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arXiv:1508.00860

How to combine three quantum states

Māris Ozols

Published Aug 4, 2015• 3 citations
arXivHTMLPDF

Abstract

We devise a ternary operation for combining three quantum states: it consists of permuting the input systems in a continuous fashion and then discarding all but one of them. This generalizes a binary operation recently studied by Audenaert et al. [arXiv:1503.04213] in the context of entropy power inequalities. Our ternary operation continuously interpolates between all such nested binary operations. Our construction is based on a unitary version of Cayley's theorem: using representation theory we show that any finite group can be naturally embedded into a continuous subgroup of the unitary group. Formally, this amounts to characterizing when a linear combination of certain permutations is unitary.

DOI
10.48550/arxiv.1508.00860

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We devise a ternary operation for combining three quantum states: it consists of permuting the input systems in a continuous fashion and then discarding all but one of them. This generalizes a binary operation recently studied by Audenaert et al. [arXiv:1503.04213] in the context of entropy power inequalities. Our ternary operation continuously interpolates between all such nested binary operations. Our construction is based on a unitary version of Cayley's theorem: using representation theory we show that any finite group can be naturally embedded into a continuous subgroup of the unitary group. Formally, this amounts to characterizing when a linear combination of certain permutations is unitary.
To simulate a quantum system with continuous degrees of freedom on a quantum computer based on quantum digits, it is necessary to reduce continuous observables (primarily coordinates and momenta) to discrete observables. We consider this problem based on expanding quantum observables in series in powers of two and three analogous to the binary and ternary representations of real numbers. The coefficients of the series (“digits”) a re, t herefore, Hermitian o perators. We investigate the corresponding quantum mechanical operators and the relations between them and show that the binary and ternary expansions of quantum observables automatically lead to renormalization of some divergent integrals and series (giving them finite values).
Quantum-mechanical concepts can be formulated in constructive finite terms without loss of their empirical content if we replace a general unitary group by a unitary representation of a finite group. Any linear representation of a finite group can be realized as a subrepresentation of a permutation representation. Thus, quantum-mechanical problems can be expressed in terms of permutation groups. This approach allows us to clarify the meaning of a number of physical concepts. Combining methods of computational group theory with Monte Carlo simulation we study a model based on representations of permutation groups.
Given two arbitrary pure states $ |\\phi>$ and $ |\\psi>$ of qubits or higher level states, we provide arguments in favor of states of the form $ \\frac{1}{\\sqrt{2}}(|\\psi> |\\phi> + i |\\phi> |\\psi>) $ instead of symmetric or anti-symmetric states, as natural candidates for optimally entangled states constructed from these states. We show that such states firstly have on the average a high value of concurrence, secondly can be constructed by a universal unitary operator independent of the input states. We also show that these states are the only ones which can be produced with perfect fidelity by any quantum operation designed for intertwining two pure states with a relative phase. A probabilistic method is proposed for producing any pre-determined relative phase into the combination of any two arbitrary states.
Abstract In this paper, we present the connection of two concepts as induced representation and partially reduced irreducible representations (PRIR) appear in the context of port-based teleportation protocols. Namely, for a given finite group G with arbitrary subgroup H, we consider a particular case of matrix irreducible representations, whose restriction to the subgroup H, as a matrix representation of H, is completely reduced to diagonal block form with an irreducible representation of H in the blocks. The basic properties of such representations are given. Then as an application of this concept, we show that the spectrum of the port-based teleportation operator acting on n systems is connected in a very simple way with the spectrum of the corresponding Jucys–Murphy operator for the symmetric group S(n−1)⊂S(n)S(n-1)\subset S(n)S(n−1)⊂S(n) S ( n - 1 ) ⊂ S ( n ) . This shows on the technical level relation between teleporation and one of the basic objects from the point of view of the representation theory of the symmetric group. This shows a deep connection between the central object describing properties of deterministic PBT schemes and objects appearing naturally in the abstract representation theory of the symmetric group. In particular, we present a new expression for the eigenvalues of the Jucys–Murphy operators based on the irreducible characters of the symmetric group. As an additional but not trivial result, we give also purely matrix proof of the Frobenius reciprocity theorem for characters with explicit construction of the unitary matrix that realizes the reduction in the natural basis of induced representation to the reduced one.
Negentropic entanglement for which number theoretic entropy characterized by p-adic prime is negative so that entanglement carries information, is in key role in TGD inspired theory of consciousness and quantum biology. The key feature of negentropic entanglement is that density matrix is proportional to unit matrix so that assumption that state function reduction corresponds to measurement of density matrix does not imply state function reduction to one-dimensional sub-space. This special kind of degenerate density matrix emerges naturally for hierarchy heff=nh interpreted in terms of a hierarchy of dark matter phases. I have already earlier considered explicit realizations of negentropic entanglement assuming that E is invariant under group of unitary or orthogonal transformations (also subgroups of unitary group can be considered - say symplectic group). One can however consider much more general options and this leads to a connection with topological quantum computation (TQC). Entanglement matrix E equal to 1/n1/2 factor times unitary matrix U (as a special case orthogonal matrix O) defines a density matrix given by ρ=UU†/n= Idn/n, which is group invariant. One has NE respected by state function reduction if NMP is assumed. This would give huge number of negentropically entangled states providing a representation for some unitary group or its subgroup (such as symplectic group). In principle any unitary representation of any Lie group would allow representation in terms of NE. In principle any unitary representation of any Lie group would allow a representation in terms of NE. In as generalized number theory vision, a natural condition is that matrix elements of E belong to algebraic extension of p-adic numbers used so that discreted algebraic subgroups of unitary or orthogonal group are selected. This realizes evolutionary hierarchy as a hierarchy of p-adic number fields and their algebraic extensions, and one can imagine that evolution of cognition proceeds by generation of negentropically entangled systems with increasing algebraic dimensions and increasing dimension reflecting itself as an increase of largest prime power dividing n and defining p-adic prime in question. One fascinating implication is ability of TGD Universe to emulate itself like Turing machine: unitary S-matrix codes for scattering amplitudes and therefore for and negentropically entangled subsystem could represent sub-matrix for S-matrix as rules representing the laws of physics in approximation that world corresponds to n-dimension Hilbert space. Also limit n→ ∞ makes sense, especially so in p-adic context where real infinity can correspond to finite number in sense of p-adic norm. Here also dimensions n given as products of powers of infinite primes can be formally considered.