a high chance of ty-ing in with fundamental experimental physics. The realization of loophole free Bell tests in 2015 was a milestone in physics. Experiments will not be able to distinguish between different interpretations of quantum mechanics since they give all the same
2We do not claim that this list of interpretations is in any way complete.
predictions. However, they can and should explore all corners of quantum mechan-ics in the hopes of finding ways to go beyond it. As such, an experiment that does not require full measurement independence while covering all other loopholes should be conducted. On a broader scale however, one of the next big steps is the explo-ration of quantum mechanics at a macroscopic scale. Performing experiments that demonstrate or fail to demonstrate quantum effects at a macroscopic scale would mark a big step towards understanding why we do not observe them in daily life and whether gravity in particular has anything to do with it. Given the link between quantum entanglement and nonlocality, the latter may prove useful when getting into the realm where not only states but also measurement apparatuses are difficult to understand. Admittedly, at this stage, this is probably wishful thinking.
In summary, the research in and around nonlocality is far from finished. Any new advance in quantum information theory, whether it be theoretical or experimental gives us new areas to look at in terms of nonlocality. Many implications of nonlo-cality are not yet fully understood, and the ones that are have not yet found their way into applications. Exploring quantum nonlocality improves our understanding of nature while at the same time suggesting practical uses, making it a valuable and rewarding field of research.
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Facets of MDL polytopes
Here, we present the parametrized facets of theMDL(2,2,2)(`,1−3`),MDL(2,2,2)(1− 3h, h) as well as ISMDL(∈,∈,∈((1−h,1−h),(h, h)). We give one representative of each family. Remember that this list is only conjectured to be complete, see Algorithm 3.1. Each column after the first contains the coefficients βabxy corre-sponding to the probabilities in the first column. The inequalities are such that P
abxyβabxyP(abxy)≤0.
The facets ofMDL(2,2,2)(`,1−3`) can be found in Table A.1, those ofMDL(2,2,2)(1− 3h, h) in Table A.2 and those of ISMDL(∈,∈,∈((1−h,1−h),(h, h)) in Table A.3.
L1L2L3L4L5L6L7L8 P(0000)9l3−7l2+2l−9l3 3l−1+8l2 3l−1−2l 3l−19l3−8l2+2l9l3−8l2+2l9l3−9l2+2ll3−2l2+l2l3−3l2+l2l3−3l2+l P(1000)9l3−6l2+ll−3l29l3−6l2+l9l3−6l2+l9l3−6l2+ll300 P(0100)9l3−9l2+2l2l−3l29l3−9l2+2l9l3−9l2+2l9l3−6l2+ll3−3l2+l6l3−5l2+l3l3−4l2+l P(1100)9l3−24l2+13l−2−3l2+4l−19l3−15l2+7l−19l3−15l2+7l−19l3−15l2+7l−1l3−9l2+6l−1−9l2+6l−19l3−15l2+7l−1 P(0010)9l3−15l2+7l−1−3l2+4l−19l3−15l2+7l−19l3−15l2+7l−19l3−12l2+6l−1l3−l218l3−21l2+8l−118l3−21l2+8l−1 P(1010)9l3−6l2+ll−3l29l3−6l2+l9l3−6l2+l9l3−15l2+7l−1l300 P(0110)9l3−9l2+2ll−3l29l3−6l2+l9l3−6l2+l9l3−15l2+7l−1l36l3−5l2+l9l3−6l2+l P(1110)9l3−8l2+2l2l−3l29l3−9l2+2l9l3−9l2+2l9l3−6l2+ll3−3l2+l4l3−4l2+l3l3−4l2+l P(0001)9l3−18l2+11l−2−3l2+5l−29l3−21l2+12l−29l3−24l2+13l−29l3−15l2+7l−1l3−4l2+4l−16l3−11l2+6l−1−9l3−3l2+5l−1 P(1001)9l3−15l2+7l−1−3l2+7l−29l3−15l2+7l−19l3−6l2+l9l3−18l2+8l−1l3−6l2+5l−109l3−6l2+l P(0101)9l3−21l2+12l−2−3l2+7l−29l3−24l2+13l−29l3−12l2+6l−19l3−15l2+7l−1l3−6l2+5l−13l3−10l2+6l−16l3−11l2+6l−1 P(1101)9l3−6l2+ll−3l29l3−6l2+l9l3−6l2+l9l3−6l2+ll39l3−6l2+l0 