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- 1. Quantum Dynamics as Generalized Conditional Probabilities quant-ph/0606022 M. S. Leifer University of Guelph (19th September 2006)
- 2. Quantum Theory as Generalized Probability Classical Quantum B(HA ) on a Hilbert (ΩA , SA ). Algebra of operators Algebra of R.V.’s on a sample space space HA . Probability distribution: P (X) ρA ∈ B(HA ) Quantum State: X= XP (X) X = Tr (XρA ) Expectation value: Expectation value: X Cartesian Product: ΩAB = ΩA × ΩB HAB = HA ⊗ HB Tensor Product: P (X, Y ) ρAB ∈ B(HAB ) Joint Distribution: Joint state: P (X) = P (X, Y ) ρA = TrB (ρAB ) Marginal Distn.: Partial Trace: Y Stochastic transition map: ΓY |X EB|A : B(HA ) → B(HB ) TPCP map P (X, Y ) ? Conditional Probability: P (Y |X) = P (X)
- 3. Generalized Probability Theory Quantum theory and classical probability are part of a more general theory, with B(H) replaced by a more general C ∗ algebra. In this talk, specialize to ﬁnite dimensional algebras of the form: O = B Cd1 ⊕ Cd2 ⊕ . . . ⊕ CdN We are mainly interested in these t wo special cases: Classical probability with a ﬁnite sample space Oc,d = B(C⊕d ) = B(C ⊕ C ⊕ . . . ⊕ C) ``Full’’ ﬁnite-dimensional quantum theory Oq,d = B(Cd )
- 4. Why quantum conditional probability? ``Practical’’ Reasons: Several probabilistic structures require cond. prob. or cond. independence for their deﬁnition, e.g. Markov Chains, Bayesian Net works. Better understand the relationships bet ween different qinfo tasks. ``Foundational” reasons: QT is actually more like a generalized theory of stochastic process than abstract Kolmogorov probability. Spacelike and timelike events are combined differently. Cond. prob. is the missing notion that relates the t wo. Could be relevant to applying QT in the absence of background causal structure.
- 5. Outline 1. Introduction 2. Stochastic Dynamics as Conditional Probability 3. Choi-Jamiolkowski Isomorphism 4. A New Isomorphism 5. Operational Interpretation 6. Application: Cloning, broadcasting & monogamy of entanglement 7. Future Directions
- 6. 1. Introduction: Uses of Cond. Prob. (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 ) = ΓY |X P (X) X (D) Conditional Shannon Entropy H(Y |X) = H(X, Y ) − H(X) = − P (X, Y ) log2 P (Y |X) X,Y (E) Reduction of complexity via conditional independence P (Z|X, Y ) = P (Z|X) ⇔ P (X, Y, Z) = P (X)P (Y |X)P (Z|X)
- 7. 1. Introduction: Uses of Cond. Prob. (A) Reconstructing a joint state from a marginal ρAB = f (ρA , ρB|A ) (B) Updating a state after a measurement E (j) (ρA ) ρA → Tr E (j) (ρA ) (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 Cerf & Adami achieve (A), (D), (E). I achieve (A), (B), (C).
