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# Conditional Density Operators in Quantum Information

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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

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### Conditional Density Operators in Quantum Information

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