P(0011)9l3−9l2+2l−9l3 3l−1+8l2 3l−1−2l 3l−19l3−9l2+2l9l3−6l2+l9l3−9l2+2ll3−2l2+l6l3−5l2+l9l3−6l2+l P(1011)9l3−6l2+l2l−3l29l3−6l2+l9l3−15l2+7l−19l3−6l2+ll3−3l2+l027l3−27l2+9l−1 P(0111)9l3−6l2+ll−3l29l3−6l2+l9l3−18l2+8l−19l3−5l2+ll39l3−6l2+l2l2−6l3 P(1111)9l3−15l2+7l−1−3l2+4l−19l3−15l2+7l−19l3−15l2+7l−19l3−6l2+ll3−l227l3−27l2+9l−10 L9L10L11L12L13L14L15L16 P(0000)l3−3l2+l4l3−4l2+l4l3−4l2+l4l3−4l2+l4l3−4l2+l3l3−4l2+l3l3−4l2+l3l3−4l2+l P(1000)00000000 P(0100)6l3−5l2+l6l3−5l2+l6l3−5l2+l6l3−5l2+l6l3−5l2+l9l3−6l2+l9l3−6l2+l6l3−5l2+l P(1100)9l3−15l2+7l−118l3−21l2+8l−118l3−21l2+8l−118l3−21l2+8l−118l3−21l2+8l−118l3−21l2+8l−118l3−21l2+8l−115l3−20l2+8l−1 P(0010)18l3−21l2+8l−118l3−21l2+8l−127l3−27l2+9l−127l3−27l2+9l−127l3−27l2+9l−127l3−27l2+9l−127l3−27l2+9l−10 P(1010)0000000l3 P(0110)9l3−6l2+l09l3−6l2+l9l3−6l2+l9l3−6l2+l9l3−6l2+l9l3−6l2+ll2−3l3 P(1110)3l3−4l2+l6l3−5l2+l6l3−5l2+l6l3−5l2+l6l3−5l2+l2l2−6l36l3−5l2+l6l3−5l2+l P(0001)3l3−10l2+6l−112l3−16l2+7l−13l3−10l2+6l−1−9l2+6l−16l3−11l2+6l−16l3−11l2+6l−13l3−10l2+6l−16l3−11l2+6l−1 P(1001)0027l3−27l2+9l−1018l3−21l2+8l−19l3−15l2+7l−118l3−21l2+8l−19l3−15l2+7l−1 P(0101)−9l2+6l−19l3−15l2+7l−1−9l2+6l−112l3−16l2+7l−1−9l2+6l−19l3−15l2+7l−19l3−15l2+7l−19l3−15l2+7l−1 P(1101)9l3−6l2+l9l3−6l2+l000000 P(0011)6l3−5l2+l6l3−5l2+l6l3−5l2+l9l3−6l2+l7l3−5l2+l8l3−5l2+l7l3−5l2+l3l3−4l2+l P(1011)009l3−6l2+l06l3−5l2+l3l3−4l2+l6l3−5l2+l6l3−5l2+l P(0111)9l3−6l2+l9l3−6l2+l9l3−6l2+ll2−3l39l3−6l2+l9l3−6l2+l9l3−6l2+l0 P(1111)27l3−27l2+9l−127l3−27l2+9l−1000003l3−l2 L17L18L19L20L21L22L23L24 P(0000)8l3−5l2+l7l3−5l2+l7l3−5l2+l7l3−5l2+l6l3−5l2+l6l3−5l2+l2l−3l22l−3l2 P(1000)09l3−6l2+l09l3−6l2+l00l−3l2l−3l2 P(0100)6l3−5l2+l6l3−5l2+l6l3−5l2+l6l3−5l2+l9l3−6l2+l9l3−6l2+ll−3l2l−3l2 P(1100)18l3−21l2+8l−1027l3−27l2+9l−1027l3−27l2+9l−127l3−27l2+9l−1−3l2+4l−1−3l2+4l−1 P(0010)027l3−27l2+9l−1027l3−27l2+9l−103l3−l2−3l2+7l−2−3l2+4l−1 P(1010)9l3−6l2+l9l3−6l2+l9l3−6l2+l9l3−6l2+l00−3l2+4l−1−3l2+4l−1 P(0110)6l3−5l2+l09l3−6l2+l0l2−3l30−3l2+4l−1l−3l2 P(1110)7l3−5l2+l6l3−5l2+l6l3−5l2+l6l3−5l2+l9l3−6l2+l9l3−6l2+ll−3l2l−3l2 P(0001)6l3−11l2+6l−118l3−21l2+8l−13l3−10l2+6l−112l3−16l2+7l−112l3−16l2+7l−19l3−15l2+7l−1−3l2+4l−1−3l2+4l−1 P(1001)18l3−21l2+8l−1018l3−21l2+8l−118l3−21l2+8l−118l3−21l2+8l−115l3−20l2+8l−1l−3l2l−3l2 P(0101)12l3−16l2+7l−13l3−l29l3−15l2+7l−118l3−21l2+8l−118l3−21l2+8l−118l3−21l2+8l−1l−3l2−3l2+4l−1 P(1101)09l3−6l2+l00002l−3l2l−3l2 P(0011)6l3−5l2+l06l3−5l2+l6l3−5l2+l6l3−5l2+l6l3−5l2+l2l−3l2l−3l2 P(1011)9l3−6l2+l18l3−21l2+8l−19l3−6l2+l010l3−6l2+l9l3−6l2+ll−3l2l−3l2 P(0111)015l3−20l2+8l−1000l3l−3l2l−3l2 P(1111)27l3−27l2+9l−19l3−15l2+7l−127l3−27l2+9l−118l3−21l2+8l−100−3l2+4l−1l−3l2 TableA.1:ConjecturedfacetsforMDL(2,2,2)(`,1−3`).
L25L26L27L28L29L30L31L32L33L34L35 P(0000)2l−3l22l−4l22l−4l22l−5l22l−5l2ll−l2l−l2l−l2l−l2l−l2 P(1000)l−3l20000000000 P(0100)l−3l2l−3l2l−3l2l−3l2l−3l2l−3l2l−2l2l−2l2l−2l2l−3l2l−3l2 P(1100)−3l2+4l−1−15l2+11l−2−3l2+4l−1−6l2+5l−1−6l2+5l−1−3l2+4l−1−6l2+5l−1−9l2+6l−1−4l2+4l−1−6l2+5l−1−6l2+5l−1 P(0010)−3l2+4l−1−6l2+5l−1−6l2+5l−1−9l2+6l−1−9l2+6l−1−6l2+5l−100−6l2+5l−1−9l2+6l−1−9l2+6l−1 P(1010)−3l2+4l−1000000l2000 P(0110)l−3l2l−3l2l−3l2l−3l2l−3l2l−3l2l2l2l−2l2l−3l2l−3l2 P(1110)l−3l22l−5l2l−l2l−2l2l−2l2l−l2l−2l2l−2l2l2l−2l2l−2l2 P(0001)−3l2+4l−1−12l2+10l−2−12l2+10l−2−15l2+11l−2−15l2+11l−23l−1−2l2+3l−1−2l2+3l−1−2l2+3l−1−3l2+4l−1−3l2+4l−1 P(1001)l−3l200000−3l2+4l−1−3l2+4l−1−3l2+4l−100 P(0101)l−3l2−3l2+4l−1−3l2+4l−1−3l2+4l−1−6l2+5l−1−3l2+4l−1−4l2+4l−1−3l2+4l−1−2l2+3l−1−3l2+4l−1−6l2+5l−1 