- 8. 1. Introduction: Cerf & Adami Cerf & Adami: Let ρAB ∈ OA ⊗ OB be a density operator, and deﬁne n 1 1 1 − 2n − 2n ρB|A = lim (ρA ⊗ IB ) ρAB (ρA ⊗ IB ) n n→∞ log2 ρB|A = log2 ρAB − log2 ρA ⊗ IB ρAB = 2log2 ρA ⊗IB +log2 ρB|A (A) Reconstruction: (C) Entropy: S(B|A) = S(A, B) − S(A) = −Tr ρAB log2 ρB|A (E) Complexity Reduction: log2 ρC|AB = ρC|A ⊗ IB ⇔ log2 ρABC = log2 ρA ⊗ IB ⊗ IC + log2 ρB|A ⊗ IC + log2 ρC|A ⊗ IB S(B : C|A) = S(C|A) − S(C|A, B) = 0
- 9. 1. Introduction: An Obvious Alternative Let ρAB ∈ OA ⊗ OB be a density operator, and deﬁne: −1 −1 ρB|A = ρA 2 IB ρAB ρA 2 ⊗ ⊗ IB Properties: ρB|A is a density operator on OA ⊗ OB . PA Maximally mixed on subsystem A: TrB (ρB|A ) = r . dA (A) Reconstruction: 1 1 ρAB = ρA ⊗ IB ρAB ρA ⊗ IB 2 2
- 10. 2. Dynamics as Conditional Probability (a) (b) (c) Y Y X Y ΓY |X ΓX|Y ΓX|Z ΓY |Z X X Z P (Y ) = Γ P (X) . 1: Distinct ways inYwhich a general joint probability distribution P (X, Y ) may arise. ) (a) X is the cause of Y . P (X, Y |X X eration of Y must be in the temporal future of the generation of X. For example, Y may be the result of sending X thro P (X, described |X a stochastic matrix ΓY |X . (b) Y is the cause of X.(X) = P The generation of Y )must be in the temp P (X, X isy channelY ) = Γ byP (X) Y re of the generation of Y . For example, X may be the result of sending Y through a noisy channel described by a stoch Y rix ΓX|Y . (c) X and Y are the result of some common cause, described by a random variable Z. ) P (X, Y They may be observe P (Y |X) = ΓY one r celike separation from|X another, provided the points where this happens (Y |X) = the forward lightcone of the p P are both in P (X) re Z was generated. P (X), Γr |X ⇔ P (X, Y ) Isomorphism: Y n the quantum case, this is analogous to preparing a density operator ρA at t1 and then subjecting the system ynamical evolution according to a TPCP map EB|A to obtain a density operator ρB = EB|A (ρA ) at t2 . lassically, there is no reason not to consider the two-time joint probability distribution P (X, Y ) that results f
- 11. 2. Dynamics as conditional probability (a) (b) (c) B B A B EA|B EB|A Source: τAB A A ρB = EB|A (ρA ) τAB ρAB =? τA = TrB (τAB ) ρB|A =? τB|A =? r Isomorphism: ρA , EB|A ⇔ τAB ?
- 12. 3. Choi-Jamiolkowski Isomorphism Recall: Kraus decomposition of CP-maps EB|A : B(HA ) → B(HB ) (µ) RB|A : HA → HB (µ) (µ)† EB|A (ρA ) = RB|A ρA RB|A µ For bipartite pure states and operators: RB|A = = αjk |j k|A ⇔ |Ψ αjk |k ⊗ |j B AB A B jk jk For mixed states and CP-maps: (µ) EB|A ⇔ τAB = Ψ(µ) Ψ(µ) RB|A ⇔ Ψ(µ) AB AB µ
- 13. 3. Choi-Jamiolkowski Isomorphism 1 Φ =√ + |j ⊗ |j Let A A AA dA j Then τAB = IA ⊗ EB|A Φ+ Φ+ AA AA EB|A (ρA ) = Φ ρA ⊗ τ A B Φ 2 + + dA AA AA Operational interpretation: Noisy gate teleportation.
- 14. 3. Choi-Jamiolkowski Isomorphism Remarks: Isomorphism is basis dependent. A basis must be chosen to deﬁne Φ+ . AA If we restrict attention to Trace Preser ving CP-maps then IA τA = TrB (τAB ) = dA This is a special case of the isomorphism we want to construct r ρA , EB|A ⇔ τAB IA where ρA = . dA
- 15. 4. A New Isomorphism r ρA , EB|A → τAB direction: 1 = ⊗ IA Φ+ ρT 2 Instead of Φ + use |Φ A AA AA AA Then τAB = IA ⊗ EB|A (|Φ Φ|AA ) r AA r τAB → ρA , EB|A direction: ρA = τ A , τA = TrB (τAB ) T Set 1 −1 −2 Let τB|A = IB τAB τA 2 ⊗ ⊗ IB τA τB|A satisﬁes TrB τB|A = PA dr A Hence, it is uniquely associated to a TPCP map via the Choi-Jamiolkowski isomorphism. EB|A : B(PA HA ) → B(HB ) r
- 16. 4. A New Isomorphism 7 (b) (c) B B (a) τA A B r EB|A τAB Source: τAB A A T τA ρ A = τA these diagrams, time ﬂows up the page. Starting from (a), the space and time axes are interchanged and the diagram ed out” to arrive at (b). This does not describe a possible experiment, since we cannot send system A backwards in
- 17. 5. Operational Interpretation Reminder about measurements: M = {M }, M > 0, M =I POVM: M P (M ) = Tr (M ρ) Probability Rule: E M (ρ) = ρ|M Update CP-map: Tr (M ρ) AM ρAM † AM † AM = M E M (ρ) = j j j j j j E M depends on details of system-measuring device interaction.
