Slide 1

Introduction to the Quantum Theory of

OPEN SYSTEMS

Reed College Physics Seminar
13 February 2013

Dedecated to the memory of 
Richard E. Crandall (1947-2012)

Nicholas Wheeler

Slide 2

Overview:

Composite Systems / Open Systems, and why they are of interest

Essential Mathematical Notions

Zurek's Decoherence Model

Evaporation of Information into Environment

A Simple Dissipation Model

Evaporation of Energy into Environment

Essentials of Quantum Measurement Theory

Idealized Perfect Meters

Imperfect Meters & their Uses

Simultaneous Measurement of Conjugate Observables

Heisenberg Uncertainty Principle in a New Light

Slide 3

"Isolated" Mechanical Systems Do Not Exist

In classical/quantum mechanics we work theoretically with "isolated" systems, subject to the assumption that all "environmental" influences can be neglected…else subsumed in "impressed potentials" (no back action). But recall

• NEWTON'S BUCKET: Local inertia coordinated with "fixed stars."

• MACH'S PRINCIPLE: Mass/inertia meaningless in one particle universe  …conjectured to arise therefore by mysterious influence of remote masses…isotropy of mass/inertia implies effect is non-local, must be attributed to most remote masses. "When the subway jerks, it's the fixed stars that throw you down."

Einstein partly motivated by Mach's idea. Solution sought now in Higgs field? More speculatively, in "holographic" model of universe?

Experimentalists—who live in the real world—go to great pains to (approximately) isolate their systems.

Have always been amazed that experiments are possible… that one can place a bit of the universe on a bench top, tickle it without disturbing the universe, draw inferences that pertain to the whole. 

Can imagine toy universes (and spheres of human inquiry) in such "procedural reductionism" is not possible.

In short: "system isolation" is—like the "point mass" concept and (arguably) all physical concepts—an idealization, which we must on occasion be prepared to abandon. And by doing so we open up new vistas:

Slide 3

Composite Systems  vs  Open Systems

Composite systems are systems assembled—either formally ("mentally") or in physical fact—from subsystems (Subscript[𝒮, 1]), (Subscript[𝒮, 2]), ... , (Subscript[𝒮, N]).

(Subscript[ψ, 1])((Subscript[x, 1]))(Subscript[ψ, 2])((Subscript[x, 2]))⋯(Subscript[ψ, N])((Subscript[x, N]))  :   Formally composite

ψ((Subscript[x, 1]),(Subscript[x, 2]),⋯ ,(Subscript[x, N]))              :   Physically composite, interactive

Open systems are composite systems assembed from system 𝒮 and an "environment" ℰ that typically has many degrees of freedom. If the pure states of 𝒮 and ℰ lived in finite-dimensional Hilbert spaces (Subscript[ℋ, 𝒮]), (Subscript[ℋ, ℰ]) and the conjunction were merely formal, would write

(Subscript[ψ, 𝒮])=(({{(Subscript[a, 1])}, {(Subscript[a, 2])}}))       (Subscript[ψ, ℰ])=(({{(Subscript[b, 1])}, {(Subscript[b, 2])}, {(Subscript[b, 3])}, {(Subscript[b, 4])}}))

and to describe the state of the composite system 𝒮⊗ℰ borrow from matrix theory a device called the "Kronecker product":

(Subscript[ψ, 𝒮])⊗(Subscript[ψ, ℰ])=(({{(Subscript[a, 1]) (Subscript[b, 1])}, {(Subscript[a, 1]) (Subscript[b, 2])}, {(Subscript[a, 1]) (Subscript[b, 3])}, {(Subscript[a, 1]) (Subscript[b, 4])}, {(Subscript[a, 2]) (Subscript[b, 1])}, {(Subscript[a, 2]) (Subscript[b, 2])}, {(Subscript[a, 2]) (Subscript[b, 3])}, {(Subscript[a, 2]) (Subscript[b, 4])}}))

But if the conjunction is physical (interactive) we must expect to write

(Subscript[ψ, 𝒮⊗ℰ])=(({{(Subscript[ψ, 1])}, {(Subscript[ψ, 2])}, {(Subscript[ψ, 3])}, {(Subscript[ψ, 4])}, {(Subscript[ψ, 5])}, {(Subscript[ψ, 6])}, {(Subscript[ψ, 7])}, {(Subscript[ψ, 8])}}))

Typially, such vectors cannot be Kronecker factored (displayed as Kronecker products). In such "non-separable" cases the states of 𝒮 and ℰ have become "entangled."

Linear operators on elements of (Subscript[ℋ, 𝒮]), (Subscript[ℋ, ℰ]) are described

𝔸=(({{(Subscript[a, 11]), (Subscript[a, 12])}, {(Subscript[a, 21]), (Subscript[a, 22])}}))       𝔹=(({{(Subscript[b, 11]), (Subscript[b, 12]), (Subscript[b, 13]), (Subscript[b, 14])}, {(Subscript[b, 21]), (Subscript[b, 22]), (Subscript[b, 23]), (Subscript[b, 24])}, {(Subscript[b, 31]), (Subscript[b, 32]), (Subscript[b, 33]), (Subscript[b, 34])}, {(Subscript[b, 41]), (Subscript[b, 42]), (Subscript[b, 43]), (Subscript[b, 44])}}))