P(1101)2l−3l2l−3l2l−3l20l−3l2l−3l20000l−3l2 P(0011)l−3l2l−2l2l−2l2l−3l2l−3l2l−2l2l−l2l−l2l−2l2l−3l2l−3l2 P(1011)l−3l200000l−2l2l−2l2l−l200 P(0111)−3l2+4l−1l−3l2l−3l2l−3l20l−3l200l−2l2l−3l20 P(1111)−3l2+4l−1−9l2+6l−1−9l2+6l−1−12l2+7l−1−9l2+6l−1−9l2+6l−103l2−l00−9l2+6l−1 L36L37L38L39L40L41L42L43L44L45L46 P(0000)l−l22l−2l22l−2l2l−2l2l−2l2l−2l2l−2l2l−2l2l−2l2l−2l2l−2l2 P(1000)0l−2l2l−2l200000000 P(0100)l−3l2l−2l2l−2l2l−3l2l−3l2l−3l2l−3l2l−3l2l−3l2l−3l2l−3l2 P(1100)−3l2+4l−1−2l2+4l−1−2l2+3l−1−9l2+6l−1−6l2+5l−1−9l2+6l−1−9l2+6l−1−9l2+6l−1−9l2+6l−1−9l2+6l−1−9l2+6l−1 P(0010)−6l2+5l−1−2l2+3l−1−2l2+3l−10−9l2+6l−1−6l2+5l−1−9l2+6l−1−9l2+6l−1−12l2+7l−100 P(1010)0−2l2+4l−1−2l2+3l−1000000l−3l2l−3l2 P(0110)l−3l2−2l2+4l−1−2l2+4l−1l−3l2l−3l2−9l2+6l−100l−3l2l−3l2l−3l2 P(1110)l−2l2l−2l2l−2l2l−3l2l−3l2l−3l2l−3l2l−3l2l−3l2l−2l2l−2l2 P(0001)−3l2+4l−1−2l2+3l−1−2l2+3l−1−6l2+5l−1−3l2+4l−1−6l2+5l−1−3l2+4l−1−6l2+5l−1−6l2+5l−13l−1−3l2+4l−1 P(1001)0−2l2+4l−1−2l2+4l−1000−9l2+6l−100−6l2+5l−1−9l2+6l−1 P(0101)−3l2+4l−1−2l2+4l−1−2l2+3l−1−6l2+5l−1−6l2+5l−13l2−l−6l2+5l−1−6l2+5l−1−9l2+6l−1−6l2+5l−1−6l2+5l−1 P(1101)l−3l2l−2l2l−2l200l−3l200l−3l200 P(0011)l−3l22l−2l22l−2l20l−3l2l−2l2l−2l2l−3l20l−2l2l−3l2 P(1011)0l−2l2l−2l2000l−3l200l−3l20 P(0111)l−3l2l−2l2l−2l200l−3l2l−3l2l−3l23l2−l00 P(1111)−9l2+6l−1−2l2l−2l200−9l2+6l−100−9l2+6l−1−9l2+6l−1−9l2+6l−1 L47L48L49L50L51L52L53L54L55L56L57L58L59L60 P(0000)l−2l2l−2l2l−2l2l−2l2l−2l2l−2l2l−2l2l−2l2l−3l2lllll P(1000)l−3l2l−3l2000000−12l2+7l−1l−1l−1l−1ll−1 P(0100)l−3l2l−3l2l−3l2l−3l2l−3l2l−3l2l−3l2l−3l2l−3l2l−1l−1ll3l−1 P(1100)00−3l2+4l−1−6l2+5l−1−9l2+6l−1−6l2+5l−1−9l2+6l−13l2−l0l−1ll−1l0 P(0010)−6l2+5l−1−9l2+6l−1−6l2+5l−1−6l2+5l−10−9l2+6l−1−12l2+7l−100l−1l−1ll−12l−1 P(1010)−9l2+6l−10000000l−3l2llll−1l P(0110)−9l2+6l−10l−3l20l−3l2l−3l2l−3l2l20llll−10 P(1110)0l−3l2l−3l2l−3l2l−3l2l−3l2l−3l2l−3l2l2ll−1ll−12l−1 P(0001)−9l2+6l−1−9l2+6l−1−3l2+4l−1−6l2+5l−1−3l2+4l−1−3l2+4l−1−3l2+4l−1−3l2+4l−1−6l2+5l−1l−1l−1l−1ll−1 P(1001)00l−3l20−9l2+6l−1−9l2+6l−13l2−l−9l2+6l−1−9l2+6l−1lllll P(0101)3l2−l0−6l2+5l−1−6l2+5l−1−6l2+5l−1−3l2+4l−1−6l2+5l−1−6l2+5l−1−9l2+6l−1lll−1ll P(1101)0l−3l20l−3l200000ll−1ll0 P(0011)l−3l20l−3l2l−3l2l−3l2l−3l2l−3l2l−3l2l−3l2ll−1ll3l−1 P(1011)0−9l2+6l−1−9l2+6l−10l−3l2l−3l2l2l−3l20lllll P(0111)l−3l2−9l2+6l−10l−3l20l−3l2000llll0 P(1111)−12l2+7l−1−6l2+5l−10−9l2+6l−1000−12l2+7l−13l2−ll−1l−1lll−1 TableA.1:(Cont)ConjecturedfacetsforMDL(2,2,2)(`,1−3`).
L61L62L63L64L65L66L67L68L69L70L71L72L73L74 P(0000)lllllllllllll0 P(1000)3l−13l−1l−12l−13l−102l−13l−1003l−100−1 P(0100)ll0ll00l0l00l0 P(1100)000003l−1003l−100000 P(0010)ll3l−1002l−13l−103l−103l−13l−100 P(1010)llll03l−1ll00l000 P(0110)00003l−13l−100000000 P(1110)ll2l−13l−1003l−10l00000 P(0001)l−12l−1l−1l−12l−10l−12l−13l−12l−12l−13l−13l−10 P(1001)2l−10l03l−1l03l−103l−10000 P(0101)2l−1002l−12l−1002l−103l−1003l−10 P(1101)0l03l−10l00l00000 P(0011)l03l−10ll0lll0000 P(1011)l3l−1lll0l00l0000 P(0111)02l−10lll00000000 P(1111)3l−12l−12l−10002l−103l−102l−13l−100 TableA.1:(Cont)ConjecturedfacetsforMDL(2,2,2)(`,1−3`).
