Superpositions, transition probabilities and
primitive observables in infinite quantum systems
Detlev Buchholz & Erling Størmer
“Quantum Sciences”
IHES
March 19, 2015
1 / 19
Motivation
Simple systems (“particles”) in quantum mechanics are
described by pure states (maximal information)
satify superposition principle (interference, entanglement)
admit statistical interpretation (transition probabilities)
Meaningful for finite systems; but problems in QFT:
localized (partial) states are never pure (Reeh-Schlieder)
inevitable loss of information due to radiation (Huygens)
appearance of horizons (Unruh . . . )
Simple systems are often to be described by non–pure states.
Question: Status of superpostions, transition probabilities etc ?
2 / 19
Motivation
Simple systems (“particles”) in quantum mechanics are
described by pure states (maximal information)
satify superposition principle (interference, entanglement)
admit statistical interpretation (transition probabilities)
Meaningful for finite systems; but problems in QFT:
localized (partial) states are never pure (Reeh-Schlieder)
inevitable loss of information due to radiation (Huygens)
appearance of horizons (Unruh . . . )
Simple systems are often to be described by non–pure states.
Question: Status of superpostions, transition probabilities etc ?
2 / 19
Motivation
Simple systems (“particles”) in quantum mechanics are
described by pure states (maximal information)
satify superposition principle (interference, entanglement)
admit statistical interpretation (transition probabilities)
Meaningful for finite systems; but problems in QFT:
localized (partial) states are never pure (Reeh-Schlieder)
inevitable loss of information due to radiation (Huygens)
appearance of horizons (Unruh . . . )
Simple systems are often to be described by non–pure states.
Question: Status of superpostions, transition probabilities etc ?
2 / 19
Motivation
Simple systems (“particles”) in quantum mechanics are
described by pure states (maximal information)
satify superposition principle (interference, entanglement)
admit statistical interpretation (transition probabilities)
Meaningful for finite systems; but problems in QFT:
localized (partial) states are never pure (Reeh-Schlieder)
inevitable loss of information due to radiation (Huygens)
appearance of horizons (Unruh . . . )
Simple systems are often to be described by non–pure states.
Question: Status of superpostions, transition probabilities etc ?
2 / 19
Motivation
time
V
space
Fig. Restricted information in V due to outgoing radiation
3 / 19
Motivation
Simple systems (“particles”) in quantum mechanics are
described by pure states (maximal information)
satify superposition principle (interference, entanglement)
admit statistical interpretation (transition probabilities)
Meaningful for finite systems; but problems in QFT:
localized (partial) states are never pure (Reeh-Schlieder)
inevitable loss of information due to radiation (Huygens)
appearance of horizons (Unruh . . . )
Simple systems are often to be described by non–pure states.
Question: Status of superpostions, transition probabilities etc ?
4 / 19
Motivation
Simple systems (“particles”) in quantum mechanics are
described by pure states (maximal information)
satify superposition principle (interference, entanglement)
admit statistical interpretation (transition probabilities)
Meaningful for finite systems; but problems in QFT:
localized (partial) states are never pure (Reeh-Schlieder)
inevitable loss of information due to radiation (Huygens)
appearance of horizons (Unruh . . . )
Simple systems are often to be described by non–pure states.
Question: Status of superpostions, transition probabilities etc ?
4 / 19
Motivation
Basics: (finite quantum systems)
observables: N ' B(H)
states: ω : N → C
type I factors
positive, linear, normalized functionals
pure states: ω 6= p1 ω1 + p2 ω2
(not mixed)
Pure states admit:
bijective lifts: ω → T Ω ⊂ H s.t. ω(A) = hΩ, A Ωi, A ∈ N
superpositions: ω1 , ω2 → TΩ1 , TΩ2 → T(c1 Ω1 + c2 Ω2 ) → ω12
resulting states are again pure (relative phases matter)
.
