A Bit of Algebra. Information algebras. Errors and their effects. Error detection and correction. E. Manny Knill:

Transcription

1 A Bit of Algebra Produced with pdflatex and inkscape Information algebras. Errors and their effects. Error detection and correction. E. Manny Knill: NIST Boulder TOC

2 P Lessons E A j Mat n j (C) ι : H m j=1 I j C j R ῑ :  m j=1 B(I j) 1l Cj At all times during information processing, there are A-dits that contain the information of interest.     E R E R E R ˆB ˆB ˆB Puzzles to apply the lessons to: Let U  be a logical unitary. Are all L such that L1l = U equivalent logical unitary? What is a useful logical hamiltonian? Consider the ground space G of a hamiltonian H. Requirements for G to be robust against local perturbations? What makes stabilizer codes special? NIST Boulder 1 TOC

3 Physical Systems and Control Assume: Physical systems are quantum. Simplify: State spaces are finite dimensional. 2 Bot TOC

4 Physical Systems and Control Assume: Physical systems are quantum. Simplify: State spaces are finite dimensional. Ideal, controllable qubits are in principle available. 2 Top Bot TOC

5 Physical Systems and Control Assume: Physical systems are quantum. Simplify: State spaces are finite dimensional. Ideal, controllable qubits are in principle available. Physical systems are in principle accessible via known interactions to ideal qubits of our choice. Simplify: No independent internal evolution of physical systems. 2 Top Bot TOC

6 Physical Systems and Control Assume: Physical systems are quantum. Simplify: State spaces are finite dimensional. Ideal, controllable qubits are in principle available. Physical systems are in principle accessible via known interactions to ideal qubits of our choice. Simplify: No independent internal evolution of physical systems. The interactions are quantum controllable. If H I is an interaction, then H I σ z can be turned on/off at will. Top 2 TOC

7 Accessible Observables Let P be a physical system. H is an accessible observable if we can measure H. The measurement need only be asymptotically exact. Claim: Accessible observables are the Hermitian operators of a -closed subalgebra of the operators of P. 3 Bot TOC

8 Accessible Observables Let P be a physical system. H is an accessible observable if we can measure H. The measurement need only be asymptotically exact. Claim: Accessible observables are the Hermitian operators of a -closed subalgebra of the operators of P. H P Q : Quantum controllable interaction on P Q. For any state ρ Q, tr Q (H P Q ρ Q ) is a quantum controllable evolution of P. Thm. 1 3 Top Bot TOC

9 Accessible Observables Let P be a physical system. H is an accessible observable if we can measure H. The measurement need only be asymptotically exact. Claim: Accessible observables are the Hermitian operators of a -closed subalgebra of the operators of P. H P Q : Quantum controllable interaction on P Q. For any state ρ Q, tr Q (H P Q ρ Q ) is a quantum controllable evolution of P. Thm. 1 If H is a quantum controllable evolution of P, then H is an accessible observable. Thm. 2 3 Top Bot TOC

10 Accessible Observables Let P be a physical system. H is an accessible observable if we can measure H. The measurement need only be asymptotically exact. Claim: Accessible observables are the Hermitian operators of a -closed subalgebra of the operators of P. H P Q : Quantum controllable interaction on P Q. For any state ρ Q, tr Q (H P Q ρ Q ) is a quantum controllable evolution of P. Thm. 1 If H is a quantum controllable evolution of P, then H is an accessible observable. Thm. 2 If H and F are quantum controllable, then so are αh + βf, {H, F } and i[h, F ]. Thm. 3 Top 3 TOC

11 Information P, H: Physical system and its Hilbert space. Â: An algebra of accessible operators. -closed by def. Given: State ρ of P. Â-accessible information ρ : X Â tr(xρ). 4 Bot TOC

12 Information P, H: Physical system and its Hilbert space. Â: An algebra of accessible operators. -closed by def. Given: State ρ of P. Â-accessible information ρ : X Â tr(xρ). Information Â-accessible information. An Information type is an isomorphism class of -algebras. 4 Top Bot TOC

13 Information P, H: Physical system and its Hilbert space. Â: An algebra of accessible operators. -closed by def. Given: State ρ of P. Â-accessible information ρ : X Â tr(xρ). Information Â-accessible information. An Information type is an isomorphism class of -algebras. Examples of -algebras: Complex n n matrices, Mat n (C). n-state qdits. Complex diagonal n n matrices, D(n, n) = n i=1 Mat 1(C). classical n-element alphabet. Generally: k i=1 Mat n i (C). c-q-dits? Top 4 TOC

14 Classification of -Algebras Every finite dimensional -algebra A is isomorphic to a direct sum of complete matrix algebras: A j Mat nj (C) The summands are given by the minimal non-trivial ideals of A. Hence the decomposition is unique up to reordering. 5 Bot TOC

15 Classification of -Algebras Every finite dimensional -algebra A is isomorphic to a direct sum of complete matrix algebras: A j Mat nj (C) The summands are given by the minimal non-trivial ideals of A. Hence the decomposition is unique up to reordering. Abstract Pauli operators: σ k u, u = x, y, z. k l : [σ k u, σ l v] = 0, σ k uσ k u = 1l, u v w : σ k uσ k v = iσ k w. Consider -algebras generated by: {σ 1 z}? {σ 1 x, σ 1 y}? {σ 1 x, σ 1 y, σ 2 z}? {σ 1 x + σ 2 x, σ 1 y + σ 2 y}? 5 Top Bot TOC

16 Classification of -Algebras Every finite dimensional -algebra A is isomorphic to a direct sum of complete matrix algebras: A j Mat nj (C) The summands are given by the minimal non-trivial ideals of A. Hence the decomposition is unique up to reordering. Abstract Pauli operators: σ k u, u = x, y, z. k l : [σ k u, σ l v] = 0, σ k uσ k u = 1l, u v w : σ k uσ k v = iσ k w. Consider -algebras generated by: {σ 1 z}? 1 1 {σ 1 x, σ 1 y}? {σ 1 x, σ 1 y, σ 2 z}? {σ 1 x + σ 2 x, σ 1 y + σ 2 y}? 5 Top Bot TOC