which on (Subscript[ℋ, 𝒮⊗ℰ]) give rise to

𝔸⊗ 𝔹 =(({{(Subscript[a, 11]) (Subscript[b, 11]), (Subscript[a, 11]) (Subscript[b, 12]), (Subscript[a, 11]) (Subscript[b, 13]), (Subscript[a, 11]) (Subscript[b, 14]), (Subscript[a, 12]) (Subscript[b, 11]), (Subscript[a, 12]) (Subscript[b, 12]), (Subscript[a, 12]) (Subscript[b, 13]), (Subscript[a, 12]) (Subscript[b, 14])}, {(Subscript[a, 11]) (Subscript[b, 21]), (Subscript[a, 11]) (Subscript[b, 22]), (Subscript[a, 11]) (Subscript[b, 23]), (Subscript[a, 11]) (Subscript[b, 24]), (Subscript[a, 12]) (Subscript[b, 21]), (Subscript[a, 12]) (Subscript[b, 22]), (Subscript[a, 12]) (Subscript[b, 23]), (Subscript[a, 12]) (Subscript[b, 24])}, {(Subscript[a, 11]) (Subscript[b, 31]), (Subscript[a, 11]) (Subscript[b, 32]), (Subscript[a, 11]) (Subscript[b, 33]), (Subscript[a, 11]) (Subscript[b, 34]), (Subscript[a, 12]) (Subscript[b, 31]), (Subscript[a, 12]) (Subscript[b, 32]), (Subscript[a, 12]) (Subscript[b, 33]), (Subscript[a, 12]) (Subscript[b, 34])}, {(Subscript[a, 11]) (Subscript[b, 41]), (Subscript[a, 11]) (Subscript[b, 42]), (Subscript[a, 11]) (Subscript[b, 43]), (Subscript[a, 11]) (Subscript[b, 44]), (Subscript[a, 12]) (Subscript[b, 41]), (Subscript[a, 12]) (Subscript[b, 42]), (Subscript[a, 12]) (Subscript[b, 43]), (Subscript[a, 12]) (Subscript[b, 44])}, {(Subscript[a, 21]) (Subscript[b, 11]), (Subscript[a, 21]) (Subscript[b, 12]), (Subscript[a, 21]) (Subscript[b, 13]), (Subscript[a, 21]) (Subscript[b, 14]), (Subscript[a, 22]) (Subscript[b, 11]), (Subscript[a, 22]) (Subscript[b, 12]), (Subscript[a, 22]) (Subscript[b, 13]), (Subscript[a, 22]) (Subscript[b, 14])}, {(Subscript[a, 21]) (Subscript[b, 21]), (Subscript[a, 21]) (Subscript[b, 22]), (Subscript[a, 21]) (Subscript[b, 23]), (Subscript[a, 21]) (Subscript[b, 24]), (Subscript[a, 22]) (Subscript[b, 21]), (Subscript[a, 22]) (Subscript[b, 22]), (Subscript[a, 22]) (Subscript[b, 23]), (Subscript[a, 22]) (Subscript[b, 24])}, {(Subscript[a, 21]) (Subscript[b, 31]), (Subscript[a, 21]) (Subscript[b, 32]), (Subscript[a, 21]) (Subscript[b, 33]), (Subscript[a, 21]) (Subscript[b, 34]), (Subscript[a, 22]) (Subscript[b, 31]), (Subscript[a, 22]) (Subscript[b, 32]), (Subscript[a, 22]) (Subscript[b, 33]), (Subscript[a, 22]) (Subscript[b, 34])}, {(Subscript[a, 21]) (Subscript[b, 41]), (Subscript[a, 21]) (Subscript[b, 42]), (Subscript[a, 21]) (Subscript[b, 43]), (Subscript[a, 21]) (Subscript[b, 44]), (Subscript[a, 22]) (Subscript[b, 41]), (Subscript[a, 22]) (Subscript[b, 42]), (Subscript[a, 22]) (Subscript[b, 43]), (Subscript[a, 22]) (Subscript[b, 44])}}))

(𝔸⊗𝔹)((Subscript[ψ, 𝒮])⊗(Subscript[ψ, ℰ]))=((Subscript[𝔸ψ, 𝒮]))⊗(𝔹 (Subscript[ψ, ℰ]))

but we must expect more generally to encounter matrices

ℂ =(({{(Subscript[c, 11]), (Subscript[c, 12]), (Subscript[c, 13]), (Subscript[c, 14]), (Subscript[c, 15]), (Subscript[c, 16]), (Subscript[c, 17]), (Subscript[c, 18])}, {(Subscript[c, 21]), (Subscript[c, 22]), □, □, □, □, □, □}, {□, □, (Subscript[c, 33]), □, □, □, □, □}, {□, □, □, (Subscript[c, 44]), □, □, □, □}, {□, □, □, □, (Subscript[c, 55]), □, □, □}, {□, □, □, □, □, (Subscript[c, 66]), □, □}, {□, □, □, □, □, □, (Subscript[c, 88]), □}, {(Subscript[c, 81]), □, □, □, □, □, □, (Subscript[c, 88])}}))

that cannot be displayed as Kronecker products.

The primary objective of the quantum theory of open systems is to describe the effect of ℰ upon the motion of 𝒮, and

The central problem is that the state of ℰ is known very imprecisely (and is anyway of no intrinsic interest). We are obliged therefore to proceed probabilistically. Two (interrelated) principal ways to proceed:

1. Devise stochastic equations of 𝒮-motion (think Brownian motion);

2. Say what one can about the motion of 𝒮⊗ℰ—thought of as a large closed system

—and obtain 𝒮-dynamics by a projective technique.

