Talk given at the workshop "Operator Structures in Quantum Information Theory" in Banff in February 2007. Focuses on the results of http://arxiv.org/abs/quant-ph/0606022

1. Conditional Density Operators in
Quantum Information
M. S. Leifer
Banff (12th February 2007)
quant-ph/0606022 Phys. Rev. A 74, 042310(2006)
quant-ph/0611233

2. Classical Conditional Probability
(Ω, S, µ) µ(A) = 0
A, B ∈ S
µ (A ∩ B)
Prob(B|A) =
µ (A)
Generally, A and B can be ANY events in the sample space.
Special Cases:
1. A is hypothesis, B is obser ved data - Bayesian Updating
2. A andB refer to properties of t wo distinct systems
3. A and B refer to properties of a system at 2 different times -
Stochastic Dynamics
2. and 3. - Ω = Ω1 × Ω2

3. Quantum Conditional Probability
In Quantum Theory
Ω → H Hilbert Space
S → {closed s-spaces of H}
-
µ → ρ Density operator
What is the analog of Prob(B|A) ?
Definition should ideally encompass:
1. Conditioning on classical data (measurement-update)
2. Correlations bet ween 2 subsystems w.r.t tensor product
3. Correlations bet ween same system at 2 times (TPCP maps)
4. Correlations bet ween ARBITRARY events, e.g. incompatible
obser vables
AND BE OPERATIONALLY MEANINGFUL!!!!!

4. Outline
1. Conditional Density Operator
2. Remarks on Conditional Independence
3. Choi-Jamiolkowski Revisited
4. Applications
5. Open Questions

5. 1. Conditional Density Operator
Classical case: Special cases 2. and 3. - Ω = Ω1 × Ω2
We’ll generally assume s-spaces finite and deal with them by defining
integer-valued random variables
Ω1 = {X = j}j∈{1,2,...N } Ω2 = {Y = k}k∈{1,2,...M }
P (X = j, Y = k) abbreviated to P (X, Y )
f (P (X, Y ))
f (P (X = j, Y = k)) abbreviated to
X
j
Marginal P (X) = P (X, Y )
Y
P (X, Y )
Conditional P (Y |X) =
P (X)

6. 1. Conditional Density Operator
Can easily embed this into QM case 2, i.e. w.r.t. a tensor product HX ⊗ HY
Write
ρXY = P (X = j, Y = k) |j j|X ⊗ |k k|Y
jk
Conditional Density operator
ρY |X = P (Y = k|X = j) |j j|X ⊗ |k k|Y
jk
ρY |X = ρ−1 ⊗ IY ρXY
Equivalently X
where ρX = TrY (ρXY ) and ρX is the Moore-Penrose
−1
pseudoinverse.
More generally, [ρX ⊗ IY , ρXY ] = 0 so how to generalize the
conditional density operator?

7. 1. Conditional Density Operator
A “reasonable” family of generalizations is:
n
1 1 1
− 2n − 2n
(n)
= ⊗ ⊗ IY
ρY |X ρX IY ρXY ρX
n
(∞) (n)
Cerf & Adami studied ρY |X = lim ρY |X
n→∞
(∞)
= exp (log ρXY − (log ρX ) ⊗ IY ) when well-defined.
ρY |X
Conditional von Neumann entropy
(∞)
S(Y |X) = S(X, Y ) − S(X) = −Tr ρXY log ρY |X
(1)
ρY |X which we’ll write ρY |X
Many people (including me) have studied
Can be characterized as a +ve operator satisfying
TrY ρY |X = Isupp(ρX )

8. 2. Remarks on Conditional Independence
(∞)
ρY |X
For the natural definition of conditional independence is entropic:
S(X : Y |Z) = 0
Equivalent to equality in Strong Subadditivity:
S(X, Z) + S(Y, Z) ≥ S(X, Y, Z) + S(Z)
Ruskai showed it is equivalent to
(∞) (∞)
= ⊗ IX
ρY |XZ ρY |Z
Hayden et. al. showed it is equivalent to
ρXY Z = HXY Z =
pj σXj Z (1) ⊗ τXj Z (2) HXj Z (1) ⊗ HYj Z (1)
j j j j
j j
Surprisingly, it is also equivalent to
ρXY Z = ρXZ ρ−1 ρY Z
ρXY |Z = ρX|Z ρY |Z Z

9. 2. Remarks on Conditional Independence
But this is not the “natural” definition of conditional independence for n = 1
ρY |XZ = ρY |Z ⊗ IX and ρX|Y Z = ρX|Z ⊗ IY are strictly
weaker and inequivalent to each other.
Open questions:
Is there a hierarchy of conditional independence relations for different
values of n ?
Do these have any operational significance in general?

10. 3. Choi-Jamiolkowsi Revisited
CJ isomorphism is well known. I want to think of it slightly differently, in
terms of conditional density operators
Trace Preser ving
ρY |X ∼ EY |X : B(HX ) → B(HY )
= Completely Positive
TPCP
=
ρ+ |j k|X ⊗ |j k|X
Let |X
X
jk
EY |X → ρY |X direction
ρY |X = IX ⊗ EY |X ρ+ |X
X
ρY |X → EY |X direction
EY |X (σX ) = TrXX ρ+ ⊗ ρY |X
|X σX
X

11. 3. Choi-Jamiolkowsi Revisited
What does it all mean? ρXY ∼ ρX , ρY |X ∼ ρX , EY |X
= =
Cases 2. and 3. from the intro are already unified in QM, i.e. both
experiments
Time evolution
EY |X
Prepare ρXY Prepare ρX
can be described simply by specifying a joint state ρXY .
Do expressions like Tr (MX ⊗ NY ρXY ) , where MX , NY are POVM
elements have any meaning in the time-evolution case?

12. 3. Choi-Jamiolkowski Revisited
Lemma: ρ = pj ρj is an ensemble decomposition of a
j
density matrix ρ iff there is a POVM M = M (j) s.t.
√ (j) √
ρM ρ
ρj =
pj = Tr M (j)
and
ρ
Tr M (j) ρ
−1 −1
= pj ρ
(j)
Proof sketch: Set M ρj ρ
2 2

13. 3. Choi-Jamiolkowski Revisited
M M-measurement of ρ
Input: ρ
P (M = j) = Tr M (j) ρ
Measurement probabilities:
ρ
M-preparation of ρ
ρ
Input: Generate a classical r.v. with p.d.f
P (M = j) = Tr M (j) ρ
M
Prepare the corresponding state:
√ (j) √
ρM ρ
ρj =
Tr M (j) ρ

14. 3. Choi-Jamiolkowski revisited
measurement measurement
N N
measurement measurement
Y Y
M N
X Y
ρXY EY |X
ρX ρT
X X
ρXY X
P (M, N ) is the same for any POVMs measurement preparation
MT
M
M and N .

15. 4. Applications
Relations bet ween different concepts/protocols in quantum
information, e.g. broadcasting and monogamy of entanglement.
Simplified definition of quantum sufficient statistics.
Quantum State Pooling.
Re-examination of quantum generalizations of Markov chains,
Bayesian Net works, etc.

16. 5. Open Questions
Is there a hierarchy of conditional independence relations?
Operational meaning of conditional density operator and
conditional independence for general n?
Temporal joint measurements, i.e. Tr (MXY ρXY ) ?
The general quantum conditional probability question.
Further applications in quantum information?