17 Classification of -Algebras Every finite dimensional -algebra A is isomorphic to a direct sum of complete matrix algebras: A j Mat nj (C) The summands are given by the minimal non-trivial ideals of A. Hence the decomposition is unique up to reordering. Abstract Pauli operators: σ k u, u = x, y, z. k l : [σ k u, σ l v] = 0, σ k uσ k u = 1l, u v w : σ k uσ k v = iσ k w. Consider -algebras generated by: {σ 1 z}? 1 1 {σ 1 x, σ 1 y}? 2 {σ 1 x, σ 1 y, σ 2 z}? {σ 1 x + σ 2 x, σ 1 y + σ 2 y}? 5 Top Bot TOC

18 Classification of -Algebras Every finite dimensional -algebra A is isomorphic to a direct sum of complete matrix algebras: A j Mat nj (C) The summands are given by the minimal non-trivial ideals of A. Hence the decomposition is unique up to reordering. Abstract Pauli operators: σ k u, u = x, y, z. k l : [σ k u, σ l v] = 0, σ k uσ k u = 1l, u v w : σ k uσ k v = iσ k w. Consider -algebras generated by: {σ 1 z}? 1 1 {σ 1 x, σ 1 y}? 2 {σ 1 x, σ 1 y, σ 2 z}? 2 2 {σ 1 x + σ 2 x, σ 1 y + σ 2 y}? 5 Top Bot TOC

19 Classification of -Algebras Every finite dimensional -algebra A is isomorphic to a direct sum of complete matrix algebras: A j Mat nj (C) The summands are given by the minimal non-trivial ideals of A. Hence the decomposition is unique up to reordering. Abstract Pauli operators: σ k u, u = x, y, z. k l : [σ k u, σ l v] = 0, σ k uσ k u = 1l, u v w : σ k uσ k v = iσ k w. Consider -algebras generated by: {σz} 1? 1 1 {σx, 1 σy} 1? 2 {σx, 1 σy, 1 σz} 2? 2 2 {σx 1 + σx, 2 σy 1 + σy} 2? 3 1 Top 5 TOC

20 Representations of -Algebras H: Hilbert space. Â: -(closed operator )algebra on H. E.g. the accessible operators. A: The corresponding information type. I.e. abstract -algebra. H decomposes canonically according to an isometry ι : H j I j C j R, where the lifting of ι to an operator -algebra isomorphism ῑ satisfies ῑ : Â B(I j ) 1l Cj. j 6 Bot TOC

21 Representations of -Algebras H: Hilbert space. Â: -(closed operator )algebra on H. E.g. the accessible operators. A: The corresponding information type. I.e. abstract -algebra. H decomposes canonically according to an isometry ι : H j I j C j R, where the lifting of ι to an operator -algebra isomorphism ῑ satisfies ῑ :  B(I j ) 1l Cj. j The commutant  of  is given by ῑ 1 ( j 1l Ij B(C j ) ) Top 6 TOC

22 Representation Examples Three spin-1/2 systems Hilbert space: Q 1 Q 2 Q 3. Real Pauli operators: σu, k u = x, y, z, k = 1, 2, 3. Â = σz 1? Â = σx, 1 σy 1? Â = Jx 12, Jy 12 = σ 1 x + σx, 2 σy 1 + σy 2? Â = J 123 x, J 123 y = σ 1 x + σ 2 x + σ 3 x, σ 1 y + σ 2 y + σ 3 y? 7 Bot TOC

23 Representation Examples Three spin-1/2 systems Hilbert space: Q 1 Q 2 Q 3. Real Pauli operators: σu, k u = x, y, z, k = 1, 2, 3. Â = σz 1? (1 4) (1 4) Â = σx, 1 σy 1? Â = Jx 12, Jy 12 = σ 1 x + σx, 2 σy 1 + σy 2? Â = J 123 x, J 123 y = σ 1 x + σ 2 x + σ 3 x, σ 1 y + σ 2 y + σ 3 y? 7 Top Bot TOC

24 Representation Examples Three spin-1/2 systems Hilbert space: Q 1 Q 2 Q 3. Real Pauli operators: σu, k u = x, y, z, k = 1, 2, 3. Â = σz 1? (1 4) (1 4) Â = σx, 1 σy 1? 2 4 Â = Jx 12, Jy 12 = σ 1 x + σx, 2 σy 1 + σy 2? Â = J 123 x, J 123 y = σ 1 x + σ 2 x + σ 3 x, σ 1 y + σ 2 y + σ 3 y? 7 Top Bot TOC

25 Representation Examples Three spin-1/2 systems Hilbert space: Q 1 Q 2 Q 3. Real Pauli operators: σu, k u = x, y, z, k = 1, 2, 3. Â = σz 1? (1 4) (1 4) Â = σx, 1 σy 1? 2 4 Â = Jx 12, Jy 12 = σ 1 x + σx, 2 σy 1 + σy 2? (1 2) (3 2) Â = J 123 x, J 123 y = σ 1 x + σ 2 x + σ 3 x, σ 1 y + σ 2 y + σ 3 y? 7 Top Bot TOC

26 Representation Examples Three spin-1/2 systems Hilbert space: Q 1 Q 2 Q 3. Real Pauli operators: σu, k u = x, y, z, k = 1, 2, 3. Â = σz 1? (1 4) (1 4) Â = σx, 1 σy 1? 2 4 Â = Jx 12, Jy 12 = σ 1 x + σx, 2 σy 1 + σy 2? (1 2) (3 2) Â = J 123 x, J 123 y = σ 1 x + σ 2 x + σ 3 x, σ 1 y + σ 2 y + σ 3 y? 4 (2 2) Top 7 TOC

27 Interpretations A j Mat n j (C) ι : H m j=1 I j C j R ῑ : Â m j=1 B(I j) 1l Cj Information: m letters in the symbol alphabet. The letter j, when realized, carries a quantum state in I j. Physics: m particle types, subject to superselection rules. Particle j s internal degrees of freedom have statespace I j. 8 TOC