Slide 4

Density Matrices & the Partial Trace

Standard quantum theory asserts that if 𝒮 is in state ψ then the expected mean of a series of A-measurements is

⟨A⟩ = (ψ |𝔸| ψ)

But it is an idealization to suppose that the states presented to the A-meter are ψ in every instance: they are "shot from imperfect ψ-guns," drawn from ensembles that are never pure. Suppose the ensemble produces

(Subscript[ψ, 1]) with probability (Subscript[p, 1])
(Subscript[ψ, 2]) with probability (Subscript[p, 2])

Then

⟨A⟩=(Subscript[p, 1])((Subscript[ψ, 1])|𝔸|(Subscript[ψ, 1]))+(Subscript[p, 2])((Subscript[ψ, 2])|𝔸|(Subscript[ψ, 2]))

Can be written

⟨A⟩ = Tr[𝔸 ρ]

where

ρ=|(Subscript[ψ, 1]))(Subscript[p, 1])((Subscript[ψ, 1])|+|(Subscript[ψ, 2]))(Subscript[p, 2])((Subscript[ψ, 2])|

is a "density matrix." Density matrices—introduced by Weyl and 
von Neumann—are in almost all contexts the preferred state descriptors (though they describe the states not of systems but of ensembles). Density matrices 

1. are manifestly hermitian,

2. are easily seen invariably to have unit trace,

3. are positive-semidefinite (all eigenvalues non-negative)

All matrices with those properties interpretable as density matrices.

REMARK: Classical mixtures can be described by recipes (how much of what). Quantum mixtures cannot: differently weighted mixtures of different states can result in the same density matrix.

When developed in its eigenbasis,

ρ=(({{(Subscript[r, 1]), 0, 0, 0, 0}, {0, (Subscript[r, 2]), 0, 0, 0}, {0, 0, (Subscript[r, 3]), 0, 0}, {0, 0, 0, (Subscript[r, 4]), 0}, {0, 0, 0, 0, (Subscript[r, 5])}}))   Impure mixture:  Tr[ρ] = 1, 0<Tr[(ρ^2)]<1

ρ=(({{0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}))            Pure case:  Tr[ρ]=Tr[(ρ^2)]=1

Need "partial trace" (notion introduced by Dirac), defined

(Subscript[Tr, 1])[𝕄]=(∑_(k=1)^2)(Transpose[(Subscript[u, k])]⊗(Subscript[𝕀, 4])).𝕄.((Subscript[u, k])⊗(Subscript[𝕀, 4]))

(Subscript[Tr, 2])[𝕄]=(∑_(k=1)^4)((Subscript[𝕀, 2])⊗Transpose[(Subscript[v, k])]).𝕄.((Subscript[𝕀, 2])⊗(Subscript[v, k]))

(Subscript[Tr, 1])[𝔸⊗𝔹]=(({{((Subscript[a, 11])+(Subscript[a, 22])) (Subscript[b, 11]), ((Subscript[a, 11])+(Subscript[a, 22])) (Subscript[b, 12]), ((Subscript[a, 11])+(Subscript[a, 22])) (Subscript[b, 13]), ((Subscript[a, 11])+(Subscript[a, 22])) (Subscript[b, 14])}, {((Subscript[a, 11])+(Subscript[a, 22])) (Subscript[b, 21]), ((Subscript[a, 11])+(Subscript[a, 22])) (Subscript[b, 22]), ((Subscript[a, 11])+(Subscript[a, 22])) (Subscript[b, 23]), ((Subscript[a, 11])+(Subscript[a, 22])) (Subscript[b, 24])}, {((Subscript[a, 11])+(Subscript[a, 22])) (Subscript[b, 31]), ((Subscript[a, 11])+(Subscript[a, 22])) (Subscript[b, 32]), ((Subscript[a, 11])+(Subscript[a, 22])) (Subscript[b, 33]), ((Subscript[a, 11])+(Subscript[a, 22])) (Subscript[b, 34])}, {((Subscript[a, 11])+(Subscript[a, 22])) (Subscript[b, 41]), ((Subscript[a, 11])+(Subscript[a, 22])) (Subscript[b, 42]), ((Subscript[a, 11])+(Subscript[a, 22])) (Subscript[b, 43]), ((Subscript[a, 11])+(Subscript[a, 22])) (Subscript[b, 44])}}))

(Subscript[Tr, 1])[𝔸⊗𝔹]=Tr[𝔸]𝔹

(Subscript[Tr, 2])[𝔸⊗𝔹]=𝔸 Tr[𝔹]

Application does not require matrix to possess Kronecker factored structure:

ℂ=(({{(Subscript[c, 11]), (Subscript[c, 12]), (Subscript[c, 13]), (Subscript[c, 14]), (Subscript[c, 15]), (Subscript[c, 16]), (Subscript[c, 17]), (Subscript[c, 18])}, {(Subscript[c, 21]), (Subscript[c, 22]), (Subscript[c, 23]), (Subscript[c, 24]), (Subscript[c, 25]), (Subscript[c, 26]), (Subscript[c, 27]), (Subscript[c, 28])}, {(Subscript[c, 31]), (Subscript[c, 32]), (Subscript[c, 33]), (Subscript[c, 34]), (Subscript[c, 35]), (Subscript[c, 36]), (Subscript[c, 37]), (Subscript[c, 38])}, {(Subscript[c, 41]), (Subscript[c, 42]), (Subscript[c, 43]), (Subscript[c, 44]), (Subscript[c, 45]), (Subscript[c, 46]), (Subscript[c, 47]), (Subscript[c, 48])}, {(Subscript[c, 51]), (Subscript[c, 52]), (Subscript[c, 53]), (Subscript[c, 54]), (Subscript[c, 55]), (Subscript[c, 56]), (Subscript[c, 57]), (Subscript[c, 58])}, {(Subscript[c, 61]), (Subscript[c, 62]), (Subscript[c, 63]), (Subscript[c, 64]), (Subscript[c, 65]), (Subscript[c, 66]), (Subscript[c, 67]), (Subscript[c, 68])}, {(Subscript[c, 71]), (Subscript[c, 72]), (Subscript[c, 73]), (Subscript[c, 74]), (Subscript[c, 75]), (Subscript[c, 76]), (Subscript[c, 77]), (Subscript[c, 78])}, {(Subscript[c, 81]), (Subscript[c, 82]), (Subscript[c, 83]), (Subscript[c, 84]), (Subscript[c, 85]), (Subscript[c, 86]), (Subscript[c, 87]), (Subscript[c, 88])}}))

