The usage of the trivial system is crucial for considering both states and effects as special cases of transformations. The following postulates differ remarkably between the two axiomatisations. Whereas this can be done for all transformations and indeed it is a theorem of the minimal axiomatisation, as in Table 1 , not necessarily it is actually the case.
What we put into discussion here, is the ontology of the unitary realisation of quantum transformations. The fact that each transformation must necessarily be ultimately unitary would be of no concern if it made no harm to the whole logical consistency of theories in physics. However, this is not the case, due to the information paradox.
Lloyd and Preskill [ 10 ] expressed the impossibility of reconciling unitarity with the following relevant facts quoting from the same reference [ 10 ]. Unitarity can be temporarily violated during the black hole evaporation process, accommodating violations of monogamy of entanglement and the no-cloning principle, and allowing assumptions 1 , 2 , and 3 to be reconciled.
Antonini and Nambiar write [ 12 ]. This is the essence of the black hole information paradox BHIP : unlike any other classical or quantum system, black holes may not conserve information, thus violating unitarity. Some physicists speculate that quantum gravity may actually be non-unitary. When this phenomenon is analyzed closer, we discover that it takes pure states to mixed states, a violation of unitarity, a fundamental property of quantum physics.
And Polchinski declared [ 13 ]:. In the following sections we will see that unitarity of the realisation of quantum transformations is a spurious postulate , since in addition to be inessential, it is also not falsifiable. The same holds for the requirement of state purity as the actual realisation of mixed states as marginal of pure entangled ones, as in most interpretations of quantum theory, e.
With these motivations we devote the entire next section to develop the theory of quantum falsification, and apply it to prove unfalsifiability of purity of quantum states, unitarity of quantum transformations, and consequently the unfalsifiability of unitary realisation of transformations and pure realisation of mixed states. In such case any effective falsification test can be achieved as a binary observation test of the form.
Then, any operator of the form. However, any of such a test would correspond to a set of binary falsification tests with the falsifier made as coarse-graining of falsifiers only, and among such tests the most efficient one being the one which coarse-grains all falsifiers into a single falsifier and all inconclusive events into a single inconclusive event.
In the following section we will see that unitarity of the realisation of quantum transformations is actually a spurious postulate, since in addition to be inessential, it is also not falsifiable. We devote the entire next section to quantum falsification theory and apply it to prove unfalsifiability of purity of quantum states, and unitarity of quantum transformations, and consequently the unfalsifiability of unitary realisation of transformations and pure realisation of mixed states.
The impossibility of falsifying purity of a state has as an immediate consequence the impossibility of falsifying the atomicity of a transformation. Footnote 4. One has. A maximally entangled state of this kind has the general form. We are now in position to prove the following theorem. The application of the operator to a fixed maximally-entangled state puts isometricity transformations in one-to-one correspondence with maximally entangled states.
Thus, being able to falsify maximal entanglement would allow to falsify isometricity. Obviously Theorem 4 exclude the possibility to falsify unitarity of a transformation, since it is a special case of isometricity.
Consider the general purification scheme in Eq. The impossibility of falsifying the unitarity of a transformation Theorem 3 with input and output systems under our control excludes the possibility of falsifying that a transformation is actually achieved unitarily, according to the scheme. Some authors argue that unobservable physics e. However, I think that we should keep cosmology as an exception.
Quantum Theory should be taken at a completely different level of consideration. It is a mature theory, it is under lab control, and, by its own nature, it categorises the same rules for experiments.
For such a theory, falsifiability , at least in principle , is a necessary requirement. The case of unitarity and the information paradox is paradigmatic in this respect, and one may legitimately ask what is the point in keeping within the theory an inessential metaphysical statement, without which the theory perfectly stands on its own legs. Somebody may argue that unitarity is dictated by a more refined theory, e. However, although this is the case for the free theory, it no longer survives the interacting one.
Footnote 6. If not falsifiable and inessential, why then unitarity is so relevant to the theory? Download Download PDF. Translate PDF. Even the passage from the micro-cosmos to the macro-cosmos, and reciprocally, can generate unsolved questions or counter-intuitive ideas.
We define a quasi-paradox as a statement which has a prima facie self-contradictory support or an explicit contradiction, but which is not completely proven as a paradox.
We present herein four elementary quantum quasi-paradoxes and their corresponding quantum Sorites paradoxes, which form a class of quantum quasi-paradoxes. Some paradoxes require the revi- 2. It is always possible to remove a par- The Sorites paradoxes are associated with Eubulides ticle from an object in such a way that what is of Miletus fourth century B.
However, repeating is not a clear frontier between visible and invisible matter, and repeating this process, at some point, the determinist and indeterminist principle, stable and unstable visible object is decomposed so that the left part matter, long time living and short time living matter. Some of the below quantum quasi-paradoxes can later be 2. Borowski E. The Harper Collins Diction- ary of Mathematics. Incoherent states, density matrix pdf Simple exercises in 2D vector calculus pdf 2.
Operators in Hilbert space pdf Hermitian operators and eigen functions Commutators, uncertainty relations Unitary transformations, operators, propagators pdf Symmetries of matrix elements Projection operators, density matrix pdf 3. Matrix formalism of QM pdf Matrix representations of vectors and operators Eigen value equations, matrix diagonalization Invariants Trace 4.
Density matrix formalism for mixed quantum states pdf Liouville equation Bi-partite systems, spin mixtures pdf Report Approximations 1. Zeeman and Stark effects pdf 3. Fermi's Golden Rule pdf 5.
WKB approximation pdf 6. Variational theory 7. Multipole elm fields ppt Report Simple Systems 1. Stationary states in finite quantum wells pdf , double wells pdf 2. Gaussian particle wave packets pdf 3.
Particle on a ring, in a 3D box pdf1 , pdf2 4. Periodic conditions, crystal lattices pdf 5. Harmonic oscillator ppt, pdf 6. Mean fiel d , central potentials pdf. Rotations and angular momentum pdf Ladder operators Rotational functions, finite rotations ppt Nuclear Shell Model pdf 3.
Fermion spins , spinors, Pauli matrices SU 2 pdf 4. Angular-momentum coupling pdf , 5. Spherical tensors pdf 6. Clebsch-Gordan coefficients, Wigner Eckhart Theorem Reports From Classical to Quantum Mechanics pdf 1.
Poisson brackets and commutators 2. Ehrenfest Theorem 2. Wave packet dynamic s 4. Identical objects particles, field quanta Non-interacting Fermions and Bosons Creation and annihilation operators
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