28 P States A j Mat n j (C) ι : H m j=1 I j C j R ῑ : Â m j=1 B(I j) 1l Cj System state is ρ. Current information content is characterized by: ρ : A A tr(âρ) Recover statistics from Π ρ for projectors Π. Abstractly: Information normalized positive linear functional of A. Physically, note: Many states of P have the same information content. 9 TOC

29 P Pure States A j Mat n j (C) ι : H m j=1 I j C j R ῑ :  m j=1 B(I j) 1l Cj State functionals ρ on  are convex closed. Pure states are extremal state functionals. Every ρ is a mixture of pure states. The pure states are minimal for this generating property. ρ pure ρ is a subsystem product state. ρ is Â-pure if ρ is pure and 1l ρ = Bot TOC

30 P Pure States A j Mat n j (C) ι : H m j=1 I j C j R ῑ :  m j=1 B(I j) 1l Cj State functionals ρ on  are convex closed. Pure states are extremal state functionals. Every ρ is a mixture of pure states. The pure states are minimal for this generating property. ρ pure ρ is a subsystem product state. ρ is Â-pure if ρ is pure and 1l ρ = 1. State space examples. A = 1l, σ z : A = 1l, σ x, σ y, σ z : 10 Top Bot TOC

31 P Pure States A j Mat n j (C) ι : H m j=1 I j C j R ῑ :  m j=1 B(I j) 1l Cj State functionals ρ on  are convex closed. Pure states are extremal state functionals. Every ρ is a mixture of pure states. The pure states are minimal for this generating property. ρ pure ρ is a subsystem product state. ρ is Â-pure if ρ is pure and 1l ρ = 1. State space examples. A = 1l, σ z : {p : 0 + (1 p) : 1} 0 p 1 A = 1l, σ x, σ y, σ z : 10 Top Bot TOC

32 P Pure States A j Mat n j (C) ι : H m j=1 I j C j R ῑ :  m j=1 B(I j) 1l Cj State functionals ρ on  are convex closed. Pure states are extremal state functionals. Every ρ is a mixture of pure states. The pure states are minimal for this generating property. ρ pure ρ is a subsystem product state. ρ is Â-pure if ρ is pure and 1l ρ = 1. State space examples. A = 1l, σ z : {p : 0 + (1 p) : 1} 0 p 1 A = 1l, σ x, σ y, σ z : Bloch sphere. Top 10 TOC

33 Algebra Extensions Consider observable algebras  ˆB. A j Mat m j (C),   = j Π Â,j B l Mat n l (C), ˆB ˆB = l Π ˆB,l ( ) How does Mat mj (C) ÂΠ Â,j ˆB ˆBΠ ˆB,l? 11 Bot TOC

34 Algebra Extensions Consider observable algebras  ˆB. A j Mat m j (C),   = j Π Â,j B l Mat n l (C), ˆB ˆB = l Π ˆB,l ( ) How does Mat mj (C) ÂΠ Â,j ˆB ˆBΠ ˆB,l? Examples: J 12 x, Jy 12 : j = 0 j = 1 J 12 z : j z = 1 j z = 0 j z = 1 11 Top Bot TOC

35 Algebra Extensions Consider observable algebras  ˆB. A j Mat m j (C),   = j Π Â,j B l Mat n l (C), ˆB ˆB = l Π ˆB,l ( ) How does Mat mj (C) ÂΠ Â,j ˆB ˆBΠ ˆB,l? Examples: J 12 x, Jy 12 : j = 0 j = 1 J 12 z : j z = 1 j z = 0 j z = 1 11 Top Bot TOC

36 Algebra Extensions Consider observable algebras  ˆB. A j Mat m j (C),   = j Π Â,j B l Mat n l (C), ˆB ˆB = l Π ˆB,l ( ) How does Mat mj (C) ÂΠ Â,j ˆB ˆBΠ ˆB,l? Examples: J 123 x, Jy 123, swap 12, swap 23 : j = 1/2, sym j = 3/2, sym J 123 x, Jy 123 : j = 1/2 j = 3/2 11 Top Bot TOC

37 Algebra Extensions Consider observable algebras  ˆB. A j Mat m j (C),   = j Π Â,j B l Mat n l (C), ˆB ˆB = l Π ˆB,l ( ) How does Mat mj (C) ÂΠ Â,j ˆB ˆBΠ ˆB,l? Examples: J 123 x, Jy 123, swap 12, swap 23 : j = 1/2, sym j = 3/2, sym J 123 x, Jy 123 : j = 1/2 j = 3/2 Top 11 TOC

38 Algebra Extensions Consider observable algebras  ˆB. A j Mat m j (C),   = j Π Â,j B l Mat n l (C), ˆB ˆB = l Π ˆB,l Type j of  occurs d(j, l) times in type l of ˆB. Constraint: j n jd(j, l) m l. 12 Bot TOC

39 Algebra Extensions Consider observable algebras  ˆB. A j Mat m j (C),   = j Π Â,j B l Mat n l (C), ˆB ˆB = l Π ˆB,l Type j of  occurs d(j, l) times in type l of ˆB. Constraint: j n jd(j, l) m l. Bratteli diagram for B : n 1 n 2... n l.  ˆB:.. d(1, 1) d(1, 2) d(1, l) A : m 1 m 2. d(j, l).. m j.... Omit connections with d(j, l) = 0... Top 12 TOC

40 Information Type Extensions Clean information type extension: A-letters embed only once. Each A-letter embeds to a different B letter. Clean extension Bratteli diagram is a matching from A to B. B : n 1 n 2... n l... n l A : m 1 m 2... m j TOC

41 Union: Union and Concatenation A, B, C,... A B D... No novel superpositions, letters identify origin. 14 Bot TOC

42 Union: Union and Concatenation A, B, C,... A B D... No novel superpositions, letters identify origin. Concatenation: A, B, C,... ABC... The factors commute in the concatenation (free commutative product). 14 Top Bot TOC

43 Union: Union and Concatenation A, B, C,... A B D... No novel superpositions, letters identify origin. Concatenation: A, B, C,... ABC... The factors commute in the concatenation (free commutative product). Conventions for concatenating A with itself: 1. String ordering with : A B.... Multiply according to (A B...)(A B...) = AA BB Labeled products: A (1) B (2) C (3) = B (2) A (1) C (3) =.... Differently labeled symbols commute. 14 Top Bot TOC