H1H2H3H4H5H6H7H8 P(0000)−h3+4h2−4h+1−9h3+18h2−11h+2−9h3+18h2−11h+2−9h3+21h2−12h+2−9h3+12h2−6h+1−9h3+12h2−6h+1−6h3+11h2−6h+1−6h3+11h2−6h+1 P(1000)−h3+6h2−5h+1−9h3+15h2−7h+1−9h3+24h2−13h+2−9h3+15h2−7h+1−9h3+6h2−h−9h3+15h2−7h+1−6h3+5h2−h0 P(0100)−h3+6h2−5h+1−9h3+21h2−12h+2−9h3+24h2−13h+2−9h3+24h2−13h+2−9h3+24h2−13h+2−9h3+15h2−7h+10−9h3+15h2−7h+1 P(1100)−h3−9h3+6h2−h−9h3+6h2−h−9h3+6h2−h−9h3+6h2−h−9h3+6h2−h−3h3+4h2−h0 P(0010)−h3+2h2−h−9h3+9h2−2h−9h3+8h2−2h−9h3+9h2−2h−9h3+18h2−8h+1−9h3+5h2−h−2h3+3h2−h−3h3+4h2−h P(1010)−h3+3h2−h−9h3+6h2−h−9h3+9h2−2h−9h3+6h2−h−9h3+15h2−7h+1−9h3+9h2−2h−6h3+5h2−h−6h3+5h2−h P(0110)−h3−9h3+6h2−h−9h3+6h2−h−9h3+6h2−h−9h3+6h2−h−9h3+6h2−h00 P(1110)−h3+9h2−6h+1−9h3+15h2−7h+1−9h3+15h2−7h+1−9h3+15h2−7h+1−9h3+15h2−7h+1−9h3+6h2−h3h3−h2−6h3+5h2−h P(0001)−h3+2h2−h−9h3+7h2−2h−9h3+8h2−2h−9h3+8h2−2h−9h3+8h2−2h−9h3+9h2−2h−6h3+5h2−h−3h3+4h2−h P(1001)−h3−9h3+6h2−h−9h3+6h2−h−9h3+6h2−h−9h3+6h2−h−9h3+6h2−h−6h3+11h2−6h+10 P(0101)−h3+3h2−h−9h3+9h2−2h−9h3+9h2−2h−9h3+9h2−2h−9h3+9h2−2h−9h3+6h2−h0−6h3+5h2−h P(1101)h2−h3−9h3+24h2−13h+2−9h3+15h2−7h+1−9h3+15h2−7h+1−9h3+15h2−7h+1−9h3+15h2−7h+1−9h3+15h2−7h+1−12h3+16h2−7h+1 P(0011)h2−h3−9h3+15h2−7h+1−9h3+15h2−7h+1−9h3+15h2−7h+1−9h3+15h2−7h+1−9h3+15h2−7h+1−3h3+4h2−h0 P(1011)−h3−9h3+6h2−h−9h3+6h2−h−9h3+6h2−h−9h3+6h2−h−9h3+15h2−7h+1−6h3+5h2−h3h3−h2 P(0111)−h3−9h3+9h2−2h−9h3+6h2−h−9h3+6h2−h−9h3+6h2−h−9h3+6h2−h0−4h3+4h2−h P(1111)−h3+3h2−h−9h3+8h2−2h−9h3+9h2−2h−9h3+9h2−2h−9h3+9h2−2h−9h3+18h2−8h+1−4h3+4h2−h−6h3+5h2−h H9H10H11H12H13H14H15H16 P(0000)−6h3+11h2−6h+1−6h3+11h2−6h+1−6h3+11h2−6h+1−6h3+11h2−6h+1−6h3+11h2−6h+1−6h3+11h2−6h+12h2−3h+1−3h3+10h2−6h+1 P(1000)00−18h3+21h2−8h+1−18h3+21h2−8h+1−9h3+15h2−7h+1−9h3+15h2−7h+13h2−4h+10 P(0100)9h3+3h2−5h+1−3h3+10h2−6h+19h2−6h+1−12h3+16h2−7h+1−9h3+15h2−7h+1−9h3+15h2−7h+13h2−4h+19h2−6h+1 P(1100)−9h3+6h2−h−9h3+6h2−h00000−9h3+6h2−h P(0010)6h3−2h2−6h3+5h2−h−7h3+5h2−h−6h3+5h2−h−8h3+5h2−h−3h3+4h2−hh2−h−6h3+5h2−h P(1010)00−6h3+5h2−h−9h3+6h2−h−3h3+4h2−h−6h3+5h2−h2h2−h0 P(0110)−9h3+6h2−h−9h3+6h2−h−9h3+6h2−h0−9h3+6h2−h00−9h3+6h2−h P(1110)−27h3+27h2−9h+1−27h3+27h2−9h+10−27h3+27h2−9h+10−15h3+20h2−8h+19h2−6h+1−27h3+27h2−9h+1 P(0001)−2h3+3h2−h−2h3+3h2−h−4h3+4h2−h−8h3+5h2−h−3h3+4h2−h−3h3+4h2−hh2−h−h3+3h2−h P(1001)00000000 P(0101)−3h3+4h2−h−6h3+5h2−h−6h3+5h2−h−6h3+5h2−h−9h3+6h2−h−6h3+5h2−h2h2−h−6h3+5h2−h P(1101)−9h3+15h2−7h+19h2−6h+1−18h3+21h2−8h+1−18h3+21h2−8h+1−18h3+21h2−8h+1h2−3h3h−3h2−9h3+15h2−7h+1 P(0011)−18h3+21h2−8h+1−18h3+21h2−8h+1−27h3+27h2−9h+10−27h3+27h2−9h+100−18h3+21h2−8h+1 P(1011)000−9h3+6h2−h03h3−h2−h20 P(0111)−9h3+6h2−h−6h3+5h2−h−9h3+6h2−h−6h3+5h2−h−9h3+6h2−h−h3−h2−9h3+6h2−h P(1111)−3h3+4h2−h−4h3+4h2−h−6h3+5h2−h−7h3+5h2−h6h3−2h2−6h3+5h2−h2h2−h−3h3+4h2−h H17H18H19H20H21H22H23H24 P(0000)−3h3+10h2−6h+1−3h3+10h2−6h+1−3h3+10h2−6h+1−12h3+16h2−7h+1−12h3+16h2−7h+1−12h3+16h2−7h+1−12h3+16h2−7h+12h2−3h+1 P(1000)−27h3+27h2−9h+1−18h3+21h2−8h+1−18h3+21h2−8h+100−18h3+21h2−8h+1−18h3+21h2−8h+12h2−2h P(0100)9h2−6h+1−9h3+15h2−7h+1−9h3+15h2−7h+1−9h3+15h2−7h+19h2−6h+1−18h3+21h2−8h+1−18h3+21h2−8h+12h2−h P(1100)000−9h3+6h2−h0002h2−2h P(0010)−6h3+5h2−h−7h3+5h2−h−6h3+5h2−h−6h3+5h2−h3h3−h2−7h3+5h2−h−10h3+6h2−h2h2−2h P(1010)−9h3+6h2−h−6h3+5h2−h−9h3+6h2−h00−6h3+5h2−h−6h3+5h2−h2h2−h P(0110)−9h3+6h2−h−9h3+6h2−h0−9h3+6h2−h−9h3+6h2−h−9h3+6h2−h02h2−2h P(1110)00−27h3+27h2−9h+1−27h3+27h2−9h+10002h2−3h+1 P(0001)−4h3+4h2−h−3h3+4h2−h−7h3+5h2−h−4h3+4h2−h−4h3+4h2−h−6h3+5h2−h−6h3+5h2−h2h2−2h P(1001)00000002h2−3h+1 P(0101)−6h3+5h2−h−9h3+6h2−h−6h3+5h2−h−6h3+5h2−h−6h3+5h2−h0−9h3+6h2−h2h2−h P(1101)−18h3+21h2−8h+1−18h3+21h2−8h+1−27h3+27h2−9h+1−18h3+21h2−8h+1−18h3+21h2−8h+1−18h3+21h2−8h+1−27h3+27h2−9h+12h2−4h+1 P(0011)−27h3+27h2−9h+1−27h3+27h2−9h+10−18h3+21h2−8h+1−27h3+27h2−9h+1−27h3+27h2−9h+102h2−3h+1 P(1011)00−9h3+6h2−h00002h2−2h P(0111)−9h3+6h2−h−9h3+6h2−h−9h3+6h2−h0−9h3+6h2−h−9h3+6h2−h−9h3+6h2−h2h2−4h+1 P(1111)−6h3+5h2−h−6h3+5h2−h−6h3+5h2−h−6h3+5h2−h−6h3+5h2−h−6h3+5h2−h3h3−h22h2−h TableA.2:ConjecturedfacetsforMDL(2,2,2)(1−3h,h).
