Download Quantum Dynamics as Generalized Conditional Probabilities

Quantum Dynamics as
Generalized Conditional
Probabilities
M. S. Leifer
QS2006, Malta (13th July 2006)
Quantum Theory as a Meta-theory
The analogy is
based on quantum
measurement
theory....
Classical
Deterministic
Theory
apply probability
theory
Probabilistic
Theory
quantize
Quantum
Theory
...but QT is not as
abstract as PT.
Quantization
cannot be applied
to an arbitrary
theory.
Causal structure is
present in QT, but
absent from PT.
Classical Probability vs. Quantum Theory
Classical
Quantum
Probability distribution: P (X)
Joint distribution:
Transition matrix:
Conditional Prob.:
P (X, Y )
ΓY |X
P (Y |X)
Quantum State:
ρA
Joint State:
ρAB
TPCP map:
EB|A
?
Why quantum conditional probability?
Conditional probabilities allow all types of correlation to
be treated on an equal footing, whether timelike,
spacelike or completely abstract.
Causal relations are not primitive in probability theory.
Some classical probabilistic structures are defined in
terms of conditional probability.
Markov Chains
Bayesian Net works
Some Bayesians take conditional probability to be the
most fundamental notion.
See textbook by D. V. Lindley
Outline
1. Introduction
i.
The Many faces of conditional probability
ii.
Suggestions for a quantum analog of conditional probability
2. Stochastic Dynamics as Conditional Probabilities
3. Choi-Jamiolkowski Isomorphism
4. A New Isomorphism
5. Operational Interpretation
6. Application: Cloning, broadcasting & monogamy of entanglement
7. Future Directions
1. Introduction
(A) Reconstructing a joint distribution from a marginal
P (X, Y ) = P (Y |X)P (X)
(B) Bayesian Updating
P (D|H)P (H)
P (H|D) =
P (D)
(C) Stochastic Dynamics
P (Y = i) =
!"
j
ΓY |X
#
ij
P (X = j)
(D) Conditional Shannon Entropy
!
H(Y |X) = −
P (X, Y ) log2 P (Y |X)
X,Y
(E) Reduction of complexity via conditional independence
P (Y |X, Z) = P (Y |Z) ⇔ P (X, Y, Z) = P (X|Z)P (Y |Z)P (Z)
1. Introduction
(A) Reconstruction of a joint state ρAB from a marginal ρA .
(B) Updating quantum states after a measurement
ρ|M
E M (ρ)
=
Tr (M ρ)
(C) TPCP dynamics
ρB = EB|A (ρA )
(D) Conditional von Neumann Entropy
!
S(B|A) = −Tr ρAB log2 ρB|A
"
(E) Reduction of complexity via conditional independence
1. Introduction
Cerf & Adami (‘97-’99):
=
ρB|A
=
2log2 ρAB −log2 ρA ⊗IB
"
! 1
1 n
n
lim ρAB
(ρA ⊗ IB ) n
n→∞
(A) Reconstruction:
ρAB = 2log2 ρA ⊗IB +log2 ρB|A
(C) Entropy:
!
S(B|A) = S(A, B) − S(A) = −Tr ρAB log2 ρB|A
"
(E) Complexity Reduction: If log2 ρB|AC = IA ⊗ log2 ρB|C
ρABC = 2
IAB ⊗log2 ρC +log2 ρA|C ⊗IB +IA ⊗log2 ρB|C
1. Introduction
(B) Updating:
M = {M }, M > 0,
POVM:
ρ|M
Update CP-map:
E (ρ) =
M =I
M
P (M ) = Tr (M ρ)
Probability Rule:
M
!
!
j
E M (ρ)
=
Tr (M ρ)
M†
M
Aj ρAj
!
M† M
Aj Aj
=M
j
M depends on details of system-measuring device interaction.
E
1. Introduction
Is there one update rule that is more “Bayes’ rule like” than the
rest?
Traditionally (see Bub ’77 for projective measurements):
√
√
Mρ M
=
Tr (M ρ)
ρ|M
According to Fuchs (’01, ’02):
ρ|M
√
√
ρM ρ
=
Tr (M ρ)
Both reduce to Bayes’ rule when the M are projection operators and
[M , ρ] = 0
2. Dynamics as conditional probability
(a)
(b)
Y
(c)
Y
ΓY |X
ΓX|Y
X
P (Y ) =
Y
ΓX|Z
X
!