44 Union: Union and Concatenation A, B, C,... A B D... No novel superpositions, letters identify origin. Concatenation: A, B, C,... ABC... The factors commute in the concatenation (free commutative product). Conventions for concatenating A with itself: 1. String ordering with : A B.... Multiply according to (A B...)(A B...) = AA BB Labeled products: A (1) B (2) C (3) = B (2) A (1) C (3) =.... Differently labeled symbols commute. Elementary information types: The Bit and the Qubit. Every information type is obtained from the Bit and the Qubit by union, concatenation and clean restriction. Top 14 TOC

45 Error operators Errors: Effects conditional on environment final states. conditional quantum operation: ρ i E iρe i E i : Error operators. (subnormalized) 15 Bot TOC

46 Error operators Errors: Effects conditional on environment final states. conditional quantum operation: ρ i E iρe i E i : Error operators. (subnormalized) Fix a family of possible environments. E = {αe : E is an error operator, α C} is linearly closed. Every E E can be a conditional effect ρ λeρe. Possible environments max(λ). True environment/quantum operation is not precisely known. Top 15 TOC

47 Effects on Information Error operator E: ρ EρE... ρ... EρE = E... E ρ [tr(aeρe ) = tr(e AEρ)] A j Mat n j (C) ι : H m j=1 I j C j R ῑ :  m j=1 B(I j) 1l Cj E (unconditionally) preserves Â-information if for every Â-pure state ρ: E... E ρ is proportional to... ρ on  C(1l 1l  ). 16 Bot TOC

48 Effects on Information Error operator E: ρ EρE... ρ... EρE = E... E ρ [tr(aeρe ) = tr(e AEρ)] A j Mat n j (C) ι : H m j=1 I j C j R ῑ :  m j=1 B(I j) 1l Cj E (unconditionally) preserves Â-information if for every Â-pure state ρ: E... E ρ is proportional to... ρ on  C(1l 1l  ). Without (1l 1lÂ): E could lose information to R. Unconditionally means: Without restricting ρ with respect to the C j. Varying proportionality constants as a function of ρ environment gains information about what we stored in Â. 16 Top Bot TOC

49 Effects on Information Error operator E: ρ EρE... ρ... EρE = E... E ρ [tr(aeρe ) = tr(e AEρ)] A j Mat n j (C) ι : H m j=1 I j C j R ῑ :  m j=1 B(I j) 1l Cj E (unconditionally) preserves Â-information if for every Â-pure state ρ: E... E ρ is proportional to... ρ on  C(1l 1l  ). Without (1l 1lÂ): E could lose information to R. Unconditionally means: Without restricting ρ with respect to the C j. Varying proportionality constants as a function of ρ environment gains information about what we stored in Â. Equivalencies: 1. E preserves Â-information. 2. E = E A + E R where E A = j 1l I j C j w.r.t. ι and E R 1l A = E1l is in the commutant  of Â. Thm. 4 Top 16 TOC

50 Example: The Even Bit Abstract bit: C 2 = C1l 0 + C1l 1 1l 2 b = 1l b = 1l b, 1l 01l 1 = 0. Even bit: Ĉ2 = [σzσ 1 z 2 = 1]σz 1 1l 0 1lĈ2 (σ 1 z + 1)/2 1l 1 1lĈ2 (1 σ 1 z)/2. C 2 Mat 1 (C) Mat 1 (C) ι : C 2 C 2 (C C) C 2 ῑ : Ĉ2 C C Conventionally: O the other spin-1/2 s? 17 Bot TOC

51 Example: The Even Bit Abstract bit: C 2 = C1l 0 + C1l 1 1l 2 b = 1l b = 1l b, 1l 01l 1 = 0. Even bit: Ĉ2 = [σzσ 1 z 2 = 1]σz 1 1l 0 1lĈ2 (σ 1 z + 1)/2 1l 1 1lĈ2 (1 σ 1 z)/2. C 2 Mat 1 (C) Mat 1 (C) ι : C 2 C 2 (C C) C 2 ῑ : Ĉ2 C C Conventionally: O Information-preserving errors: σ 1 z, σ 2 z, swap 12, σ 3 x,... 00: 01: 10: 11: : α1l β1l 0 0 Not information preserving: σ 1 xσ 2 x, σ 1 x, the other spin-1/2 s? Top 17 TOC

52 Loss and Error Detection E = 1lÂE + (1l 1lÂ)E = 1lÂE + 1l R E : 1lÂE: A-dit conserving. 1l R E: A-dit destroying. A j Mat n j (C) ι : H m j=1 I j C j R ῑ : Â m j=1 B(I j) 1l Cj 18 Bot TOC

53 Loss and Error Detection E = 1lÂE + (1l 1lÂ)E = 1lÂE + 1l R E : 1lÂE: A-dit conserving. 1l R E: A-dit destroying. A j Mat n j (C) ι : H m j=1 I j C j R ῑ : Â m j=1 B(I j) 1l Cj Destruction/loss of A-dit is often readily detectable. E.g. measure 1l R. Always start with state in supp(1lâ). 18 Top Bot TOC

54 Loss and Error Detection E = 1lÂE + (1l 1lÂ)E = 1lÂE + 1l R E : 1lÂE: A-dit conserving. 1l R E: A-dit destroying. A j Mat n j (C) ι : H m j=1 I j C j R ῑ :  m j=1 B(I j) 1l Cj Destruction/loss of A-dit is often readily detectable. E.g. measure 1l R. Always start with state in supp(1lâ). E is detectable if 1lÂE1l is (uncond.) Â-information preserving. Equivalently: 1. 1lÂE1l  2. E = j 1l I j C j + F, where 1lÂF 1l = 0. The set of detectable errors is linearly closed. Top 18 TOC

