Published: 9th March 2011|
|Preface Farid Ablayev, Bob Coecke and Alexander Vasiliev|
|On Computational Power of Quantum Read-Once Branching Programs Farid Ablayev and Alexander Vasiliev||1|
|Fast and robust two- and three-qubit swapping gates on multi-atomic ensembles in quantum electrodynamic cavity Sergey N. Andrianov and Sergey A. Moiseev||13|
|Invited Presentation: Three qubit entanglement within graphical Z/X-calculus Bob Coecke and Bill Edwards||22|
|Invited Presentation: The GHZ/W-calculus contains rational arithmetic Bob Coecke, Aleks Kissinger, Alex Merry and Shibdas Roy||34|
|Extended Abstract: Quantum nonlocality from the probabilistic viewpoint Yuri I. Ozhigov||49|
This volume contains the proceedings of the Workshop on High Productivity Computations (HPC 2010) which took place on June 21-22 in Kazan, Russia. This workshop was held as a satellite workshop of the 5th International Computer Science Symposium in Russia (CSR 2010).
HPC 2010 was intended to organize the discussions about high productivity computing means and models, including but not limited to high performance and quantum information processing.
The programme committee of HPC 2010 consisted of
This volume contains a selection of the papers presented at HPC 2010, including an invited contribution by Bob Coecke, consisting of two parts; each part gives an application of two distinct compositional diagrammatic accounts on multipartite quantum entanglement.
We thank the programme committee members for their effort. We would like to thank CSR 2010 organizers for giving us the opportunity to hold this workshop as an affiliated event and for taking care of the organizational matters. Finally, we are grateful to Electronic Proceedings in Theoretical Computer Science (EPTCS) for accepting to publish this volume, and in particular Rob van Glabbeek for his help.
Kazan and Oxford, December 2010
Classical physics obeys the principle of locality of all interactions which transmit information. The information in the non-quantum world cannot travel faster than light, the relativistic limit of information transfer velocity. However, in quantum mechanics the situation is more complex. There are two types of information, which we can call of the user’s type and of the administrative type. It turns out that the relativistic limit concerns only information of the user’s type, that is, information created by experimentalists or observers. Information of the administrative type can travel instantaneously. Formally, this is not completely unexpected, since the Coulomb field also spreads instantaneously. But only recent experiments have shown the unexpected feature of nonlocality, which can be formulated using the language of probability theory.
Nonlocality experiments were done for the first time in 1980s by A. Aspect and by A. Zeilinger (see references in ). In these experiments the entanglement is not just of the angstrom scale, like in a hydrogen molecule, but (meanwhile) can span several hundreds of kilometers.
More precisely, in these experiments photon states of the form
are detected on distances of several hundreds of kilometers. Imagine that the first photon is detected by an observer called Alice, and the second one by Bob. We assume that Alice has two possibilities to set her detector (i.e. bases to measure against in the state space of her qubit), and so has Bob, namely, Alice can measure the σx or σz observable, and Bob the -( σx+σz ) ∕ √2 or ( σz-σx ) ∕ √2 observable. Each of these operators has eigenvalues 1 or -1. We will assume that in simultaneous measurements Alice has received a value X or Y and Bob a or b correspondingly, according to the above-stated order. We take the convention that 1 means that the photon occurs with horizontal polarization relative to the detector position, and that (-1) means that the photon occurs with vertical polarization.
Within quantum mechanics, the state |Ψ> of two photons is represented by a single vector in the Hilbert space of states. It means that there is such space of elementary outcomes Ω that all random variables X,Y,a,b are random variables over this space, that is functions of elementary outcomes: X(ω), Y(ω), a(ω), b(ω), where ω = ( ω1,ω2 ) ∈Ω = Ω1 ×Ω2 and the sets Ω1, Ω2 correspond to choices of Alice and Bob respectively.
If we assume that the so-called locality1 is available for us, we will come to an inequality that can be violated by the experiment on detecting biphotons. Thus, we have come to a conclusion that nonlocality of quantum mechanics follows from that experiment.
Now we will consider the nonlocality in more detail. It means that the random variables concerning Bob, depend not only on his components of an elementary outcome, but also on the components, belonging to Alice and vice versa, that is all outcomes X, Y, a, b depend both on ω1, and on ω2.
How can it be realized? Only as follows: there is some object ω' which travels from Alice to Bob and back, transferring the information on the other half of an elementary outcome of corresponding experiment. If this object ω' submits to restriction of relativism, and cannot move faster than light we can deduce restrictions on instant of emission of biphoton from a source and instant of detecting of arrival of each of photons by Alice and Bob. Let Δt be the natural uncertainty of the moment of emission of biphoton by a source about which we assume that all biphotons which time of emission lies out of this range do not play any role for statistics of the given experiment. Presence of such interval is a direct consequence of the uncertainty relation ”energy-time”. Now we will assume that watches of Alice and Bob, and a source of biphotons are precisely synchronized, and we will introduce a random variable δt, that equals to difference of the moment of operation of the detector and the moment of a choice of its position (that is a choice between X and Y and between a and b). Then if a material object ω', transferring the information on the other half of an elementary outcome, obeys relativism the inequality should be carried out
where d is a distance between Alice (or Bob), and a source of biphotons, c is the velocity of light.
Experiments testify that this inequality is violated for biphotons, detected on distances in several hundreds of kilometers that has absolutely fundamental consequences for the quantum theory. Really, the violation of (1) says that ω' cannot be a hidden variable of any of photons. That is ω' directly transfers the information on orientation of detectors from Alice to Bob or vice versa. This effect is usually called “quantum nonlocality” .
The quantum theory is completely coordinated with a principle of relativism according to which no information can move with the speed surpassing a velocity of light. Formally it is expressed that the statistics of measurements of Alice do not depend in any way on, whether Bob measures his photon or not. That is by means of entangled quantum states it is impossible to transmit the information generated by participants of experiment to each other. But we have just found out that this restriction does not extend on the information on elementary outcomes in the experiment on measurement of quantum states!
From this it is possible to derive only one conclusion. There is the management system, interaction with which determines the reality. This interaction exactly corresponds to interaction of a user with a computer. A user, that is an experimenter, defines conditions (position of detectors) then the management system, working with elementary outcomes, gives out the result of experiment. Thus the time spent by the management system on the coordination of conditions set by various users, is not the real physical time. (However, it is impossible to confirm this concerning the general time which the management system spends for processing of users conditions; moreover, constructivism just assumes that the management system strictly submits to Markov principle, that is it operates as a computer.) Therefore restriction of relativism, which is fair only for users, does not extend on actions with elementary outcomes. It is also exactly the scheme of physical constructivism.
The conclusion is that the quantum theory for the systems consisting of more than one particle should use a constructive mathematical apparatus at which the space of elementary outcomes is considered explicitly, for example, like it is done in heuristics of collective behavior; thus the user access to it is defined by means of the algorithm each step of which corresponds to that knowledge, which is accessible to the experimenter at present. In this case elementary outcomes correspond to samples of photons which appear connected by bonds, that is objects which belong to the management system.
1Shortly speaking, locality means that the result of measurement of Alice does not depend in any way on orientation of the detector of Bob and vice versa.