Combined systems in PT-symmetric quantum mechanics

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1 Combined systems in PT-symmetric quantum mechanics Brunel University London 15th International Workshop on May 18-23, 2015, University of Palermo, Italy - 1 -

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5 Combined systems in PT-symmetric quantum mechanics May 2015 Transition probability The simplest situation in which a probabilistic idea arises in quantum theory is also the simplest situation in which the concept of distance arises. PH CP 1 ^ η α ξ α η α ^ ξ α αβ L ξ α θ η α ξ η ^ η α ^ ξ α S 2 The transition probability for the states ξ and η determines an angle θ: cos θ = η ξ ξ η ξ ξ η η. (3) This angle defines a distance between the states ξ and η in PH.

6 Combined systems in PT-symmetric quantum mechanics May 2015 Suppose we set θ = ds and ξ = ψ, η = ψ + dψ. A Taylor expansion then shows that ds 2 = 4 ψ ψ d ψ dψ ψ dψ d ψ ψ ψ ψ, (4) 2 an expression known to geometers as the Fubini-Study metric.

7 Combined systems in PT-symmetric quantum mechanics May 2015 Combined systems in Hermitian quantum mechanics CP 3 1 CP CP 1 Z S=0 ψ ψ(a B) ψ A ψ B CP S Z =0 ψ A ψ B ψ A ψb ψ A 01 ψ B C S=1 ψ [A ψ B] Z Q e p Quantum entanglement and Segre variety. CP m CP n CP (m+1)(n+1) 1. (5)

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9 Combined systems in PT-symmetric quantum mechanics May 2015 Quantum Hamiltonian dynamics Define the state-space coordinates ({q i }, {p i }) by the expansion coefficients of the normalised state vector ψ of an n-level system in terms of the energy eigenstates { E i }: ψ = n 1 i=1 Then the Schrödinger equation pi e iq i E i + ( 1 n 1 i=1 p i )1 2 E n, (6) i t ψ t = Ĥ ψ t (7) can be written in the Hamiltonian form: H(q, p) H(q, p) q i = and ṗ i =, (8) p i q i where the function H(q, p) = ψ Ĥ ψ ψ ψ is given by the expectation of the Hamiltonian operator in the state (6). (9)

10 Combined systems in PT-symmetric quantum mechanics May 2015 Nonlinear observables and nonlinear quantum dynamics If we replace the Hamiltonian H = ψ Ĥ ψ by an arbitrary function of the state ψ, then we obtain a nonlinear Schrödingier equation (Kibble, Weinberg). More generally, let ϕ(x) be a general observable function on the state space. Then ϕ(x) admits an expansion of the form ϕ(x) = ϕ k (x), (10) k=0 where the functions ϕ k (x) are homogeneous polynomials in Π α β (x) of degree k: and where ϕ k (x) = ϕ α 1α 2 α k β 1 β 2 β k Π β 1 α 1 (x)π β 2 α 2 (x) Π β k α k (x), (11) ˆΠ(x) = x x x x is a normalised projection operator onto the pure state x. The coefficients ϕ are totally symmetric trace-free tensors of rank 2k. (12)

11 Combined systems in PT-symmetric quantum mechanics May 2015 A characteristic equation for the function ϕ k (x) is given by 2 ϕ k = k(n + 1)ϕ k, (13) where 2 is the Laplacian operator on the state space. Thus a general nonlinear observable ϕ(x) can alternatively be expressed in the form of a linear operator ˆΦ given by ( ) ˆΦ = ϕ, ϕ α 1 β 1, ϕ α 1α 2 β 1 β 2, ϕ α 1α 2 α 3 β 1 β 2 β 3,, (14) acting on symmetric tensor products of Hilbert spaces,

12 Combined systems in PT-symmetric quantum mechanics May 2015 Mixture and mixed state A probabilistic mixture on the state space is represented by a density function ρ(x) 0 that integrates to one. The information encoded in ρ(x) is equivalent to that in the density tensors (Mielnik): ρ α β = Π α β(x)ρ(x)dv x, ρ α 1α 2 β 1 β 2 = Π α 1 β 1 (x)π α 2 β 2 (x)ρ(x)dv x, (15) ρ α 1α 2 α 3 β 1 β 2 β 3 =. Π α 1 β 1 (x)π α 2 β 2 (x)π α 3 β 3 (x)ρ(x)dv x, and the expectation of a nonlinear observable ϕ(x) is given by ρ(x)ϕ(x)dv x = ρ α β ϕ β α + ρ α 1α 2 β 1 β 2 ϕ β 1β 2 α 1 α 2 + ρ α 1α 2 α 3 β 1 β 2 β 3 ϕ α 1α 2 α 3 β 1 β 2 β 3 +. (16)

13 Combined systems in PT-symmetric quantum mechanics May 2015 Superluminal communication in nonlinear quantum mechanics For linear quantum mechanics, measurements on an entangled system yield in an identical reduced density matrix ρ α β. However, the form of reduced density tensors ρ α 1α 2 β 1 β 2, is usually different. It then appears that access to nonlinearity results in instantaneous communication. z u ρ z ρ u

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16 Combined systems in PT-symmetric quantum mechanics May 2015 Combined systems in PT-symmetric quantum mechanics The fundamental issue in Lee et al. Phys. Rev. Lett. 112, (2014); Chen et al. Phys. Rev. A90, (2014) is that the analysis is performed at the Hilbert space level, without the understanding of the state space formalism. The nonlinear evolution generated by a PT-symmetric Hamiltonian on the original (or unphysical) Hilbert space is mapped into a linear evolution on the physical state space. For a combined system, the state space is represented by the Segre map: CP m CP n CP (m+1)(n+1) 1. One can combine state spaces of Hilbert spaces characterised by different Hamiltonians (i.e. Hamiltonians that give rise to different CP T inner product).

17 Combined systems in PT-symmetric quantum mechanics May 2015 The second issue is that even if one examines the situation on the unphysical Hilbert space, the implication of nonlinearity is not clear. There are loopholes in the conventional derivation of superluminal communication associated with nonlinear evolution equation. The third issue is the assertion that a global CP T inner product implies that PT symmetry cannot describe a real physical system. The Hilbert space one starts out from is not the physical one and one cannot attribute to it an realism interpretation. The fourth issue is the assumption that if PT-symmetric quantum mechanics can be mapped to a Hermitian theory, then there is nothing new in it. For a finite system, this is nearly true, except that for a PT-symmetric system there are phase transitions (exceptional points) that are absent in any Hermitian system; as such, they (PT & Hermitian) cannot be equivalent at all levels.