55 Example: The Even Bit Abstract bit: C 2 = C1l 0 + C1l 1 1l 2 b = 1l b = 1l b, 1l 01l 1 = 0. Even bit: Ĉ2 = [σzσ 1 z 2 = 1]σz 1 1l 0 1lĈ2 (σ 1 z + 1)/2 1l 1 1lĈ2 (1 σ 1 z)/2. Conventionally: O Information-preserving errors: σ 1 z, σ 2 z, swap 12, σ 3 x,... 00: 01: 10: 11: : α1l β1l 0 0 Detectable errors: σ 1 z, σ 2 z, swap 12, σ 1 x, σ 2 x... 00: 01: 10: 11: : α1l 0 0 β1l 19 TOC

56 Environmental Interactions P Unitary U P E. The following are equivalent: E A j Mat n j (C) ι : H m j=1 I j C j R ῑ :  m j=1 B(I j) 1l Cj. 1l j = 1lι 1 (I j ) 1. U P E is detectable by  (w.r.t. P E). 2. For all φ E, ψ E, E φ U P E ψ E is detectable by  (w.r.t. P ). { E φ U P E ψ E } φ,ψ : Set of conditional effects inducible by E via U P E. 20 Bot TOC

57 Environmental Interactions P Unitary U P E. The following are equivalent: E A j Mat n j (C) ι : H m j=1 I j C j R ῑ :  m j=1 B(I j) 1l Cj. 1l j = 1lι 1 (I j ) 1. U P E is detectable by  (w.r.t. P E). 2. For all φ E, ψ E, E φ U P E ψ E is detectable by  (w.r.t. P ). { E φ U P E ψ E } φ,ψ : Set of conditional effects inducible by E via U P E. Hamiltonian H P E is Zeno detectable if j 1l jh P E 1l j Â. Detectable Zeno detectable. Useful for controlling U P E = 1l iɛh P E + O(ɛ 2 ). Top 20 TOC

58 P Zeno Error Control E A j Mat n j (C) ι : H m j=1 I j C j R ῑ :  m j=1 B(I j) 1l Cj. 1l j = 1lι 1 (I j C j ) Zeno error control I: ρ UρU 1lÂUρU 1l 21 Bot TOC

59 P Zeno Error Control E A j Mat n j (C) ι : H m j=1 I j C j R ῑ :  m j=1 B(I j) 1l Cj. 1l j = 1lι 1 (I j C j ) Zeno error control I: ρ UρU 1lÂUρU 1l U 1l iɛh + O(ɛ 2 ), H detectable, ρ = 1lÂρ1lÂ... 1lÂUρU 1l =... ρ + O(ɛ 2 ) 21 Top Bot TOC

60 P Zeno Error Control E A j Mat n j (C) ι : H m j=1 I j C j R ῑ :  m j=1 B(I j) 1l Cj. 1l j = 1lι 1 (I j C j ) Zeno error control I: ρ UρU 1lÂUρU 1l U 1l iɛh + O(ɛ 2 ), H detectable, ρ = 1lÂρ1lÂ... 1lÂUρU 1l =... ρ + O(ɛ 2 ) Proof: Let X Â. tr ( ) X 1lÂUρU 1l = tr ( ) X 1lÂU 1lÂρ1l U 1l = tr ( ) X 1lÂU1l ρ 1lÂU 1l = tr ( 1lÂU 1l X 1lÂU1l ρ ) = tr ( 1lÂU 1lÂ1lÂU1l X ρ ) + O(ɛ 2 ) [1l iɛh is detectable] = tr ( 1lÂ(1l iɛh)1lâ(1l + iɛh)1lâ Xρ ) + O(ɛ 2 ) = tr ( 1l Xρ ) + O(ɛ 2 ) = tr (Xρ) + O(ɛ 2 ) Top 21 TOC

61 P Zeno Error Control E A j Mat n j (C) ι : H m j=1 I j C j R ῑ : Â m j=1 B(I j) 1l Cj. 1l j = 1lι 1 (I j C j ) Zeno error control II: ρ UρU j 1l juρu 1l j 22 Bot TOC

62 P Zeno Error Control E A j Mat n j (C) ι : H m j=1 I j C j R ῑ : Â m j=1 B(I j) 1l Cj. 1l j = 1lι 1 (I j C j ) Zeno error control II: ρ UρU j 1l juρu 1l j U 1l iɛh + O(ɛ 2 ), H Zeno detectable, ρ = 1lÂρ1lÂ... j 1l juρu 1l j =... ρ + O(ɛ 2 ) Proof: Follow the proof for Zeno error control I. Top 22 TOC

63 Implementing Zeno Example: Co-spin-1/2 in three spin-1/2 systems: Â = [Jtot = 1/2] swap 12, swap 23 Zeno by measurement: while(true) j von-neumann measure Jtot if(j 1/2) exit 23 Bot TOC

64 Implementing Zeno Example: Co-spin-1/2 in three spin-1/2 systems: Â = [Jtot = 1/2] swap 12, swap 23 Zeno by measurement: while(true) j von-neumann measure Jtot if(j 1/2) exit Zeno by forced decoherence to center or via commutant: D: ρ [Jtot = 1/2]ρ[Jtot = 1/2] + [Jtot = 3/2]ρ[Jtot = 3/2] while(true) Apply D 23 Top Bot TOC

65 Implementing Zeno Example: Co-spin-1/2 in three spin-1/2 systems: Â = [Jtot = 1/2] swap 12, swap 23 Zeno by measurement: while(true) j von-neumann measure Jtot if(j 1/2) exit Zeno by forced decoherence to center or via commutant: D: ρ [Jtot = 1/2]ρ[Jtot = 1/2] + [Jtot = 3/2]ρ[Jtot = 3/2] while(true) Apply D Zeno by randomization in commutant: while(true) Randomly choose U {e iθû Ĵ} θ,û Apply ρ UρU Top 23 TOC

66 Error-Correcting Subsystems  is E-correcting if a recovery operation R s.t. for E E: R (E... E ) preserves Â-information. A j Mat n j (C) ι : H m j=1 I j C j R ῑ :  m j=1 B(I j) 1l Cj 24 Bot TOC

67 Error-Correcting Subsystems  is E-correcting if a recovery operation R s.t. for E E: R (E... E ) preserves Â-information. Alternatives for  is E-correcting : a. E is correctable by Â. b.  is an E-error-correcting subsystem. A j Mat n j (C) ι : H m j=1 I j C j R ῑ :  m j=1 B(I j) 1l Cj 24 Top Bot TOC