transition probabilites: ω1 , ω2 → ω1 · ω2 = |hΩ1 , Ω2 i|2
expectation values of observables ω1 · ω2 = ω1 (P2 ) = ω2 (P1 )
5 / 19
Motivation
Fact: Algebras of observables M are not always factors of type I
Task: Characterization of “simple systems” on non–type I factors M
and determination of their properties
Observation: Algebras M of interest are hyperfinite factors
Ingredients for solution:
funnels of type I algebras (replace B(H) for finite systems)
generic states on funnels (replaces concept of pure states)
primitive observables (replace minimal projections)
Message: Generic states on funnels describe “simple systems”
6 / 19
Funnels
Hyperfinite factors M generated by
N1 ⊂ N2 · · · ⊂ Nn · · · type I∞ factors with common identity
T
s.t. Nn0 Nn+1 infinite dimensional (hence type I∞ ), n ∈ N,
S
N = n Nn dense in M in strong operator topology
Remarks:
N called proper sequential type I∞ funnel [Takesaki]
C*–algebras generated by funnels N are isomorphic
different funnels generating M are related elements of In M
Present context:
N not closed, allowing unified analysis of states of any infinite type
7 / 19
Funnels
Physical interpretation:
..... N n ... N 2
N1
.....
Observables localized in increasing regions
relativistic QFT’s having split property (semilocal nets)
non–relativistic QFT’s
infinite lattice systems . . .
8 / 19
Generic states
States ω : N → C, GNS–representation (π, H, Ω)
locally normal, i.e. weakly continuous on unit balls of Nn , n ∈ N,
faithful, i.e. ω(A∗ A) = 0 implies A = 0
generic, viz. representing vector Ω cyclic for Nn0
T
Nn+1 , n ∈ N
Remark: Generic vector states form dense Gδ set [Dixmier, Marechal]
Definition
Let ω be generic. Its orbit under non–mixing operations is given by
.
ωN = {ωA = ω ◦ Ad A : A ∈ N , ωA (1) = 1} ,
.
where Ad A (B) = A∗ B A, B ∈ N .
9 / 19
Generic states
Physical interpretation:
Generic state ω describes “global background” in which physical
operations are performed (“state of the world”). These operations
produce the corresponding orbit ωN .
Examples:
vacuum state in QFT
thermal equilibrium states in QFT
Hadamard states in QFT on curved spacetime
10 / 19
Superpositions
Fix generic state ω (any type), orbit ωN . Norm distance
.
kωA − ωB k = sup |ωA (C) − ωB (C)| ,
C∈ N 1
ωA , ωB ∈ ωN .
Proposition
There exists a lift from ωN to rays in N which is
1
bijective: ωA = ωB iff B = t A for t ∈ T
2
locally continuous: if kωAm − ωA k → 0 for (bounded) Am , A ∈ Nn ,
n fixed, then tm Am → A in the strong operator topology
3
locally complete: if kωAl − ωAm k → 0 for (bounded) Al , Am ∈ Nn ,
there is A ∈ Nn such that tm Am → A and kωAm − ωA k → 0.
Note that N is a pre–Hilbert space, i.e. result analogous to lifting pure
states to rays in Hilbert space
11 / 19
Superpositions
Fix generic state ω (any type), orbit ωN . Norm distance
.
kωA − ωB k = sup |ωA (C) − ωB (C)| ,
C∈ N 1
ωA , ωB ∈ ωN .
Proposition
There exists a lift from ωN to rays in N which is
1
bijective: ωA = ωB iff B = t A for t ∈ T
2
locally continuous: if kωAm − ωA k → 0 for (bounded) Am , A ∈ Nn ,
n fixed, then tm Am → A in the strong operator topology
3
locally complete: if kωAl − ωAm k → 0 for (bounded) Al , Am ∈ Nn ,
there is A ∈ Nn such that tm Am → A and kωAm − ωA k → 0.
Note that N is a pre–Hilbert space, i.e. result analogous to lifting pure
states to rays in Hilbert space
11 / 19
Superpostions
Physical interpretation:
3
ωN maximal set of states arising from local non-mixing operations
1
superpositions of states in ωN are meaningful,
ωA , ωB → T A, T B → T (cA A + cB B) → ω(cA A+cB B) ,
and relative phase between cA , cB ∈ C matters
Mixtures:
. P
Conv ωN =
m
pm
ωAm : ωAm ∈ ωN ,
pm
> 0,
P
m pm
=1
Proposition
Let ωA =
PM
m=1 pm
ωAm ∈ ωN ; then ωA1 = · · · = ωAM = ωA .