(Subscript[Tr, 1])[ℂ]=(({{(Subscript[c, 11])+(Subscript[c, 55]), (Subscript[c, 12])+(Subscript[c, 56]), (Subscript[c, 13])+(Subscript[c, 57]), (Subscript[c, 14])+(Subscript[c, 58])}, {(Subscript[c, 21])+(Subscript[c, 65]), (Subscript[c, 22])+(Subscript[c, 66]), (Subscript[c, 23])+(Subscript[c, 67]), (Subscript[c, 24])+(Subscript[c, 68])}, {(Subscript[c, 31])+(Subscript[c, 75]), (Subscript[c, 32])+(Subscript[c, 76]), (Subscript[c, 33])+(Subscript[c, 77]), (Subscript[c, 34])+(Subscript[c, 78])}, {(Subscript[c, 41])+(Subscript[c, 85]), (Subscript[c, 42])+(Subscript[c, 86]), (Subscript[c, 43])+(Subscript[c, 87]), (Subscript[c, 44])+(Subscript[c, 88])}}))

(Subscript[Tr, 2])[ℂ]=(({{(Subscript[c, 11])+(Subscript[c, 22])+(Subscript[c, 33])+(Subscript[c, 44]), (Subscript[c, 15])+(Subscript[c, 26])+(Subscript[c, 37])+(Subscript[c, 48])}, {(Subscript[c, 51])+(Subscript[c, 62])+(Subscript[c, 73])+(Subscript[c, 84]), (Subscript[c, 55])+(Subscript[c, 66])+(Subscript[c, 77])+(Subscript[c, 88])}}))

Enables us to "projectively" extract description of (Subscript[ρ, 𝒮]) from that of (Subscript[ρ, 𝒮⊗ℰ]).

Slide 5

Zurek's Decoherence Model

The effects most characteristic of quantum mechanics (wave mechanics) are interference effects:

(((({{(Subscript[ψ, 1])}, {(Subscript[ψ, 2])}})))^adj)(({{a, z}, {(OverBar[z]), b}}))(({{(Subscript[ψ, 1])}, {(Subscript[ψ, 2])}}))=a (Subscript[ψ, 1]) (Subscript[(OverBar[ψ]), 1])+z (Subscript[ψ, 2]) (Subscript[(OverBar[ψ]), 1])+(OverBar[z]) (Subscript[ψ, 1]) (Subscript[(OverBar[ψ]), 2])+b (Subscript[ψ, 2]) (Subscript[(OverBar[ψ]), 2])

=Tr[(({{a, z}, {(OverBar[z]), b}})).(({{(Subscript[(OverBar[ψ]), 1])(Subscript[ψ, 1]), (Subscript[(OverBar[ψ]), 2])(Subscript[ψ, 1])}, {(Subscript[(OverBar[ψ]), 1])(Subscript[ψ, 2]), (Subscript[(OverBar[ψ]), 2])(Subscript[ψ, 2])}}))]

Those effects are seen to derive from off-diagonal elements of the density matrix. 

"Decoherence" is the name given to processes that kill those off-diagonal terms, and thus reduce quantum statistics to ordinary statistics.

Decoherence comes about spontaneously as a result of interaction with the environment (information initially resident in 𝒮 evaporates into ℰ, as was discussed here just one year ago by Max Schlosshauer), and quantum mechanics degenerates into a probabilistic version of classical mechanics.

A tractable model of such a process was devised in 1982 by Wohciech Zurek

—then at the threshold of his illustrious career—of which I present a bare-bones instance.

Zurek takes 𝒮 to be a qubit (2-state system), and ℰ to be a population of N qubits:

I assign to N the absurdly small value of N = 2. 

One expects the composite Hamiltonian to be of the form

ℍ=(Subscript[ℍ, 𝒮])+(Subscript[ℍ, int])+(Subscript[ℍ, ℰ])

but we set (Subscript[ℍ, 𝒮])=(Subscript[ℍ, ℰ])=𝕆 and—writing

𝕀=(({{1, 0}, {0, 1}}))       σ=(({{1, 0}, {0, -1}}))

assign to (Subscript[ℍ, int]) the structure

(Subscript[ℍ, int])=σ⊗((Subscript[g, 1])σ⊗𝕀+(Subscript[g, 2])𝕀⊗σ)

=(({{(Subscript[g, 1])+(Subscript[g, 2]), 0, 0, 0, 0, 0, 0, 0}, {0, (Subscript[g, 1])-(Subscript[g, 2]), 0, 0, 0, 0, 0, 0}, {0, 0, -(Subscript[g, 1])+(Subscript[g, 2]), 0, 0, 0, 0, 0}, {0, 0, 0, -(Subscript[g, 1])-(Subscript[g, 2]), 0, 0, 0, 0}, {0, 0, 0, 0, -(Subscript[g, 1])-(Subscript[g, 2]), 0, 0, 0}, {0, 0, 0, 0, 0, -(Subscript[g, 1])+(Subscript[g, 2]), 0, 0}, {0, 0, 0, 0, 0, 0, (Subscript[g, 1])-(Subscript[g, 2]), 0}, {0, 0, 0, 0, 0, 0, 0, (Subscript[g, 1])+(Subscript[g, 2])}}))

where the gs are coupling constants. It is the diagonal structure of (Subscript[ℍ, int]) that makes the model tractable. 