X
ΓY |Z
Z
ΓY |X P (X)
P
(X,
Y
)
G. 1: Distinct ways in which a general joint probability distribution P (X, Y ) may arise. (a) X is the cause of Y .
X
!
neration of Y must be in the temporal future of the generation of X. For example, Y may
be the result of sending X thro
Y )be in the temp
Pchannel
(X, Ydescribed
) = Γby
P (X) matrix ΓY |X . (b) Y is the cause ofPX.(X)
oisy
a stochastic
The =
generationPof(X,
X must
Y |X
ure of the generation of Y . For example, X may be the result of sending Y through a noisy channel described by a stoch
Y
trix ΓX|Y . (c) X and Y are the result of some common cause, described by a random variable
Z. They may be observe
r
acelike separation from one another, provided the points where this happens are both in the forward lightcone of the p
Y |X
ere Z was generated.
P (Y |X) = Γ
P (X, Y )
P (Y |X) =
P (X)
r a density operator ρA at t1 and then subjecting the system
In the quantum case, this is analogous to preparing
(P (X), Γ |Xto
) obtain
⇔ aPdensity
(X, Yoperator
)
Isomorphism:
dynamical evolution
according to a TPCP map EYB|A
ρB = EB|A (ρA ) at t2 .
Classically, there is no reason not to consider the two-time joint probability distribution P (X, Y ) that results f
2. Dynamics as conditional probability
(a)
(b)
B
(c)
A
B
B
EA|B
EB|A
A
Source: τAB
A
ρB = EB|A (ρA )
τAB
τA = TrB (τAB )
ρAB =?
r
ρB|A , EB|A
,?
τB|A =?
!
r
ρ
,
E
Isomorphism:
A B|A
"
⇔ τAB ?
3. Choi-Jamiolkowski Isomorphism
For bipartite pure states and operators:
RB|A =
!
jk
αjk |j!B "k|A ⇔ |Ψ!AB =
!
jk
αjk |k!A ⊗ |j!B
For mixed states and CP-maps:
EB|A (ρA ) =
!
µ
(µ)
(µ)†
RB|A ρA RB|A
⇒ τAB
#
! ""
(µ)
=
"Ψ
µ
AB
$
"
(µ) "
Ψ "
AB
3. Choi-Jamiolkowski Isomorphism
#
! +"
1
|j"A ⊗ |j"A!
Let !Φ AA! = √
dA j
Then τAB
!" + #
$ +"
%
= IA ⊗ EB|A! "Φ AA! Φ "AA!
EB|A (ρA ) =
2
dA
!
Φ
"
+"
AA
" +#
"
!
! ρA ⊗ τ A B Φ
AA!
Operational interpretation: Noisy gate teleportation.
3. Choi-Jamiolkowski Isomorphism
Remarks:
Isomorphism is basis dependent.
! +"
A basis must be chosen to define !Φ
If we restrict attention to Trace Preser ving CP-maps then
1
τA = TrB (τAB ) =
IA
dA
This is a special case of the isomorphism we want to construct
!
where
r
ρA , EB|A
1
ρA =
IA .
dA
"
⇔ τAB
AA!
.
4. A New Isomorphism
!
r
ρA , EB|A
"
→ τAB direction:
! +"
1
# +$
!
"
!
T
2
Instead of Φ
use |Φ!
! #Φ
⊗
I
=
ρ
!
!
A
AA
A
AA
AA!
r
Then τAB = IA ⊗ EB|A! (|Φ"AA! #Φ|AA! )
!
"
r
τAB → ρA , EB|A
direction:
T
Set ρA = τA ,
− 21
Let σB|A = τA
σB|A
τA = TrB (τAB )
⊗
− 21
IB τAB τA
is a density operator, satisfying
It is uniquely associated to a TPCP map
⊗ IB
!
"
1
TrB σB|A = r PA
dA
r
EB|A
: L (PA HA ) → L (HB )
via the Choi-Jamiolkowski isomorphism.
4. A New Isomorphism
7
(b)
(c)
B
(a)
B
τA
A
B
r
EB|A
τAB
Source: τAB
A
τA
A
T
ρ A = τA
n these diagrams, time flows up the page. Starting from (a), the space and time axes are interchanged and the diagram
hed out” to arrive at (b). This does not describe a possible experiment, since we cannot send system A backwards in
5. Operational Interpretation
Reminder about measurements:
M = {M }, M > 0,
POVM:
ρ|M
Update CP-map:
E (ρ) =
M =I
M
P (M ) = Tr (M ρ)
Probability Rule:
M
!
!
j
E M (ρ)
=
Tr (M ρ)
M†
M
Aj ρAj
!
M† M
Aj Aj
=M
j
M depends on details of system-measuring device interaction.
E
5. Operational Interpretation
Lemma: ρ =
!
M
P (M )ρ|M is an ensemble decomposition of a
density matrix ρ iff there is a POVM M = {M } s.t.