H25H26H27H28H29H30H31H32H33H34 P(0000)−6h3+11h2−6h+12h2−3h+12h2−3h+12h2−3h+13h2−7h+2h2−3h3h2−3h31−3h1−3h12h2−10h+2 P(1000)−6h3+8h2−2h2h2−2h2h2−h2h2−4h+13h2−4h+1−9h3+6h2−h06h2−5h+100 P(0100)−6h3+5h2−h2h2−h2h2−h2h2−4h+13h2−4h+1−18h3+21h2−8h+106h2−5h+13h2−4h+13h2−4h+1 P(1100)−6h3+8h2−2h2h2−h2h2−2h2h2−h3h2−h0−9h3+6h2−h03h2−h3h2−h P(0010)−6h3+5h2−h2h2 1−3h+2h2−h 1−3h−h2h2−h2h2−2h3h2−2h−15h3+20h2−8h+1−h32h2−h2h2−h2h2−h P(1010)−6h3+11h2−6h+12h2−2h2h2−h2h2−h3h2−h−9h3+15h2−7h+1−6h3+5h2−h3h2−h00 P(0110)−6h3+11h2−6h+12h2−h2h2−4h+12h2−h3h2−h0003h2−h3h2−h P(1110)−6h3+11h2−6h+12h2 1−3h+2h2−h 1−3h−h2h2−3h+12h2−4h+13h2−4h+1−18h3+21h2−8h+1−9h3+6h2−h9h2−6h+19h2−6h+19h2−6h+1 P(0001)−6h3+8h2−2h2h2−2h2h2−h2h2−2h3h2−2h−7h3+5h2−h−6h3+5h2−h2h2−h−h4h2−2h P(1001)−6h3+11h2−6h+12h2−3h+12h2−4h+12h2−h3h2−h−9h3+6h2−h−9h3+6h2−h000 P(0101)−6h3+5h2−h2h2−3h+12h2−h2h2−h3h2−h−6h3+5h2−h03h2−h3h2−h3h2−h P(1101)−6h3+14h2−7h+12h2−3h+12h2−3h+12h23h2−4h+10−27h3+27h2−9h+19h2−6h+13h2−4h+13h2−4h+1 P(0011)−6h3+8h2−2h2h2−h2h2−h2h2−3h+13h2−4h+1−27h3+27h2−9h+1−18h3+21h2−8h+106h2−5h+16h2−5h+1 P(1011)−6h3+8h2−2h2h2−3h+12h2−h2h2−4h+13h2−h−9h3+6h2−h−9h3+15h2−7h+13h2−h00 P(0111)−6h3+5h2−h2h2−h2h2−h2h2−4h+13h2−h003h2−h3h2−h3h2−h P(1111)−6h3+7h2−2h2h2−2h2h2−2h2h2−h3h2−2h−6h3+5h2−h−15h3+20h2−8h+12h2−hh2−hh2−h H35H36H37H38H39H40H41H42H43H44H45H46H47 P(0000)12h2−10h+23h2−4h+13h2−4h+13h2−4h+13h2−4h+13h2−4h+13h2−4h+13h2−4h+13h2−4h+13h2−4h+11−h3h2−4h+13h2−4h+1 P(1000)03h2−h3h2−h3h2−h3h2−h3h2−h3h2−h3h2−h3h2−h3h2−h−h09h2−6h+1 P(0100)3h2−4h+13h2−h3h2−h3h2−h3h2−h3h2−h3h2−h009h2−6h+11−h3h2−4h+19h2−6h+1 P(1100)3h2−h3h2−2h3h2−2h3h2−h3h2−h3h2−h3h2−4h+13h2−h00−h3h2−4h+10 P(0010)2h2−h3h2−2h3h2−h3h2−2h3h2−2h3h2−2h3h2−2hh2−hh2−h2h2−h−h3h2−h2h2−h P(1010)03h2−h3h2−h3h2−h3h2−h3h2−h3h2−h3h2−h3h2−h3h2−h−h03h2−h P(0110)3h2−h3h2−h3h2−4h+13h2−h3h2−h3h2−h3h2−h000−h3h2−h0 P(1110)9h2−6h+13h2−4h+13h2−4h+13h2−h3h2−h3h2−h3h2−2h3h2−h2h2−h2h2−h−hh2−h0 P(0001)4h2−2h3h2−h3h2−h3h2−2h3h2−h3h2−h3h2−2h3h2−h3h2−h3h2−2h−hh2−h2h2−h P(1001)03h2−4h+13h2−4h+13h2−h3h2−h3h2−4h+13h2−h3h2−4h+13h2−4h+13h2−h1−h00 P(0101)3h2−h3h2−h3h2−h3h2−h3h2−h3h2−4h+13h2−h003h2−h−h3h2−h3h2−h P(1101)15h2−11h+23h2−4h+13h2−4h+13h2−h3h2−4h+13h2−4h+13h2−2h9h2−6h+1001−h3h2−h0 P(0011)6h2−5h+13h2−4h+13h2−h3h2−4h+13h2−4h+13h2−h3h2−h2h2−hh2−h9h2−6h+1−h2h2−h0 P(1011)03h2−h3h2−h3h2−4h+13h2−h3h2−h3h2−4h+13h2−h3h2−h3h2−4h+1−h00 P(0111)3h2−h3h2−4h+13h2−h3h2−4h+13h2−h3h2−h3h2−4h+1000−h3h2−h0 P(1111)5h2−2h3h2−h3h2−h3h2−h3h2−h3h2−2h3h2−h2h2−h2h2−h3h2−h−h2h2−h3h2−h H48H49H50H51H52H53H54H55H56H57H58H59H60 P(0000)1−h3h2−4h+13h2−4h+13h2−4h+13h2−4h+13h2−4h+13h2−4h+13h2−4h+13h2−4h+13h2−4h+13h2−4h+13h2−4h+13h2−4h+1 P(1000)1−h3h2−h00000009h2−6h+19h2−6h+19h2−6h+19h2−6h+1 P(0100)1−h9h2−6h+1009h2−6h+19h2−6h+16h2−5h+13h2−4h+13h2−4h+16h2−5h+16h2−5h+16h2−5h+16h2−5h+1 P(1100)1−h03h2−h000006h2−5h+10000 P(0010)−h2h2−hh2−hh2−h2h2−h2h2−h3h2−h3h2−h3h2−h2h2−h3h2−h3h2−h3h2−h P(1010)−h3h2−h3h2−h3h2−h3h2−h3h2−h0003h2−h3h2−h3h2−h0 P(0110)−h0000003h2−h3h2−h3h2−h12h2−7h+100 P(1110)−h2h2−h03h2−h3h2−h3h2−h002h2−h0009h2−6h+1 P(0001)−h4h2−2h3h2−h3h2−h4h2−2h3h2−hh2−hh2−hh2−h2h2−h2h2−h2h2−h2h2−h P(1001)−h06h2−5h+16h2−5h+13h2−h00000000 P(0101)−h3h2−h003h2−h3h2−h3h2−h3h2−h3h2−h3h2−h3h2−h3h2−h3h2−h P(1101)−h09h2−6h+1006h2−5h+19h2−6h+16h2−5h+109h2−6h+1h−3h29h2−6h+19h2−6h+1 P(0011)−h03h2−h2h2−h9h2−6h+1009h2−6h+13h2−h9h2−6h+1000 P(1011)−h6h2−5h+13h2−h3h2−h6h2−5h+13h2−h0000003h2−h P(0111)−h00002h2−h3h2−h3h2−h3h2−h03h2−h3h2−h3h2−h P(1111)−h3h2−h2h2−h2h2−h3h2−h3h2−h2h2−h2h2−h2h2−h3h2−h−h23h2−h2h2−h TableA.2:(Cont)ConjecturedfacetsforMDL(2,2,2)(1−3h,h).