68 Error-Correcting Subsystems  is E-correcting if a recovery operation R s.t. for E E: R (E... E ) preserves Â-information. Alternatives for  is E-correcting : a. E is correctable by Â. b.  is an E-error-correcting subsystem. A j Mat n j (C) ι : H m j=1 I j C j R ῑ :  m j=1 B(I j) 1l Cj Equivalently: quantum operation R = l R l... R l s.t. for E E, l: R l E preserves Â-information. Thm Top Bot TOC

69 Error-Correcting Subsystems  is E-correcting if a recovery operation R s.t. for E E: R (E... E ) preserves Â-information. Alternatives for  is E-correcting : a. E is correctable by Â. b.  is an E-error-correcting subsystem. A j Mat n j (C) ι : H m j=1 I j C j R ῑ :  m j=1 B(I j) 1l Cj Equivalently: quantum operation R = l R l... R l s.t. for E E, l: R l E preserves Â-information. Thm. 5 A correctability check? 1. Linear space V of ops R s.t. RE1l  for all E E. 2. Check whether 1l span + {R R : R V} by SDP, Thm. 6 Top 24 TOC

70 Example: The Majority Bit Abstract bit: C 2 = C1l 0 + C1l 1 1l 2 b = 1l b = 1l b, 1l 01l 1 = 0. Majority bit: Ĉ2 = [ σz 1 + σz 2 + σz 3 = 3]σz 1. 1l 0 1lĈ2 (σ 1 z + 1)/2 1l 1 1lĈ2 (1 σ 1 z)/2 C 2 Mat 1 (C) Mat 1 (C) ι : C 2 C 2 C 2 (C C) C 6 ῑ : Ĉ2 C C Conventionally: O Bot TOC

71 Example: The Majority Bit Abstract bit: C 2 = C1l 0 + C1l 1 1l 2 b = 1l b = 1l b, 1l 01l 1 = 0. Majority bit: Ĉ2 = [ σz 1 + σz 2 + σz 3 = 3]σz 1. 1l 0 1lĈ2 (σ 1 z + 1)/2 1l 1 1lĈ2 (1 σ 1 z)/2 C 2 Mat 1 (C) Mat 1 (C) ι : C 2 C 2 C 2 (C C) C 6 ῑ : Ĉ2 C C Conventionally: O One-spin errors: E = span(σ k u : u = I, x, y, z; k = 1, 2, 3) Recovery operator: Measure Majority, reset to outcome. R 000 = R 001, R 010, R 100 = , , R 110, R 101, R 011 = , , R 11 = Top 25 TOC

72 Protecting Subsystems ˆB is E-protecting if a protecting operation R s.t. for E E: (E... E ) R preserves ˆB-information. A j Mat n j (C) ι : H m j=1 I j C j R ῑ : Â m j=1 B(I j) 1l Cj 26 Bot TOC

73 Protecting Subsystems ˆB is E-protecting if a protecting operation R s.t. for E E: (E... E ) R preserves ˆB-information. A j Mat n j (C) ι : H m j=1 I j C j R ῑ : Â m j=1 B(I j) 1l Cj A protectability check? 1. Linear space V of ops R s.t. ER1l ˆB ˆB for all E E. 2. Check whether 1l span + {R R : R V} by SDP,Thm Top Bot TOC

74 Protecting Subsystems ˆB is E-protecting if a protecting operation R s.t. for E E: (E... E ) R preserves ˆB-information. A j Mat n j (C) ι : H m j=1 I j C j R ῑ : Â m j=1 B(I j) 1l Cj A protectability check? 1. Linear space V of ops R s.t. ER1l ˆB ˆB for all E E. 2. Check whether 1l span + {R R : R V} by SDP,Thm. 6 Where is my information? Repetitive error correction with Â: Â E R E R E R 26 Top Bot TOC

75 Protecting Subsystems ˆB is E-protecting if a protecting operation R s.t. for E E: A j Mat n j (C) ι : H m j=1 I j C j R ῑ : Â m j=1 B(I j) 1l Cj (E... E ) R preserves ˆB-information. A protectability check? 1. Linear space V of ops R s.t. ER1l ˆB ˆB for all E E. 2. Check whether 1l span + {R R : R V} by SDP,Thm. 6 Where is my information? Repetitive error correction with Â: Â Â Â Â E R E R E R 26 Top Bot TOC

76 Protecting Subsystems ˆB is E-protecting if a protecting operation R s.t. for E E: (E... E ) R preserves ˆB-information. A j Mat n j (C) ι : H m j=1 I j C j R ῑ : Â m j=1 B(I j) 1l Cj A protectability check? 1. Linear space V of ops R s.t. ER1l ˆB ˆB for all E E. 2. Check whether 1l span + {R R : R V} by SDP,Thm. 6 Where is my information? Repetitive error correction with Â: Â E R E R E R ˆB where ˆB Â Â Â ˆB Â is E-protecting via R. ˆB Top 26 TOC

77 Example: The Full Majority Bit Abstract bit: B 2 = C1l 0 + C1l 1 1l 2 b = 1l b = 1l b, 1l 01l 1 = 0. Full Majority bit: ˆB 2 = [σz 1 + σz 2 + σz 3 > 0], [σz 1 + σz 2 + σz 3 < 0]. B 2 Mat 1 (C) Mat 1 (C) ι : C 2 C 2 C 2 (C C 4 C C 4 ) 0 ῑ : ˆB 2 C C 1l 0 [σz 1 + σz 2 + σz 3 > 0] 1l 1 [σz 1 + σz 2 + σz 3 < 0] or perhaps O span( 000, 001, 010, 100 ) 1 span( 111, 110, 101, 011 ) 27 Bot TOC