P (M ) = Tr (M ρ)
Proof sketch: M = P (M )ρ
ρ|M
− 12
√
√
ρM ρ
=
Tr (M ρ)
ρ|M ρ
− 12
5. Operational Interpretation
M
M -measurement of ρ
Input:
ρ
Measurement probabilities:
ρ
Updated state:
ρ|M
ρ
P (M ) = Tr (M ρ)
√
√
Mρ M
=
Tr (M ρ)
M -preparation of ρ
Input: Generate a classical r.v. with p.d.f
M
P (M ) = Tr (M ρ)
Prepare the corresponding state:
ρ|M
√
√
ρM ρ
=
Tr (M ρ)
5. Operational Interpretation
9
(b) N -measurement (c) N -measurement
(a)
M
A
M -measurement
N -measurement
N
N
B
B
N
B
τA
r
EB|A
τAB
τAB
P (M, N ) is the same in (a) and (c) for
any POVMs M and N .
A
τA
A
T
ρA = τA
M
M
M -measurement
M T -preparation
of the isomorphism are useful for the applications that follow. Firstly, t
l trace for tripartite states. To describe this, it is useful to introduce the
6. Application: Broadcasting & Monogamy
r a linear map EBC|A : L(HA ) → L(HB ⊗ HC ). The reduced map EB|A
For any TPCP map EBC|A : L (HA ) → L (HB ⊗ HC ) the
e map with the
partial
reduced
mapstrace,
are: i.e. EB|A = TrC ◦ EB|A .
r
r (ρA , EBC|A
), the isomorphism can be used to arrive at a tripartite stat
EB|A reduced
= TrC ◦ state
EBC|Aτ . EC|A
= isTrthe
B ◦E
BC|A bipartite state
gives the bipartite
This
same
AB
r
ism to the pair (ρA , EB|A
). This is summarized in the following diagram:
The following commutativity properties hold:
ρABC


TrC *
ρAB
r
(ρA , EBC|A
)

Tr
* C
r
(ρA , EB|A
).
ρAB , ρAC incompatible with being the
Therefore,
2
states
r
tativity property concerns M -measurements.
Starting
with
a
pair
(ρ
,
E
A B
reduced states of a global state ρABC .
at a bipartite
state τAB , and
then
an M -measurement can be applied t
√
r
r
EB|Anormalization
, EC|A incompatible
reduced
maps
with
being
the admitted.
reduced
⊗IB τAB M A2⊗I
,
where
the
factor
has
been
Thi
B
r
√ T
maps of a global Tmap EBC|A .
by first performing an M -measurement on ρA to obtain the pair ( M A
6. Application: Broadcasting & Monogamy
A TPCP-map EA! A!! |A : L (HA ) → L (HA! ⊗ HA!! ) is
broadcasting for a state ρA if
EA! |A (ρA ) = ρA!
EA!! |A (ρA ) = ρA!!
A TPCP-map EA! A!! |A : L (HA ) → L (HA! ⊗ HA!! ) is
cloning for a state ρA if
EA! A!! |A (ρA ) = ρA! ⊗ ρA!!
Note: For pure states cloning = broadcasting.
A TPCP-map is universal broadcasting if it is broadcasting for
every state.
6. Application: Broadcasting & Monogamy
No cloning theorem (Dieks ’82, Wootters & Zurek ‘82):
There is no map that is cloning for t wo nonorthogonal and nonidentical pure
states.
No broadcasting theorem (Barnum et. al. ‘96):
There is no map that is broadcasting for t wo noncommuting density operators.
Clearly, this implies no universal broadcasting as well.
Note that the maps EA! |A , EA!! |A are valid individually, but they
cannot be the reduced maps of a global map EA! A!! |A .
6. Application: Broadcasting & Monogamy
The maps EA! |A , EA!! |A must be related to incompatible states
τAA! , τAA!!
Theorem: If EA! A!! |A is universal broadcasting, then both τAA! , τAA!!
must be pure and maximally entangled.
Ensemble broadcasting {(p, ρ1 ), ((1 − p), ρ2 )} s.t. [ρ1 , ρ2 ] != 0
!
"
r
pρ1 + (1 − p)ρ2 , EA
⇔ τAA! A!!
! A!! |A
Theorem: There is a local operation on A that transforms both τAA!
and τAA!! into pure, entangled states with nonzero probability
of success.
7. Future Directions
Quantitative relations bet ween approximate ensemble
broadcasting and monogamy inequalities for entanglement.
More generally, useful in analyzing any qinfo protocol involving
the action of a TPCP-map on a particular ensemble rather than
the whole Hilbert space.
Can the various analogs of conditional probability be unified?
Can quantum theory be developed using an analog of conditional
probability as the fundamental notion?
Can we eliminate background causal structures entirely from the
formalism of quantum theory?