The global unitary propagator is

𝕌[t]=Exp[-i (Subscript[ℍ, int])t]

(({{(e^(-i (t (Subscript[g, 1])+t (Subscript[g, 2])))), 0, 0, 0, 0, 0, 0, 0}, {0, (e^(-i (t (Subscript[g, 1])-t (Subscript[g, 2])))), 0, 0, 0, 0, 0, 0}, {0, 0, (e^(i (t (Subscript[g, 1])-t (Subscript[g, 2])))), 0, 0, 0, 0, 0}, {0, 0, 0, (e^(i (t (Subscript[g, 1])+t (Subscript[g, 2])))), 0, 0, 0, 0}, {0, 0, 0, 0, (e^(i (t (Subscript[g, 1])+t (Subscript[g, 2])))), 0, 0, 0}, {0, 0, 0, 0, 0, (e^(i (t (Subscript[g, 1])-t (Subscript[g, 2])))), 0, 0}, {0, 0, 0, 0, 0, 0, (e^(-i (t (Subscript[g, 1])-t (Subscript[g, 2])))), 0}, {0, 0, 0, 0, 0, 0, 0, (e^(-i (t (Subscript[g, 1])+t (Subscript[g, 2]))))}}))

Assuming the initial state of the global system to possess the factored structure

(Subscript[ψ, initial])=(({{(Subscript[a, 1]) (Subscript[b, 1])}, {(Subscript[a, 1]) (Subscript[b, 2])}, {(Subscript[a, 1]) (Subscript[b, 3])}, {(Subscript[a, 1]) (Subscript[b, 4])}, {(Subscript[a, 2]) (Subscript[b, 1])}, {(Subscript[a, 2]) (Subscript[b, 2])}, {(Subscript[a, 2]) (Subscript[b, 3])}, {(Subscript[a, 2]) (Subscript[b, 4])}}))

with (Subscript[a, 1]^2)+(Subscript[a, 2]^2)==1,(Subscript[b, 1]^2)+(Subscript[b, 2]^2)+(Subscript[b, 3]^2)+(Subscript[b, 4]^2)==1 we construct first (Subscript[ρ, initial]), then

ρ[t]=𝕌[t](Subscript[ρ, initial])(Subscript[𝕌, adj])[t]

and finally the partial trace that describes the moving state of 𝒮:

𝕊[t]=(Subscript[Tr, ℰ])[ρ[t]]=(({{(Subscript[S, 11])(t), S(t)}, {(SuperStar[S])(t), (Subscript[S, 22])(t)}}))

The diagonal elements are found to be constant

(Subscript[S, 11])(t)=(Subscript[a, 1])(Subscript[a, 1])                 (Subscript[S, 22])(t)=(Subscript[a, 2])(Subscript[a, 2])

and to preserve the unit trace condition, while the off-diagonal elements are a moving complex number (and its conjugate):

S(t)=(Subscript[a, 1 ])(Subscript[a, 2]){f(t)+i g(t)}

where

f(t)=Cos[2 t ((Subscript[g, 1])+(Subscript[g, 2]))] (Subscript[b, 1]^2)+Cos[2 t ((Subscript[g, 1])-(Subscript[g, 2]))] (Subscript[b, 2]^2)+Cos[2 t ((Subscript[g, 1])-(Subscript[g, 2]))] (Subscript[b, 3]^2)+Cos[2 t ((Subscript[g, 1])+(Subscript[g, 2]))] (Subscript[b, 4]^2)
g(t)=-Sin[2 t ((Subscript[g, 1])+(Subscript[g, 2]))] (Subscript[b, 1]^2)-Sin[2 t ((Subscript[g, 1])-(Subscript[g, 2]))] (Subscript[b, 2]^2)+Sin[2 t ((Subscript[g, 1])-(Subscript[g, 2]))] (Subscript[b, 3]^2)+Sin[2 t ((Subscript[g, 1])+(Subscript[g, 2]))] (Subscript[b, 4]^2)

are dictated by the initial state of ℰ (and the coupling constants).

Assign random norm-preserving values to the b-coordinates and to the coupling constants, find S(t) traces figures like this on the complex plane:

Evidently S(t) does not vanish. But look to its time-average over a brief inertial interval…

and over a brief interval at a later time (note the reduced scale):

Introduce a 3rd environmental qubit and get

which when time-averaged over an initial interval gives

and when time-averaged over a brief later interval gives

As more qubits are introduced into the environment the time-averaged extinction of the off-diagonal element becomes rapidly
  • more complex in its details
  • swifter
  • more perfect.
  
So we have

(({{(Subscript[p, 1]), ■}, {■, (Subscript[p, 2])}}))⟹(({{(Subscript[p, 1]), 0}, {0, (Subscript[p, 2])}}))    Environmentally induced

and if we introduce a term (Subscript[ℍ, 𝒮]) into the global Hamiltonian can set the diagonal terms into motion. Long way from here to a convincing account of the emergence of the classical world… but its a start.

EVAPORATION OF INFORMATION:

Environmentally-induced extinction of off-diagonal elements (decoherence) is accompanied by a transfer of information from 𝒮 to ℰ. 