ωN are the extreme points of Conv ωN in complete analogy to case
of pure states
12 / 19
Transition probabilities
Definition
Transition probability for ωA , ωB ∈ ωN :
.
ωA · ωB = |ω(A∗ B)|2
Remark: comparison with Uhlmann transition probability
U
ωA · ωB ≤ ωA · ωB = supΩA ,ΩB |hΩA , ΩB i|2 .
Proposition
Let ωA , ωB ∈ ωN .
1
0 ≤ ωA · ωB ≤ 1 (notion of orthogonality), ωA · ωB = ωB · ωA
2
ωA · ωB ≤ 1 −
3
ωA , ωB 7→ ωA · ωB locally continuous
4
there exist complete
families of orthogonal states ωAm ∈ ωN ,
P
m ∈ N, i.e.
ω
·
ω
B
Am = 1 for any ωB ∈ ωN .
m
1
4
kωA − ωB k2 where equality holds iff ω is pure
13 / 19
Transition probabilities
Definition
Transition probability for ωA , ωB ∈ ωN :
.
ωA · ωB = |ω(A∗ B)|2
Remark: comparison with Uhlmann transition probability
U
ωA · ωB ≤ ωA · ωB = supΩA ,ΩB |hΩA , ΩB i|2 .
Proposition
Let ωA , ωB ∈ ωN .
1
0 ≤ ωA · ωB ≤ 1 (notion of orthogonality), ωA · ωB = ωB · ωA
2
ωA · ωB ≤ 1 −
3
ωA , ωB 7→ ωA · ωB locally continuous
4
there exist complete
families of orthogonal states ωAm ∈ ωN ,
P
m ∈ N, i.e.
ω
·
ω
B
Am = 1 for any ωB ∈ ωN .
m
1
4
kωA − ωB k2 where equality holds iff ω is pure
13 / 19
Algebra of states
Input: Span ωN . By polarization formula
ω(B ∗ · A) =
3
1X k
i ω (B + i k A)∗ · (B + i k A) ∈ Span ωN .
4
k =0
Note: Expression not invariant under substitutions A 7→ tA A, B 7→ tB B.
Definition
Let ωA , ωB ∈ ωN . Corresponding product ωA × ωB ∈ Span ωN given by
.
ωA × ωB (C) = ω(A∗ B)ω(B ∗ CA) ,
C∈N,
is well defined (phases tA , tB cancel).
14 / 19
Algebra of states
Proposition
1
The map ωA , ωB 7→ ωA × ωB extends linearily in both entries to an
associative product, i.e. Span ωN is an algebra.
2
The antilinear involution † : Span ωN → Span ωN given by
P
. P
( m cm ωAm )† = ( m c m ωAm )
.
is consistent with the product, i.e. C = Span ωN is a *–algebra
3
C is an N –bimodule
Further structure:
spectral theorem exists in C (in particular: unique decomposition
of elements of Conv ωN into orthogonal states)
there exists spatial isomorphism C ↔ C H where C H ⊂ B(H) is
N –bimodule of finite rank operators.
Question: How is the type of ω, respectively of M, encoded in the structure
of the “skeleton” C of C −ω = M∗ ?
15 / 19
Algebra of states
Proposition
1
The map ωA , ωB 7→ ωA × ωB extends linearily in both entries to an
associative product, i.e. Span ωN is an algebra.
2
The antilinear involution † : Span ωN → Span ωN given by
P
. P
( m cm ωAm )† = ( m c m ωAm )
.
is consistent with the product, i.e. C = Span ωN is a *–algebra
3
C is an N –bimodule
Further structure:
spectral theorem exists in C (in particular: unique decomposition
of elements of Conv ωN into orthogonal states)
there exists spatial isomorphism C ↔ C H where C H ⊂ B(H) is
N –bimodule of finite rank operators.