H61H62H63H64H65H66H67H68H69H70H71H72 P(0000)3h2−4h+13h2−4h+13h2−4h+13h2−4h+13h2−4h+13h2−4h+115h2−11h+26h2−5h+16h2−5h+16h2−5h+16h2−5h+16h2−5h+1 P(1000)h−3h203h2−h3h2−h06h2−5h+103h2−h3h2−h003h2−h P(0100)6h2−5h+115h2−11h+23h2−h6h2−5h+16h2−5h+16h2−5h+16h2−5h+10006h2−5h+10 P(1100)00003h2−h03h2−h0006h2−5h+13h2−h P(0010)−h23h2−h4h2−2h3h2−h3h2−h2h2−h3h2−h2h2−h2h2−h2h2−h3h2−h2h2−h P(1010)3h2−h12h2−7h+13h2−h3h2−h03h2−h03h2−h3h2−h3h2−h03h2−h P(0110)03h2−h02h2−h0000003h2−h0 P(1110)002h2−h09h2−6h+109h2−6h+12h2−h2h2−h3h2−h2h2−h3h2−h P(0001)2h2−h5h2−2h4h2−2h4h2−2hh2−h2h2−h5h2−2h5h2−2h3h2−h02h2−h3h2−h P(1001)000000006h2−5h+1006h2−5h+1 P(0101)3h2−h3h2−h3h2−h3h2−h3h2−h3h2−h3h2−h0003h2−h0 P(1101)9h2−6h+16h2−5h+12h2−h9h2−6h+16h2−5h+106h2−5h+1006h2−5h+13h2−h9h2−6h+1 P(0011)09h2−6h+106h2−5h+19h2−6h+109h2−6h+102h2−h03h2−h3h2−h P(1011)12h2−7h+106h2−5h+1003h2−h06h2−5h+103h2−h00 P(0111)3h2−h3h2−h6h2−5h+13h2−h3h2−h3h2−h3h2−h002h2−h00 P(1111)3h2−h2h2−h3h2−h3h2−h2h2−h3h2−h2h2−h3h2−h3h2−h3h2−h3h2−h3h2−h H73H74H75H76H77H78H79H80H81H82H83H84H85 P(0000)6h2−5h+16h2−5h+16h2−5h+16h2−5h+16h2−5h+16h2−5h+16h2−5h+11−2h6h2−5h+11−2h6h2−5h+16h2−5h+16h2−5h+1 P(1000)3h2−h3h2−h000001−3h9h2−6h+11−3h009h2−6h+1 P(0100)09h2−6h+109h2−6h+16h2−5h+19h2−6h+16h2−5h+11−3h9h2−6h+11−3h9h2−6h+19h2−6h+19h2−6h+1 P(1100)3h2−h03h2−h00000003h2−h3h2−h0 P(0010)2h2−h3h2−h2h2−h3h2−h03h2−h3h2−h−h3h2−h−h2h2−h02h2−h P(1010)03h2−h3h2−h003h2−h0−h3h2−h−h003h2−h P(0110)02h2−h0003h2−h3h2−h012h2−7h+103h2−h9h2−6h+13h2−h P(1110)00000000009h2−6h+1h−3h20 P(0001)3h2−h5h2−2h02h2−h2h2−h5h2−2h2h2−h−h3h2−h−h2h2−h2h2−h3h2−h P(1001)6h2−5h+109h2−6h+1003h2−h00000012h2−7h+1 P(0101)03h2−h03h2−h3h2−h3h2−h3h2−h−h003h2−h3h2−h3h2−h P(1101)9h2−6h+106h2−5h+109h2−6h+109h2−6h+10h−3h21−3h9h2−6h+19h2−6h+10 P(0011)3h2−h6h2−5h+10006h2−5h+19h2−6h+10006h2−5h+10h−3h2 P(1011)3h2−h03h2−h009h2−6h+10000012h2−7h+10 P(0111)003h2−h03h2−h0003h2−h−hh−3h23h2−h9h2−6h+1 P(1111)2h2−h03h2−h3h2−h3h2−h03h2−h0−h2−h3h2−h3h2−h0 H86H87H88H89H90H91H92 P(0000)6h2−5h+11−2h6h2−5h+11−3h1−3h1−3h0 P(1000)9h2−6h+1−2h6h2−2h000−1 P(0100)9h2−6h+1−2h6h2−2h001−3h0 P(1100)0−2h6h2−2h−h000 P(0010)3h2−h−2h6h2−2h−h−h00 P(1010)0−2h6h2−5h+10000 P(0110)0−2h6h2−5h+10000 P(1110)9h2−6h+11−2h6h2−5h+11−3h000 P(0001)3h2−h−2h6h2−2h−h−h−h0 P(1001)0−2h6h2−5h+10000 P(0101)0−2h6h2−5h+100−h0 P(1101)9h2−6h+11−2h6h2−5h+11−3h000 P(0011)01−2h6h2−2h1−3h000 P(1011)3h2−h1−2h6h2−2h0000 P(0111)3h2−h1−2h6h2−2h0000 P(1111)2h2−h−2h6h2−3h−h−h00 TableA.2:(Cont)ConjecturedfacetsforMDL(2,2,2)(1−3h,h).
IS1IS2IS3IS4IS5IS6IS7IS8IS9 P(0000)h4−4h2+4h−1h4−2h3+h2h4−2h3+h2h4−2h3+h2h4−2h3+h2h4−2h3+h2h4−2h3+h2h3−2h2+hh3−2h2+h P(1000)h4−2h3+h2h4−4h3+2h2h4−2h3+h2h4−2h3+h2h4−2h3+h2h4−2h3+h2h4−4h2+4h−1h3−4h2+2hh3−4h2+2h P(0100)h4−2h3+h2h4−4h2+4h−1h4−2h3+h2h4−2h3+h2h4−2h3+h2h4−2h3+h2h4−4h3+2h2h3−4h2+2hh3−4h2+2h P(1100)h4−4h3+2h2h4−2h3+h2h4−2h3+h2h4−2h3+h2h4−2h3+h2h4−2h3+h2h4−2h3+h2h3−2h2+hh3−2h2+h P(0010)h4−h3h4−3h3+3h2−hh4−h3h4−h3h4−3h3+3h2−hh4−3h3+3h2−hh4−h3h3−h2h3−h2 P(1010)h4−h3h4−3h3+3h2−hh4−3h3+3h2−hh4−3h3+3h2−hh4−h3h4−h3h4−h3h3−3h2+3h−1h3−3h2+3h−1 P(0110)h4−3h3+3h2−hh4−h3h4−h3h4−h3h4−3h3+3h2−hh4−3h3+3h2−hh4−3h3+3h2−hh3−3h2+3h−1h3−3h2+3h−1 P(1110)h4−3h3+3h2−hh4−h3h4−3h3+3h2−hh4−3h3+3h2−hh4−h3h4−h3h4−3h3+3h2−hh3−h2h3−h2 P(0001)h4−h3h4−h3h4−h3h4−3h3+3h2−hh4−h3h4−3h3+3h2−hh4−3h3+3h2−hh3−h2h3−3h2+3h−1 