The von Neumann entropy of an ensemble of states is defined

Entropy=-Tr[ρ log ρ]=-(∑_k) (Subscript[λ, k])log (Subscript[λ, k])

The most general 2 × 2 density matrix has the form

ρ=(({{a, ((ze)^(+iϕ))}, {(ze^(-iϕ)), 1-a}}))

where z ranges on 0 ⩽ z ⩽ (Subscript[z, max]) and (Subscript[z, max])=(Sqrt[a(1-a)]) arises from the requirement that all eigenvalues of ρ by non-negative: equivalently, that

Tr[(ρ^2)] ⩽ 1

The eigenvalues of ρ are

(Subscript[λ, 1])=(1/2)-(Sqrt[(1/4) (((-1+2 a))^2)+(z^2)])
(Subscript[λ, 2])=(1/2)+(Sqrt[(1/4) (((-1+2 a))^2)+(z^2)])

which we use to construct the function Entropy[a,z]

The curves rise as a advances 0 ⟶ (1/2),  descent as it continues its advance (1/2) ⟶ 1.

Entropy is a unitary invariant, so the global entropy is t-independent. 

But as z descends 0 ⟵ z the entropy is seen to increase: decoherence caused information initially resident in 𝒮 to evaporate into ℰ.

This means that the dynamical motion of (Subscript[ρ, 𝒮]) must be non-unitary.

Further evidence of non-unitarity emerges when one looks to the evolution of Tr[((Subscript[ρ, 𝒮])^2)]. Traces of all powers are unitary invariants, but we find

Evolution of Tr[((Subscript[ρ, 𝒮])^2)]

Quantum computers are unitarily controlled quantum interference devices. Now very easy to see how environmental noise—very difficult to prevent—can render them inoperative.

Slide 6

Lindblad's Equation & a Dissipation Model

In the 1970s people (Kossakowski, Sudarshan, Lindblad, others) addressed this question: How to describe—intrinsically, without explicit reference to ℰ—the non-unitary motion of

(Subscript[ρ, 𝒮])=(Subscript[Tr, ℰ])[(Subscript[ρ, 𝒮⊗ℰ])]

that is induced by the unitary motion of (Subscript[ρ, 𝒮⊗ℰ]) ?

The "Lindblad master equation" can be written

(∂_t) (Subscript[ρ, 𝒮])=-i[(Subscript[ℍ, 𝒮]),(Subscript[ρ, 𝒮])]+(∑_k)(Subscript[𝔸, k])(Subscript[ρ, 𝒮])(SuperPlus[(Subscript[𝔸, k])])-(∑_k)(((SuperPlus[(Subscript[𝔸, k])])(Subscript[𝔸, k])(Subscript[ρ, 𝒮])+(Subscript[ρ, 𝒮])(SuperPlus[(Subscript[𝔸, k])])(Subscript[𝔸, k]))/2)

where all environmental effects are folded into the structure of the matrices (Subscript[𝔸, k])[t].  I concentrate on one detail: "Kraus operations"

ρ ⟶ ρ=(∑_k)(Subscript[𝔸, k])ρ (SuperPlus[(Subscript[𝔸, k])])

•  manifestly preserve hermiticity
•  preserve unit trace if the "Kraus operators" (Subscript[𝔸, k]) satisfy

(∑_k)(SuperPlus[(Subscript[𝔸, k])])(Subscript[𝔸, k])=𝕀

•  preserve positive-definiteness if the (Subscript[𝔸, k]) satisfy a certain difficult-to-state side condition.

Kraus operations describe the most general property-preserving thing one  can do to a density matrix, and reduce to unitary transformations iff the (∑_k) contains but a single term.

DISSIPATIVE  EXAMPLE: Let

ρ=(({{(Subscript[p, 1]), z}, {z, (Subscript[p, 2])}})),    (Subscript[𝔸, 1])[t]=(({{(e^(-Γ t)), 0}, {0, 1}})),     (Subscript[𝔸, 2])[t]=(({{0, 0}, {(Sqrt[1-(e^(-2Γ t))]), 0}}))

and verify that  (SuperPlus[(Subscript[𝔸, 1])])[t](Subscript[𝔸, 1])[t]+(SuperPlus[(Subscript[𝔸, 2])])[t](Subscript[𝔸, 2])[t]=𝕀.  Compute

ρ[t]=(Subscript[𝔸, 1])[t]ρ(SuperPlus[(Subscript[𝔸, 1])])[t]+(Subscript[𝔸, 2])[t]ρ(SuperPlus[(Subscript[𝔸, 2])])[t]

=(({{(e^(-2 t Γ)) (Subscript[p, 1]), (e^(-t Γ)) z}, {(e^(-t Γ)) z, (1-(e^(-2 t Γ))) (Subscript[p, 1])+(Subscript[p, 2])}}))

which becomes

ρ[∞]=(({{0, 0}, {0, 1}}))

asymptotically. The

Expected energy[t]=Tr[(({{(Subscript[E, 1]), 0}, {0, (Subscript[E, 0])}}))ρ[t]]

proceeds

(Subscript[p, 1])(Subscript[E, 1])+(Subscript[p, 2])(Subscript[E, 0]) ⟶ (Subscript[E, 0])

and the

Energy dissipated   =(Subscript[p, 1])((Subscript[E, 1 ])-(Subscript[E, 0]))

The model illustrates the general principle that

Dissipation    ⟹    Decoherence

but not conversely (as Zurek's model demonstrates).


