Quantum Entanglement Simulation
Ceola Roefaro
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As a result of the presence of temporal correlations, a PDO is not necessarily a positive operator, although it still is trace-one and Hermitian. This means that it presents negative expectation values of projectors. For example, R12 has the singlet state as an eigenstate, with eigenvalue \(- \frac12\)22. This is interpreted as the signature of correlations in time23.
Open timelike curve circuit (pictorial representation). Qubits Q1 and Q2 are initially in a singlet state. Qubit Q2 enters a chronology-violating region, emerging as qubit Q3. In the chronology-violating region, qubits Q1 and Q2 must be in a singlet state, and so are qubits Q1 and Q3. Furthermore, since Q2 and Q3 are, respectively, the past and future copy of the same qubit, they are maximally correlated. This situation violates monogamy of entanglement: this is why it cannot be described by ordinary density operators, but it can be represented by PDOs
Our experiment consists of a simulation of the OTC. The simulation consists in reconstructing all the statistics contained in the PDO R123, which represents the OTC in our model, by constructing different sub-ensembles of entangled photon pairs, on which different measurements are realised. This experimental demonstration is therefore a proposal for a paradigm to realise a tomographic reconstruction of a PDO.
To perform the reconstruction of the PDO R123 we exploit different measurements to collect the three-point and the two-point correlations on the two photons. The three-point and two-point measurements are properly chosen in order to form a minimal quorum allowing for a full tomographic reconstruction27 of R123. This is needed because, in our experimental simulation, it would be impossible to perform a standard three-qubit quantum tomography procedure able to reconstruct R123, since the measurement occurring on photon A at time t1 (Q2) would obviously affect photon A at time t2 (Q3) and the outcome of the measurement on it. To avoid this, we restrict ourselves to a particular sub-sample of the standard three-qubit tomographic measurements quorum in which the sequential measurement on photon A involves commuting observables, avoiding the issues derived from the measurement temporal ordering. The remaining information needed for the PDO reconstruction is obtained from the two-point correlation measurements.
Our predictions are well confirmed by the simulation results. To the best of our knowledge, this result, shown in Fig. 3 compared to theoretical expectation, is the first tomographic reconstruction of a PDO.
Our proposal shows a radically different way of generalising quantum theory to describe chronology-violating regions containing an OTC, whose features we have simulated experimentally. R123 is a viable descriptor of the physical situation where a qubit enters an OTC after having been entangled with another qubit. This is because, as we have demonstrated, it provides the same expected values for all the possible measurements that can be performed on those two qubits. It is a linear description in the sense that two different PDOs are related via a linear transformation. By proposing to use a PDO to describe the three qubits in the chronology-violating region, we depart from standard quantum mechanics, because we use a non-positive operator to describe the state of the qubits. Our proposal hints to a different way of formulating quantum theory, where, to describe a physical system with a certain dynamics, one gives the PDO as a faithful description of that physical situation. We implicitly define a PDO as faithful if it correctly describes the correlations between observables in different qubits. Now, once that step is taken, is it still possible to preserve some notion of linearity even when describing situations where properties like entanglement monogamy are violated? We conjecture that the answer is yes, because any two PDOs describing such different physical situations (e.g. two OTCs with different initial states) can be related by a linear transformation. This notion of linearity is, however, different from the linearity of quantum mechanical evolution. It would be interesting to understand the physical meaning of linear transformations between PDOs describing OTCs with different initial conditions, which we leave for a future paper. Also, a promising development of this proposal is a consistent general treatment of both OTCs and CTCs via PDOs. This could lead to a theory that retains linearity of quantum mechanics in a more general sense, but relaxes certain assumptions about the states of physical systems. Another interesting point is that, in the treatment of CTCs offered by10, there is no violation of monogamy of entanglement. Extending this work to cover this type of CTCs is an interesting future step.
More generally, some models of quantum gravity might require spacetime to be quantised, whereby the distinction between timelike and spacelike degrees of freedom may become blurred below certain scales. This has prompted a number of proposals, for example, to modify the commutation relations of observables of different subsystems28, or to incorporate indefinite causal order1,29,30. The pseudo-density formalism, in the light of what is proposed in this paper, might be a candidate to generalise the notion of quantum states to these scenarios.
C.M. and V.V. (both responsible for the theoretical framework) proposed the experiment, and planned it together with F.P., A.A., I.P.D., M.Gram. and M.Gen. (responsible for the laboratories). The experimental realization was achieved (supervised by I.P.D., M.Gram. and M.Gen.) by S.V., E.R., A.A. and F.P. The manuscript was prepared with inputs by all the authors, who also had a fruitful systematic discussion during the whole work development.
A common answer is that we live in an infinite multiverse of universes, so we shouldn't be surprised that at least one universe has turned out as ours. But another is that our universe is a computer simulation, with someone (perhaps an advanced alien species) fine-tuning the conditions.
The latter option is supported by a branch of science called information physics, which suggests that space-time and matter are not fundamental phenomena. Instead, the physical reality is fundamentally made up of bits of information, from which our experience of space-time emerges. By comparison, temperature "emerges" from the collective movement of atoms. No single atom fundamentally has temperature.
This leads to the extraordinary possibility that our entire universe might in fact be a computer simulation. The idea is not that new. In 1989, the legendary physicist, John Archibald Wheeler, suggested that the universe is fundamentally mathematical and it can be seen as emerging from information. He coined the famous aphorism "it from bit."
In 2003, philosopher Nick Bostrom from Oxford University in the UK formulated his simulation hypothesis. This argues that it is actually highly probable that we live in a simulation. That's because an advanced civilisation should reach a point where their technology is so sophisticated that simulations would be indistinguishable from reality, and the participants would not be aware that they were in a simulation.
Physicist Seth Lloyd from the Massachusetts Institute of Technology in the US took the simulation hypothesis to the next level by suggesting that the entire universe could be a giant quantum computer.
Any virtual reality world will be based on information processing. That means everything is ultimately digitized or pixelated down to a minimum size that cannot be subdivided further: bits. This appears to mimic our reality according to the theory of quantum mechanics, which rules the world of atoms and particles. It states there is a smallest, discrete unit of energy, length and time. Similarly, elementary particles, which make up all the visible matter in the universe, are the smallest units of matter. To put it simply, our world is pixelated.
Another curiosity in physics supporting the simulation hypothesis is the maximum speed limit in our universe, which is the speed of light. In a virtual reality, this limit would correspond to the speed limit of the processor, or the processing power limit. We know that an overloaded processor slows down computer processing in a simulation. Similarly, Albert Einstein's theory of general relativity shows that time slows in the vicinity of a black hole.
Perhaps the most supportive evidence of the simulation hypothesis comes from quantum mechanics. This suggest nature isn't "real": particles in determined states, such as specific locations, don't seem to exist unless you actually observe or measure them. Instead, they are in a mix of different states simultaneously. Similarly, virtual reality needs an observer or programmer for things to happen.
This could, however, also be explained by the fact that within a virtual reality code, all "locations" (points) should be roughly equally far from a central processor. So while we may think two particles are millions of light years apart, they wouldn't be if they were created in a simulation.
I have predicted the exact range of expected frequencies of the resulting photons based on information physics. The experiment is highly achievable with our existing tools, and we have launched a crowdfunding site) to achieve it.
There are other approaches too. The late physicist John Barrow has argued that a simulation would build up minor computational errors which the programmer would need to fix in order to keep it going. He suggested we might experience such fixing as contradictory experimental results appearing suddenly, such as the constants of nature changing. So monitoring the values of these constants is another option.
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