- 18. 5. Operational Interpretation Lemma: ρ = P (M )ρ|M is an ensemble decomposition of a M density matrix ρ iff there is a POVM M = {M } s.t. √ √ ρM ρ = P (M ) = Tr (M ρ) and ρ|M Tr (M ρ) −1 −1 Proof sketch: Set M = P (M )ρ ρ|M ρ 2 2
- 19. 5. Operational Interpretation M-measurement of ρ M Input: ρ Measurement probabilities: P (M ) = Tr (M ρ) ρ √ √ Updated state: Mρ M = ρ|M Tr (M ρ) ρ M-preparation of ρ Input: Generate a classical r.v. with p.d.f P (M ) = Tr (M ρ) M √ √ Prepare the corresponding state: ρM ρ = ρ|M Tr (M ρ)
- 20. 5. Operational Interpretation 9 (b) N -measurement (c) N -measurement N N (a) B B N M M -measurement N -measurement A B τA r EB|A τAB T A A τA ρA = τA τAB P (M, N ) is the same in (a) and (c) for any POVMs M and N . M M M T -preparation M -measurement
- 21. C. Commutativity properties of the isomorphism 6. Application: Broadcasting & Monogamy tivity of the isomorphism are useful for the applications that follow. Firstly, the partial trace for tripartite states. To describe this, it is useful to introduce the co For any TPCP map EBC|A : B(HA ) → B(HA ⊗ HC ) the reduced 4. For a linear map EBC|A : L(HA ) → L(HB ⊗ HC ). The reduced map EB|A : L maps are: ing the map with the partial trace, i.e. EB|A = TrC ◦ EB|A . a pair (ρA , EBC|A ), EB|A isomorphism can EC|A = Trto ◦arrive at a tripartite state τ the = TrC ◦ EBC|A be used B EBC|A r r C gives the bipartite reduced state τAB . This is the same bipartite state th morphism to the pair (ρA , EB|A ). This is summarized in the following diagram: r The following commutativity properties hold: (ρA , EBC|A ) r ρABC Tr TrC C (ρA , EB|A ). r ρAB mmutativity property concerns M -measurements. Startingbeing thepair (ρA , EB|A Therefore, 2 states ρAB , ρAC incompatible with with a r rrive at a bipartite state τof a,global then an M. -measurement can be applied to s reduced states AB and state ρABC √ M A ⊗IB τAB M A ⊗IB , where the rnormalization factor has been admitted. This is r √T 2 reduced maps EB|A , EC|A incompatible with being the reduced tains by ﬁrst maps of a global map EBC|A . performing an M T -measurement on ρA to obtain the pair ( M A ρA r e isomorphism. This is summarized in the following diagram:
- 22. 6. Application: Broadcasting & Monogamy : B(HA ) → B(HA ⊗ HA ) is broadcasting A TPCP-map EA A |A for a state ρA if EA |A (ρA ) = ρA (ρA ) = ρA EA |A : B(HA ) → B(HA ⊗ HA ) is cloning for a A TPCP-map EA A |A state ρA if (ρA ) = ρA ⊗ ρA EA A |A Note: For pure states cloning = broadcasting. A TPCP-map is universal broadcasting if it is broadcasting for every state.
- 23. 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.
- 24. 6. Application: Broadcasting & Monogamy The maps EA |A , EA must be related to incompatible states |A 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 pρ1 + (1 − p)ρ2 , EA A r ⇔ τAA 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.
- 25. 7 Future Directions . Quantitative relations bet ween approximate ensemble broadcasting and monogamy inequalities for entanglement. Dynamics of systems initially correlated with the environment. Quantum pooling (joint work with R. Spekkens). How are the different analogs of conditional probability related? Is their a heirarchy of conditional independence relations? Can quantum theory be formulated using an analog of conditional probability as the fundamental notion?

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