By reaching beyond textbook quantum mechanics we have been able to illuminate illuminate two fundamental facts of Nature:

                                    • Evaporation of information
• Evaporation of energy

Slide 7

Generalized Quantum Measurement Theory:
Simultaneous Measurement of Conjugate Observables

In classical physics "measurement" has posed a technical problem (and inspired much ingenious invention) but has not been considered to pose a conceptual problem.

In quantum physics it has—since it involves cross-talk between macroscopic classical detectors and a microscopic quantum world—posed a deep conceptual problem from the outset…a problem made more acute in recent times by the emergence of macroscopic quantum systems.

Bohr considered the classical/quantum disjunction to be fundamental (though the correct placement of the partition remained problematic).

Increasing weight has been given to the view that quantum theory should contain within itself not only the "classical world" but also an account of those physical events that we call "measurements." 

A clean account of an idealized quantum measurement process was provided by the young John von Neumann (1903-1957)

who was 29 when his Mathematical Foundations of Quantum Mechanics was published (1932), but whose ideas had circulated for several years preciously.

The upshot of von Neumann's  PROJECTION POSTULATE—which continues to serve perfectly well in almost all formal/practical contexts, though it pertains only figuratively to such common detectors as photographic plates (whch destroy the incident photons)—is illustrated in the following figures:

In the final pages of Mathematical Foundations … von Neumann sketches a dynamical model of the hypothetically abrupt & irreversible projective measurement process.

ψ [q]   :   Initial state of system 𝒮
ϕ [x]   :   Initial state of detector 𝒟, assumed to be a narrow Gaussian centered at x = 0
Ψ[q, x, 0] = ψ[q] ϕ[x]   :   Initial state of 𝒮⊗𝒟

Unitary evolution by 𝕌 (t) = Exp[-i ℍ t] with ℍ=-(1/τ)ℚ⊗ ℙ, where ℙ is momentum operator of "pointer."

Solve Schrödinger equation, get Ψ[q, x, t] = ψ[t] ϕ[x - q t/τ] which at time t = τ (end of measurement) becomes

Ψ[q,x,τ]=ψ[q]ϕ[x-q ]

Read "classical" meter projectively, get x=(Subscript[a, i]). Means

(Subscript[ψ, out])(q)≈ψ[q]× narrow Gaussian[q-(Subscript[a, 1])]

Unavoidable imprecision of detector means prepared state is actually a compact mixture. [I discussed "Quantum measurement with imprefect devices" at a Reed Physics seminar on 16 February 2000.] Here is a representation of what such a meter actually accomplishes:

The results of A-measurements will be ambiguous if the spectrum of 𝔸 is degenerate. The projection postulate carries the implication that such ambiguity can be refined/resolved by subsequent  B-measurements if but only if 𝔸 and 𝔹 commute:

In 1927 the 26-year-old Werner Heisenberg (1902-1976)

saw that certain emergent difficulties in the quantum theory then under development by Bohr & Co. could be resolved if one accepted a principle of measurement-disturbance, which he formulated

Δ x Δ p ≈ ℏ

and later illustrated by analysis of the precision achievable with a "Heisenberg microscope":

REMARK: Heisenberg's rather vague analysis has been subjected to heavy criticism over the years, and in 2012 experimental results were reported that
surpassed the limit claimed by Heisenberg: see L. A. Rozema et al, "Violation  of Heisenberg's measurement-disturbance relationship by weak measurements," Phys. Rev. Letters 109, 100404 -1.

In 1925, Born & Jordan (and independently Dirac) came to the realization that the foundation of the "matrix mechanics" pioneered by Heisenberg earlier that same year resided in the commutation relation

[𝕏,ℙ]=iℏ𝕀

Several theorists undertook in 1928 to establish a link between that commutation relation and Heisenberg's Δ x Δ p ≈ ℏ. The definitive result was achieved by Schrödinger in 1932, who perfected ideas introduced by
E. U. Condon and by H. P. Robertson in 1929, and that had been brought to his attention by Arnold Sommerfeld.

and can in the most familiar terms be phrased

(((ΔXΔP))^2) ⩾ (((ℏ/2))^2) + (([⟨(([𝕏ℙ+ℙ𝕏])/2)⟩ - ⟨𝕏⟩⟨ℙ⟩])^2) ⩾ (((ℏ/2))^2)

Schrödinger and his predecessors built on a fact that emerges when the fundamental commutator is conjoined to implications of the projection postulate:

Conjugate observables cannot possess simultaneous eigenstates.

"Simultaneous" is used here in a sense distinct from that understood by Heisenberg when he spoke of a "measurement-disturbance relationship."  Schrödinger spoke (much more precisely than Heisenberg) of a relationship necessarily latent in

A-data collected on Monday, B-data collected on Tuesday

and went on to describe the minimal uncertainty states ("coherent states") that cause the quantum correlation term to vanish.

In 1965, two obscure engineers—E. Arthurs & J. L Kelly—at Bell Labs published in a relatively obscure journal (the Bell Labs Technichal Journal) a difficult-to-read article with the perplexing title "On the simultaneous measurement of a pair of conjugate observables." Their motivation was "determining the limitations of coherent quantum mechanical amplifiers, etc." 

Their essential idea (unstated): What one cannot do with perfect detectors one can attempt to do in a best-possible way with imperfect detectors.