Question: How is the type of ω, respectively of M, encoded in the structure
of the “skeleton” C of C −ω = M∗ ?
15 / 19
Primitive observables
Question: (When) are the transition probabilities observable?
Recall: Non–mixing operations, V ∈ N ,
ω 7→ (1/ω(V ∗ V )) ω ◦ Ad V .
Restrict to unitaries U ∗ U = 1 = UU ∗ (observable) inducing transitions
ωA 7→ ωA ◦ Ad U = ωUA ,
ωA ∈ ωN .
Standard examples: Effects of temporary perturbations of dynamics
Transition probability between initial and final states:
ωA · (ωA ◦ Ad U) = ωA · ωUA = |ωA (U)|2
Alternative interpretation: “Fidelity” of operation U in given state ωA
ωA · ωUA can be determined by measurements of U in state ωA .
16 / 19
Primitive observables
Question: (When) are the transition probabilities observable?
Recall: Non–mixing operations, V ∈ N ,
ω 7→ (1/ω(V ∗ V )) ω ◦ Ad V .
Restrict to unitaries U ∗ U = 1 = UU ∗ (observable) inducing transitions
ωA 7→ ωA ◦ Ad U = ωUA ,
ωA ∈ ωN .
Standard examples: Effects of temporary perturbations of dynamics
Transition probability between initial and final states:
ωA · (ωA ◦ Ad U) = ωA · ωUA = |ωA (U)|2
Alternative interpretation: “Fidelity” of operation U in given state ωA
ωA · ωUA can be determined by measurements of U in state ωA .
16 / 19
Primitive observables
Definition
A primitive observable is fixed by a unitary U ∈ N . For any ωA ∈ ωN ,
ωA 7→ ωUA (result of operation)
ωA · ωUA = |ωA (U)|2 (transition probability/fidelity of operation)
Primitive observables replace (generalize) minimal projections
Standard expectation values of observables can be recovered:
Proposition
Given projection E ∈ N , (finite number of) states ωA ∈ ωN , and ε > 0.
There are unitaries U ∈ N
√
1
|ωA · ωUA − ωA (E)2 | < ε, i.e. “standard probatilities ≈ fidelities”
2
ωUA (1 − E) < ε (compare von Neumann projection postulate)
17 / 19
Primitive observables
Definition
A primitive observable is fixed by a unitary U ∈ N . For any ωA ∈ ωN ,
ωA 7→ ωUA (result of operation)
ωA · ωUA = |ωA (U)|2 (transition probability/fidelity of operation)
Primitive observables replace (generalize) minimal projections
Standard expectation values of observables can be recovered:
Proposition
Given projection E ∈ N , (finite number of) states ωA ∈ ωN , and ε > 0.
There are unitaries U ∈ N
√
1
|ωA · ωUA − ωA (E)2 | < ε, i.e. “standard probatilities ≈ fidelities”
2
ωUA (1 − E) < ε (compare von Neumann projection postulate)
17 / 19
Primitive observables
Question: Is ωA · ωB operationally defined for each pair ωA , ωB ∈ ωN ?
(This would require that there are unitaries U ∈ N such that kωB − ωUA k < ε.)
Theorem (Connes, Haagerup, Størmer)
Let ω be of type IIIλ and let
1
0 ≤ λ < 1. There are ωA , ωB ∈ ωN s.t. infU kωB − ωUA k > ε(λ).
2
λ = 1. Then infU kωB − ωUA k = 0 for any ωA , ωB ∈ ωN .
Concept of transition probabilities (operationally) meaningful for
pure states ω on N [Kadison]
generic states ω on N of type III1 [Connes, Størmer]
These are exactly the two cases of interest in infinite quantum systems.
18 / 19
Summary
Generic states ω on funnels: generalization of concept of pure states
excitations – non–mixing operations ω 7→ ωA
superpositions – bijective lifts ωA 7→ T A
transition probabilities – product ωA · ωB = |ω(A∗ B)|2
primitive observables – replace projections ωA · ωUA = |ωA (U)|2
Meaningful framework for simple (elementary) quantum systems
19 / 19