Conditional Density Operators in  Quantum Information
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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

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 Conditional Density Operators in Quantum Information Presentation Transcript

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