Arthurs & Kelly proceeded in direct imitation of von  Neumann's dynamical argument:

ψ[q]           :   Initial state of system 𝒮
(Subscript[ϕ, 1])[x]          :   Initial state of detector (Subscript[𝒟, 1])
(Subscript[ϕ, 2])[y]          :   Initial state of detector (Subscript[𝒟, 2])
Ψ[q, x, y, 0]   :   Initial state of composite system 𝒮⊗ (Subscript[𝒟, 1])⊗ (Subscript[𝒟, 2])

ℍ=(1/τ){-(Subscript[λ, 1])ℚ⊗(Subscript[ℙ, 1])⊗𝕀-(Subscript[λ, 2])ℙ⊗𝕀⊗(Subscript[ℙ, 2])}   :   Interaction Hamiltonian
ℚ and ℙ refer to 𝒮,  (Subscript[ℙ, 1]) refers to (Subscript[𝒟, 1]),  (Subscript[ℙ, 2]) refers to (Subscript[𝒟, 2])

Solve Schrödinger equation (and "elementry" but tricky business)

{(∂_t)+(1/τ)(Subscript[λ, 1])q(∂_x) -i(1/τ)(Subscript[λ, 2])(∂_q)(∂_y)}Ψ[q,x,y,y]=0

subject to assumption that initial states of detectors are "balanced Gaussians"

(Subscript[ϕ, 1])[x]=((((2/(π b))))^(1/4))Exp[-(x^2)/b]     and      (Subscript[ϕ, 2])[y]=(((((2b)/π)))^(1/4))Exp[-b (y^2)]

Read detectors at time t= τ, find pointers at positions (Subscript[x, m]) and (Subscript[y, m]), giving

(Subscript[Ψ, out])[q,(Subscript[x, m]) ,(Subscript[y, m])]=((((1/(π b))))^(1/4))Exp[-((((q-(Subscript[x, m])))^2)/(2b))+i q (Subscript[y, m])]

These are eigenstates neither of position nor of momentum  but are Schrödinger's minimal uncertainty "coherent" states, that kill the quantum correlation. 

In the "phase space formulation of quantum mechanics" they acquire—via Wigner's construction (1932)

(Subscript[P, ψ])[q,p]=(((π ℏ))^(-1))∫(OverBar[ψ[q+ξ]])(e^(2(i/ℏ)p ξ))ψ[q-ξ]dq

—the representation

P[q,p]=Exp[-((((q-(Subscript[x, m])))^2)/b)-b(((p-(Subscript[y, m])))^2)]

where  (((Δq))^2)=(1/2)b, (((Δp))^2)=(1/2)(b^(-1)) ⟹   Δq Δp=(1/2).

Wigner quasi-distribution with (Subscript[x, m])=2, (Subscript[y, m])=1, b=0.5

Within a few months of the appearance of the Arthurs-Kelly paper two applied physicists at Stanford demonstrated (C. Y. She & H. Heffner, PR  152, 1103 (1966)) that their results could be obtained from a maximal entropy principle. But otherwise, the Arthurs-Kelly paper—now recognized to be a classic—was almost totally neglected.  Their results have now been reproduced by a great many alternative lines of argument.

Slide 8

Conclusion

At some early point in my career I dismissed—perhaps too facilely—literature relating to foundational matters in physics as "so much inconclusive philosophical verbiage that could be safely ignored," a waste of time to pursue.  That is no longer so. It was arguably John S. Bell who changed all of that, who rendered the foundational studies subject to explicit calculation and experiment. 

Insights have in recent decades been developed and are today being cultivated that lie beyond the reach of contemporary textbooks but that—after they have been distilled—will, I am convinced, be prominent in the quantum texts (and philosophical monographs) of the future. I speak of developments that have been inspired by recognition of the role that information-theoretic notions are destined to play, by the expanding importance of entanglement physics, by the technical requirements of quantum computation. 

My effort here has been to demonstrate that some of those developments (many of which have deep historical roots, however exotically strange they may at first sight appear) are already quite accessible, even to undergraduates.

Things to Read

Heinz-Peter  Breuer & Francesco Petruccione, The Theory of Open Quantum Systems (Oxford, 2006).

Stephen M. Barnett, Quantum Information (Oxford, 2009).

Maximilian Schlosshauer, Decoherence and the Quantum-to-Classical Transition (Springer, 2008).

Domenico  Guilini, Erich Joos, Claus  Kiefer, Joachim Kupsch, Ion-Olimpiu Stamatescu & Heinzx-Dieter Zeh, Decoherence and the Appearance of a Classical World in Quantum Theory, Springer (1996).

Christopher Gerry & Peter Knight, Introductory Quantum Optics (Cambridge, 2005).

Michael A. Nielsen & Isaac L. Chuang, Quantum Computation and Quantum Information (Cambridge, 2000).

Wojciech H. Zurek (editor), Complexity, Entropy and the Physics of Information; Volume VIII, Santa Fe Institute Studies in the Sciences of Complexity (Addison-Wesley, 1990), proceedings of a workshop held 
29 May-10 June 1989 to which many of the leading figures in the field made contributions.

Wojciech H. Zurek, "Decoherence and the transition from quantum to classical—Revisited," Los Alamos Science, Number 27 (2002, available on the web).

E. Arthurs & J. L. Kelly Jr, "On the simultaneous measurement of a pair of conjugate observables," Bell System Tech. J. 44, 725 (1965).

N. A. Wheeler, "Simultaneous measurement of noncommuting quantum observables" (October, 2012); pdf file available by request.

N. A. Wheeler, "Harmonic oscillator—revisited: coherent states," 
(December, 2012); pdf file available by request.

Those books and papers provide references to a vast and rapidly growing literature. Consult the web for lists of the publications by (for example) 
   • Wojciech Zurek
   • William Wooters
   • Asher Peres
   • Reinhard Werner
   • Anton Zeilinger