Dorje C. Brody - UniPaphhqp15/pdf/brody.pdf · Dorje C. Brody Brunel University London 15th...

Combined systems in PT-symmetric quantum mechanics

Dorje C. Brody

Brunel University London

15th International Workshop onPseudo-Hermitian Hamiltonians in Quantum Physics

May 18-23, 2015, University of Palermo, Italy

- 1 -

Combined systems in PT-symmetric quantum mechanics - 5 - 18 May 2015

Transition probability

The simplest situation in which a probabilistic idea arises in quantum theory isalso the simplest situation in which the concept of ‘distance’ arises.

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ξ η

ξηαα

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ξ α

ηα

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ξ

α

αη

θ

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PH

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The transition probability for the states |ξ⟩ and |η⟩ determines an angle θ:

cos21

2θ =

⟨η|ξ⟩⟨ξ|η⟩⟨ξ|ξ⟩⟨η|η⟩

. (3)

This angle defines a distance between the states |ξ⟩ and |η⟩ in PH.

Pseudo-Hermitian Hamiltonians in Quantum Physics Dorje C. Brody


Suppose we set θ = ds and |ξ⟩ = |ψ⟩, |η⟩ = |ψ⟩ + |dψ⟩.

A Taylor expansion then shows that

ds2 = 4⟨ψ|ψ⟩⟨dψ|dψ⟩ − ⟨ψ|dψ⟩⟨dψ|ψ⟩

⟨ψ|ψ⟩2, (4)

an expression known to geometers as the Fubini-Study metric.



Combined systems in Hermitian quantum mechanics

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SZ=0 ψBAψ

(Aψ B)

CP2Z

Z

ψ[A ψB]

ψ

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ψAψB

ψBAψψAψB

S=1C

S=0

Q e p

Quantum entanglement and Segre variety.

CPm × CPn ↪→ CP(m+1)(n+1)−1 . (5)



Quantum Hamiltonian dynamics

Define the state-space coordinates ({qi}, {pi}) by the expansion coefficients ofthe normalised state vector |ψ⟩ of an n-level system in terms of the energyeigenstates {|Ei⟩}:

|ψ⟩ =n−1∑i=1

√pi e

−iqi|Ei⟩ +

(1−

n−1∑i=1

pi

)12

|En⟩, (6)

Then the Schrodinger equation

i∂

∂t|ψt⟩ = H|ψt⟩ (7)

can be written in the Hamiltonian form:

qi =∂H(q, p)

∂piand pi = −∂H(q, p)

∂qi, (8)

where the function

H(q, p) =⟨ψ|H|ψ⟩⟨ψ|ψ⟩

(9)

is given by the expectation of the Hamiltonian operator in the state (6).



Nonlinear observables and nonlinear quantum dynamics

If we replace the Hamiltonian H = ⟨ψ|H|ψ⟩ by an arbitrary function of thestate |ψ⟩, then we obtain a nonlinear Schrodingier 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) =∞∑k=0

ϕk(x), (10)

where the functions ϕk(x) are homogeneous polynomials in Παβ(x) of degree k:

ϕk(x) = ϕα1α2···αkβ1β2···βk Π

β1α1(x)Πβ2

α2(x) · · ·Πβk

αk(x), (11)

and where

Π(x) =|x⟩⟨x|⟨x|x⟩

(12)

is a normalised projection operator onto the pure state |x⟩.

The coefficients ϕ······ are totally symmetric trace-free tensors of rank 2k.



A characteristic equation for the function ϕk(x) is given by

∇2ϕk = −k(n + 1)ϕk, (13)

where ∇2 is the Laplacian operator on the state space.

Thus a general nonlinear observable ϕ(x) can alternatively be expressed in theform of a linear operator Φ given by

Φ =(ϕ, ϕα1β1 , ϕ

α1α2β1β2

, ϕα1α2α3β1β2β3, · · ·

), (14)

acting on symmetric tensor products of Hilbert spaces,



Mixture and mixed state

A probabilistic mixture on the state space is represented by a density functionρ(x) ≥ 0 that integrates to one.

The information encoded in ρ(x) is equivalent to that in the density tensors(Mielnik):

ραβ =

∫Παβ(x)ρ(x)dVx,

ρα1α2β1β2=

∫Πα1β1(x)Πα2

β2(x)ρ(x)dVx, (15)

ρα1α2α3β1β2β3=

∫Πα1β1(x)Πα2

β2(x)Πα3

β3(x)ρ(x)dVx,

...

and the expectation of a nonlinear observable ϕ(x) is given by∫ρ(x)ϕ(x)dVx = ραβ ϕ

βα + ρα1α2β1β2

ϕβ1β2α1α2+ ρα1α2α3β1β2β3

ϕα1α2α3β1β2β3+ · · · . (16)



“Superluminal” communication in nonlinear quantum mechanics

For linear quantum mechanics, measurements on an entangled system yield in anidentical reduced density matrix ραβ.

However, the form of reduced density tensors ρα1α2β1β2, · · · is usually different.

It then appears that access to nonlinearity results in instantaneouscommunication.

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Combined systems in PT-symmetric quantum mechanics

The fundamental issue in

Lee et al. Phys. Rev. Lett. 112, 130404 (2014);

Chen et al. Phys. Rev. A90, 054301 (2014)

is that the analysis is performed at the Hilbert space level, without theunderstanding 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:

CPm × CPn ↪→ CP(m+1)(n+1)−1 .

One can combine state spaces of Hilbert spaces characterised by differentHamiltonians (i.e. Hamiltonians that give rise to different CPT inner product).



The second issue is that even if one examines the situation on the unphysicalHilbert 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’ CPT inner product implies thatPT symmetry cannot “describe a real physical system”.

• The Hilbert space one starts out from is not the physical one and one cannotattribute to it an ‘realism’ interpretation.

The fourth issue is the assumption that if PT-symmetric quantum mechanicscan 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 systemthere are phase transitions (exceptional points) that are absent in any Hermitiansystem; as such, they (PT & Hermitian) cannot be equivalent at all levels.


Dorje C. Brody - UniPaphhqp15/pdf/brody.pdf · Dorje C. Brody Brunel University London 15th...

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