78 Example: The Full Majority Bit Abstract bit: B 2 = C1l 0 + C1l 1 1l 2 b = 1l b = 1l b, 1l 01l 1 = 0. Full Majority bit: ˆB 2 = [σz 1 + σz 2 + σz 3 > 0], [σz 1 + σz 2 + σz 3 < 0]. B 2 Mat 1 (C) Mat 1 (C) ι : C 2 C 2 C 2 (C C 4 C C 4 ) 0 ῑ : ˆB 2 C C 1l 0 [σz 1 + σz 2 + σz 3 > 0] 1l 1 [σz 1 + σz 2 + σz 3 < 0] or perhaps O span( 000, 001, 010, 100 ) 1 span( 111, 110, 101, 011 ) One-spin errors: E = span(σ k u : u = I, x, y, z; k = 1, 2, 3) Protecting operator: Measure Majority, set to outcome. R 000 = R 001, R 010, R 100 = , , R 110, R 101, R 011 = , , R 11 = Top 27 TOC

79 Protecting vs Correcting Subsystems ˆB is E-protecting if a protecting operation R s.t. for E E: (E... E ) R preserves ˆB-info. Â is E-correcting if a recovery operation R s.t. for E E: R (E... E ) preserves Â-info. Protecting subsystems: Encapsulate post-error decoding procedures. Conceptual information tracking. Stabilizer code: Non-Clifford protecting subsystems. Error-correcting subsystems: Linear test for E-correctability: Convenient for search. Encapsulate pre-error encoding procedures. Stabilizer code: Pauli logical ops, Clifford encodings. 28 TOC

80 Correction by Isomorphism Given: ˆB Â, -isomorphism γ :  ˆB. A j Mat n j (C) ι : H m j=1 I j C j R ῑ :  m j=1 B(I j) 1l Cj Error E is Â-information preserving via ˆB if for every Â-pure state ρ, E γ(...)e ρ is proportional to... ρ on  C1l R 29 Bot TOC

81 Correction by Isomorphism Given: ˆB Â, -isomorphism γ :  ˆB. A j Mat n j (C) ι : H m j=1 I j C j R ῑ :  m j=1 B(I j) 1l Cj Error E is Â-information preserving via ˆB if for every Â-pure state ρ, E γ(...)e ρ is proportional to... ρ on  C1l R Define: E( ˆB) = {E : E is Â-information preserving via ˆB}. Then  is E( ˆB)-correcting. Thm. 7 Hint: R measures ˆB, then transfers ˆB to  with  -info depolarized. 29 Top Bot TOC

82 Correction by Isomorphism Given: ˆB Â, -isomorphism γ :  ˆB. A j Mat n j (C) ι : H m j=1 I j C j R ῑ :  m j=1 B(I j) 1l Cj Error E is Â-information preserving via ˆB if for every Â-pure state ρ, E γ(...)e ρ is proportional to... ρ on  C1l R Define: E( ˆB) = {E : E is Â-information preserving via ˆB}. Then  is E( ˆB)-correcting. Thm. 7 Hint: R measures ˆB, then transfers ˆB to  with  -info depolarized. Converse: Assume that for all ψ supp(1lâ), E ψ {0}. If  is E-correcting, then there exists an isomorphic ˆB such that E E( ˆB). Thm. 8 Hint: Construct the isomorphism from j E j(...)e j. 29 Top Bot TOC

83 Correction by Isomorphism Given: ˆB Â, -isomorphism γ :  ˆB. A j Mat n j (C) ι : H m j=1 I j C j R ῑ :  m j=1 B(I j) 1l Cj Error E is Â-information preserving via ˆB if for every Â-pure state ρ, E γ(...)e ρ is proportional to... ρ on  C1l R Define: E( ˆB) = {E : E is Â-information preserving via ˆB}. Then  is E( ˆB)-correcting. Thm. 7 Hint: R measures ˆB, then transfers ˆB to  with  -info depolarized. Converse: Assume that for all ψ supp(1lâ), E ψ {0}. If  is E-correcting, then there exists an isomorphic ˆB such that E E( ˆB). Thm. 8 Hint: Construct the isomorphism from j E j(...)e j. In both cases: ˆB is E( ˆB)-protecting by construction. Top 29 TOC

84 Correctability iff Detectability(E*E) If  is E-correcting, then E E is detectable. Proof: Let i R i(...)r i be the recovery operation. For E E, R i E1l Â. Check detectability of E F for E, F E: 1lÂE F 1l = i = i 1lÂE R i R if 1l ( 1lÂE R i) (Ri F 1lÂ), a sum of products of elements of Â. 30 TOC

85 Correctability iff Detectability(E*E) If E E is detectable, then  is E-correcting. Proof sketch: Define ι E : X  EXE. Thm. 9 Verify: ι E (X)ι F (Y ) = ι E (1lÂ)ι F (XY ) = ι E (XY )ι F (1lÂ). Define ι E (X) = j E jxe j, where the E j form a basis of E. Verify: ι E (X)ι E (Y ) = ι E (1lÂ)ι E (XY ). Consequently P = ι E (1lÂ) ι E (Â). P 1 ι E (...) is a -homomorphism. Construct the recovery operation: Extend the transpose of P 12 ι E (1lÂ... 1lÂ)P 12 to supp(p ). 31 TOC

86 Detectability(E*E): Special Cases Assume E is a detectable -algebra. E is correctable, because E E E. A j Mat n j (C) ι : H m j=1 I j C j R ῑ : Â m j=1 B(I j) 1l Cj 32 Bot TOC

87 Detectability(E*E): Special Cases Assume E is a detectable -algebra. E is correctable, because E E E. A j Mat n j (C) ι : H m j=1 I j C j R ῑ : Â m j=1 B(I j) 1l Cj Maximal detectable -algebras. Let D be the space of detectable errors. Every maximal -subalgebra E of D is correctable.... but not all of them simultaneously. 32 Top Bot TOC

88 Detectability(E*E): Special Cases Assume E is a detectable -algebra. E is correctable, because E E E. A j Mat n j (C) ι : H m j=1 I j C j R ῑ : Â m j=1 B(I j) 1l Cj Maximal detectable -algebras. Let D be the space of detectable errors. Every maximal -subalgebra E of D is correctable.... but not all of them simultaneously. Minimum distance. Let E 1 = E 1 be the set of weight 1 errors. Â has minimum distance (at least) d if E1 d 1 D. Minimum distance 2d + 1 d-error-correcting: (E1 d ) E1 d D. 32 Top Bot TOC