P(1001)h4−3h3+3h2−hh4−3h3+3h2−hh4−h3h4−3h3+3h2−hh4−h3h4−3h3+3h2−hh4−h3h3−3h2+3h−1h3−h2 P(0101)h4−h3h4−h3h4−3h3+3h2−hh4−h3h4−3h3+3h2−hh4−h3h4−3h3+3h2−hh3−3h2+3h−1h3−h2 P(1101)h4−3h3+3h2−hh4−3h3+3h2−hh4−3h3+3h2−hh4−h3h4−3h3+3h2−hh4−h3h4−h3h3−h2h3−3h2+3h−1 P(0011)h4−2h3+h2h4−2h3+h2h4−4h2+4h−1h4−2h3+h2h4−2h3+h2h4−4h3+2h2h4−2h3+h2h3−4h2+2hh3−2h2+h P(1011)h4−2h3+h2h4−2h3+h2h4−2h3+h2h4−4h3+2h2h4−4h2+4h−1h4−2h3+h2h4−2h3+h2h3−2h2+hh3−4h2+2h P(0111)h4−2h3+h2h4−2h3+h2h4−2h3+h2h4−4h2+4h−1h4−4h3+2h2h4−2h3+h2h4−2h3+h2h3−2h2+hh3−4h2+2h P(1111)h4−2h3+h2h4−2h3+h2h4−4h3+2h2h4−2h3+h2h4−2h3+h2h4−4h2+4h−1h4−2h3+h2h3−4h2+2hh3−2h2+h IS10IS11IS12IS13IS14IS15IS16IS17IS18IS19 P(0000)h3−2h2+hh3−2h2+hh−h2h−h2−h2+2h−1h3−3h2+3h−1h3−3h2+3h−1h3−3h2+3h−1h3−3h2+3h−1h4−3h3+3h2−h P(1000)h3−4h2+2hh3−4h2+2hh−h2h−h2−h2+2h−1h3−h2h3−h2h3−h2h3−h2h4−h3 P(0100)h3−4h2+2hh3−4h2+2hh−h2h−h2−h2+2h−1h3−h2h3−h2h3−h2h3−h2h4−3h3+3h2−h P(1100)h3−2h2+hh3−2h2+hh−h2h−h2−h2+2h−1h3−3h2+3h−1h3−3h2+3h−1h3−3h2+3h−1h3−3h2+3h−1h4−h3 P(0010)h3−3h2+3h−1h3−3h2+3h−1−h2−h2+2h−1h−h2h3−4h2+2hh3−4h2+2hh3−2h2+hh3−2h2+hh4−2h3+h2 P(1010)h3−h2h3−h2−h2−h2+2h−1h−h2h3−2h2+hh3−2h2+hh3−4h2+2hh3−4h2+2hh4−2h3+h2 P(0110)h3−h2h3−h2−h2−h2+2h−1h−h2h3−2h2+hh3−2h2+hh3−4h2+2hh3−4h2+2hh4−2h3+h2 P(1110)h3−3h2+3h−1h3−3h2+3h−1−h2−h2+2h−1h−h2h3−4h2+2hh3−4h2+2hh3−2h2+hh3−2h2+hh4−2h3+h2 P(0001)h3−h2h3−3h2+3h−1−h2+2h−1−h2h−h2h3−4h2+2hh3−2h2+hh3−4h2+2hh3−2h2+hh4−4h3+2h2 P(1001)h3−3h2+3h−1h3−h2−h2+2h−1−h2h−h2h3−2h2+hh3−4h2+2hh3−2h2+hh3−4h2+2hh4−2h3+h2 P(0101)h3−3h2+3h−1h3−h2−h2+2h−1−h2h−h2h3−2h2+hh3−4h2+2hh3−2h2+hh3−4h2+2hh4−2h3+h2 P(1101)h3−h2h3−3h2+3h−1−h2+2h−1−h2h−h2h3−4h2+2hh3−2h2+hh3−4h2+2hh3−2h2+hh4−4h2+4h−1 P(0011)h3−2h2+hh3−4h2+2hh−h2h−h2−h2h3−h2h3−3h2+3h−1h3−3h2+3h−1h3−h2h4−3h3+3h2−h P(1011)h3−4h2+2hh3−2h2+hh−h2h−h2−h2h3−3h2+3h−1h3−h2h3−h2h3−3h2+3h−1h4−3h3+3h2−h P(0111)h3−4h2+2hh3−2h2+hh−h2h−h2−h2h3−3h2+3h−1h3−h2h3−h2h3−3h2+3h−1h4−h3 P(1111)h3−2h2+hh3−4h2+2hh−h2h−h2−h2h3−h2h3−3h2+3h−1h3−3h2+3h−1h3−h2h4−h3 IS20IS21IS22IS23IS24IS25IS26IS27IS28 P(0000)h4−3h3+3h2−hh4−3h3+3h2−hh4−3h3+3h2−h−h2h3−h2h3−h2h3−h2h3−h2h3−4h2+2h P(1000)h4−h3h4−3h3+3h2−hh4−3h3+3h2−h−h2h3−3h2+3h−1h3−3h2+3h−1h3−3h2+3h−1h3−3h2+3h−1h3−2h2+h P(0100)h4−3h3+3h2−hh4−h3h4−h3−h2h3−3h2+3h−1h3−3h2+3h−1h3−3h2+3h−1h3−3h2+3h−1h3−2h2+h P(1100)h4−h3h4−h3h4−h3−h2h3−h2h3−h2h3−h2h3−h2h3−4h2+2h P(0010)h4−2h3+h2h4−4h3+2h2h4−2h3+h2h−h2h3−4h2+2hh3−4h2+2hh3−2h2+hh3−2h2+hh3−h2 P(1010)h4−2h3+h2h4−2h3+h2h4−4h3+2h2h−h2h3−2h2+hh3−2h2+hh3−4h2+2hh3−4h2+2hh3−3h2+3h−1 P(0110)h4−2h3+h2h4−2h3+h2h4−4h2+4h−1h−h2h3−2h2+hh3−2h2+hh3−4h2+2hh3−4h2+2hh3−3h2+3h−1 P(1110)h4−2h3+h2h4−4h2+4h−1h4−2h3+h2h−h2h3−4h2+2hh3−4h2+2hh3−2h2+hh3−2h2+hh3−h2 P(0001)h4−2h3+h2h4−2h3+h2h4−2h3+h2h−h2h3−4h2+2hh3−2h2+hh3−4h2+2hh3−2h2+hh3−h2 P(1001)h4−4h2+4h−1h4−2h3+h2h4−2h3+h2h−h2h3−2h2+hh3−4h2+2hh3−2h2+hh3−4h2+2hh3−3h2+3h−1 P(0101)h4−4h3+2h2h4−2h3+h2h4−2h3+h2h−h2h3−2h2+hh3−4h2+2hh3−2h2+hh3−4h2+2hh3−3h2+3h−1 P(1101)h4−2h3+h2h4−2h3+h2h4−2h3+h2h−h2h3−4h2+2hh3−2h2+hh3−4h2+2hh3−2h2+hh3−h2 P(0011)h4−h3h4−3h3+3h2−hh4−h3−h2+2h−1h3−3h2+3h−1h3−h2h3−h2h3−3h2+3h−1h3−2h2+h P(1011)h4−h3h4−h3h4−3h3+3h2−h−h2+2h−1h3−h2h3−3h2+3h−1h3−3h2+3h−1h3−h2h3−4h2+2h P(0111)h4−3h3+3h2−hh4−3h3+3h2−hh4−h3−h2+2h−1h3−h2h3−3h2+3h−1h3−3h2+3h−1h3−h2h3−4h2+2h P(1111)h4−3h3+3h2−hh4−h3h4−3h3+3h2−h−h2+2h−1h3−3h2+3h−1h3−h2h3−h2h3−3h2+3h−1h3−2h2+h TableA.3:ConjecturedfacetsforISMDL(2,2,2)((1−h,1−h),(h,h)).
IS29IS30IS31IS32IS33IS34IS35IS36 P(0000)h3−4h2+2hh3−4h2+2hh3−4h2+2hh4−h3h4−h3h4−h3h4−h3h4−4h3+2h2 P(1000)h3−2h2+hh3−2h2+hh3−2h2+hh4−h3h4−h3h4−3h3+3h2−hh4−3h3+3h2−hh4−2h3+h2 P(0100)h3−2h2+hh3−2h2+hh3−2h2+hh4−3h3+3h2−hh4−3h3+3h2−hh4−h3h4−h3h4−2h3+h2 P(1100)h3−4h2+2hh3−4h2+2hh3−4h2+2hh4−3h3+3h2−hh4−3h3+3h2−hh4−3h3+3h2−hh4−3h3+3h2−hh4−4h2+4h−1 P(0010)h3−h2h3−3h2+3h−1h3−3h2+3h−1h4−2h3+h2h4−4h2+4h−1h4−2h3+h2h4−2h3+h2h4−3h3+3h2−h P(1010)h3−3h2+3h−1h3−h2h3−h2h4−4h2+4h−1h4−2h3+h2h4−2h3+h2h4−2h3+h2h4−3h3+3h2−h P(0110)h3−3h2+3h−1h3−h2h3−h2h4−4h3+2h2h4−2h3+h2h4−2h3+h2h4−2h3+h2h4−h3 P(1110)h3−h2h3−3h2+3h−1h3−3h2+3h−1h4−2h3+h2h4−4h3+2h2h4−2h3+h2h4−2h3+h2h4−h3 P(0001)h3−3h2+3h−1h3−h2h3−3h2+3h−1h4−2h3+h2h4−2h3+h2h4−2h3+h2h4−4h2+4h−1h4−3h3+3h2−h P(1001)h3−h2h3−3h2+3h−1h3−h2h4−2h3+h2h4−2h3+h2h4−4h3+2h2h4−2h3+h2h4−h3 P(0101)h3−h2h3−3h2+3h−1h3−h2h4−2h3+h2h4−2h3+h2h4−4h2+4h−1h4−2h3+h2h4−3h3+3h2−h P(1101)h3−3h2+3h−1h3−h2h3−3h2+3h−1h4−2h3+h2h4−2h3+h2h4−2h3+h2h4−4h3+2h2h4−h3 P(0011)h3−4h2+2hh3−4h2+2hh3−2h2+hh4−3h3+3h2−hh4−h3h4−3h3+3h2−hh4−h3h4−2h3+h2 P(1011)h3−2h2+hh3−2h2+hh3−4h2+2hh4−h3h4−3h3+3h2−hh4−3h3+3h2−hh4−h3h4−2h3+h2 P(0111)h3−2h2+hh3−2h2+hh3−4h2+2hh4−3h3+3h2−hh4−h3h4−h3h4−3h3+3h2−hh4−2h3+h2 P(1111)h3−4h2+2hh3−4h2+2hh3−2h2+hh4−h3h4−3h3+3h2−hh4−h3h4−3h3+3h2−hh4−2h3+h2 TableA.3:(Cont)ConjecturedfacetsforISMDL(2,2,2)((1−h,1−h),(h,h)).
Papers published in the course of
this thesis
Arbitrarily Small Amount of Measurement Independence Is Sufficient to Manifest Quantum Nonlocality
Gilles Pütz,1,*Denis Rosset,1 Tomer Jack Barnea,1Yeong-Cherng Liang,2and Nicolas Gisin1
1Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland
2Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland (Received 25 July 2014; published 6 November 2014)
The use of Bell’s theorem in any application or experiment relies on the assumption of free choice or, more precisely, measurement independence, meaning that the measurements can be chosen freely. Here, we prove that even in the simplest Bell test—one involving 2 parties each performing 2 binary-outcome measurements—an arbitrarily small amount of measurement independence is sufficient to manifest quantum nonlocality. To this end, we introduce the notion of measurement dependent locality and show that the corresponding correlations form a convex polytope. These correlations can thus be characterized efficiently, e.g., using a finite set of Bell-like inequalities—an observation that enables the systematic study of quantum nonlocality and related applications under limited measurement independence.
DOI:10.1103/PhysRevLett.113.190402 PACS numbers: 03.65.Ud, 03.67.−a
Since Bell’s seminal work[1], quantum nonlocality has gathered more and more interest, not only from a founda-tional point of view but also as a resource in several tasks like quantum key distribution[2,3], randomness expansion [4,5], randomness extraction[6], or robust certification[7]
and quantification[8]of quantum entanglement. It has led to the notion of device independence (see, e.g.,[9]), where the violation of a Bell inequality alone certifies properties that are useful to the task at hand, e.g., nondeterminism of the outputs. In such a scenario, it is enough to consider black boxes that the parties give an input to and get an outcome from instead of having to consider the complex physical description of the implementation.
However, an important assumption has to be made in order for violations of Bell inequalities to exclude any local, and, in particular, deterministic explanation. Let us consider an adversarial scenario in which the boxes were in the hands of an adversary Eve before being given to the parties performing the experiment or protocol. The inputs for the boxes are chosen by local random number generators. If the adversary could influence these random number generators, then she can prepare the boxes with local strategies that appear to be nonlocal to the parties. The violation of a Bell inequality therefore does not imply that the outcomes of the boxes are unknown to the adversary unless we assume that the inputs are independent of the adversary and that she cannot gain any information about them. This assumption is commonly referred to as measurement independence [10,11]. Similar, but slightly stronger assumptions [12]
are the assumptions of free choice[13]and free will[14].
Ensuring measurement independence in a Bell test seems impossible. However, if we abandon measurement inde-pendence completely and place no restriction at all on the adversary’s influence, then it is impossible to show and exploit quantum nonlocality[15].
In light of this, relaxations of this assumption have gathered attention and been studied in recent works. Hall [10], Barrett and Gisin[11], and, recently, Thinh, Sheridan, and Scarani[16]studied how different possible relaxations influence well-known Bell inequalities. Colbeck and Renner[13]introduced the idea of randomness amplifica-tion, in which a quantum protocol produces random out-comes even though complete free choice is not given. This was further developed by Gallego et al. [17] and others [18–22].
A common denominator of these works is that they study and use well-known Bell inequalities. On the contrary, in this Letter, we derive Bell-like inequalities that are spe-cifically suited for a measurement dependent scenario.
Using these we show that quantum nonlocality allows for correlations that cannot be explained by any local models exploiting measurement dependence, even when the dependence is arbitrarily strong, as long as some measurement independence is retained (in the sense that we explain more precisely below).
Bell locality.—In a Bell test, two spacelike separated parties, usually referred to as Alice and Bob, have access to two boxes. They can give these boxes an input, denoted by the random variablesXandY, respectively, and each of the boxes gives back an outcome,AandBas depicted in Fig.1.
In a quantum mechanical scenario, each box is given by a quantum system and the inputs determine measurements that are performed on this system. By performing many runs, the parties collect data that allow them to estimate
In a quantum mechanical scenario, each box is given by a quantum system and the inputs determine measurements that are performed on this system. By performing many runs, the parties collect data that allow them to estimate