89 Detectability(E*E): Special Cases Assume E is a detectable -algebra. E is correctable, because E E E. A j Mat n j (C) ι : H m j=1 I j C j R ῑ : Â m j=1 B(I j) 1l Cj Maximal detectable -algebras. Let D be the space of detectable errors. Every maximal -subalgebra E of D is correctable.... but not all of them simultaneously. Minimum distance. Let E 1 = E 1 be the set of weight 1 errors. Â has minimum distance (at least) d if E1 d 1 D. Minimum distance 2d + 1 d-error-correcting: (E1 d ) E1 d D. Erasure codes on n qudits. Let E 1 = {one-qudit errors}. supp(e): The set of qudits on which E acts non-trivially. {E : supp(e) S} detectable erasure S is correctable. {E : supp(e) S} is a -algebra. Stabilizer codes: Erasure is uncond. Cliff.-correctable. Top 32 TOC

90 P Lessons E A j Mat n j (C) ι : H m j=1 I j C j R ῑ :  m j=1 B(I j) 1l Cj At all times during information processing, there are A-dits that contain the information of interest.     E R E R E R ˆB ˆB ˆB Puzzles to apply the lessons to: Let U  be a logical unitary. Are all L such that L1l = U equivalent logical unitary? What is a useful logical hamiltonian? Consider the ground space G of a hamiltonian H. Requirements for G to be robust against local perturbations? What makes stabilizer codes special? 33 TOC

91 Appendix: Accessible Observables Theorem 1. Let H P Q be a controllable interaction between physical system P and qubit Q. For any state ρ Q, tr Q (H P Q ρ Q ) is a quantum controllable evolution of P. Proof sketch: Use the following asymptotic identities for N : e itr Q (HP Q ρ Q )t = N e itr Q (HP Q ρ Q )t/n j=1 N (1l itr Q (H P Q ρ Q )t/n) j=1 N ( = tr Qj (1l ih P Q ) j t/n)ρqj j=1 ( N tr Qj tr Qj e ihp Q ) j t/n ρqj j=1 N = tr Q1...Q e ihp Q j t/n. N j=1 Theorem 2. If H is a quantum controllable evolution of P, then H is an accessible observable. Proof sketch: Prepare qubits Q j in state +, apply evolution under Hσ Q j z for each qubit for time δ. Given that P is in a λ-eigenstate of H, The product of the + evolves by angle 2δλ. Choose δ small enough to resolve even the largest eigenvalue of H, and enough qubits to distinguish even the closest pair sufficiently well. A σx measurement of the qubits and counting the number of 1 outcomes will approximately determine λ. Alternatively, use phase estimation strategies. Theorem 3. If H and F are quantum controllable, then so are αh + βf, {H, F } and i[h, F ]. Proof sketch: The identities needed are: e i(αh+βf ) [H,F ]t e (e iαh/n e iβf/n ) N (e if t/n e iht/n e if t/n e iht/n ) N 2 e i{h,f }t = Q 0 e [HσQ y,f σq x ]t 0 Q That is, couple each of a large number N of qubits each initially in state ρ successively to P with H P Q for a short amount of time, t/n. To see that it is quantum controllable, that is, to implement evolution under H P σz Q, just use quantum controllability of HP Q. That is, modify the couplings in the strategy above by replacing H P Q j P Q with H j Q σ z. 34 TOC

92 Appendix: Information Preservation Theorem 4. Define h E : (A Â) E AE and let ι be the fundamental isomorphism as defined in the slides. The following are equivalent: 1. E preserves Â-information. 2. E = E A + E R where E A = j 1l I j C j w.r.t. ι and E R 1l A = E1l is in Â. Proof sketch: That 2. implies 1. can be checked by direct calculation, taking care to note that because ρ is assumed to be Â-pure, it is a product state ψ σ on some I j C j. The proportionality constant is given by tr(c j ρc ). That 2. is equivalent to 3. is also immediate, j since the form of E A characterizes the commutant of  and we can compute E R = E(1l 1l A ). It remains to show that 1. implies 2. Define E A = E1lÂ, E R = E E A. Â-pure states cannot become mixed with respect  without violating the proportionality condition required for information preservation. Now note that E A must preserve the isotypical subspaces of  1l R. Otherwise a single-letter state of  1l R is mapped to a state with other letter contributions. So E A = E j, where E j is an operator of I j C j (via ι). If E j is not a product operator, a pure state of I j can become mixed, so E j must be a product operator. That E j must be of the form 1l Ij C j follows, otherwise any non-eigenstate of the first factor of E j violates the proportionality condition. Theorem 5. Let E be a completely positive superoperator E(ρ) = l E l ρe. Then E preserves l Â-information iff each E l does. Proof sketch: By (implicit) definition, E preserves Â-information iff for every ρ with support in 1l Â, l E l... E l ρ is proportional to... ρ on Âx =  C(1l 1l  ). If each E l preserves Â-information, then (for fixed ρ) each summand in this definition satisfies the proportionality condition, so the sum does also, that is E preserves Â-information. For the converse, consider an isotypical subspace V = I C of Â. The E l must preserve them as otherwise one of the projectors of Âx onto another isotypical subspace will have nonzero expectation for a state supported on V after applying E. Similarly, each E l must preserve Â-pure states, that is, map states of the form ψ φ to α ψ φ for some φ and α. Otherwise the projector onto the orthogonal complement of ψ C, an operator of Âx, could gain non-zero expectation. It follows that each E l satisfies the second of the above equivalent conditions (Thm. 4) for for Â-information preservation. 35 TOC

93 Appendix: SDPs Theorem 6. Let V be a linear space of operators. There exists a semi-definite program (SDP) to determine whether there exists a quantum operation R = i R i... R i with R i V. Feasible solutions to the SDP give such quantum operations. Proof sketch: Let B V be the full space of operators. Let M : T = B B B be the linear, crossed multiplication map defined by M(X Y ) = Y X. Let operators be endowed with the usual trace inner product, X, Y = tr(x Y ). We can consider T as operators mapping B to itself via the application map X Y : Z Xtr(Y Z). The hermitian transpose of X Y is Y X. Completely positive superoperators expressible in the form i R i... R i with R i V can be identified with positive members i R i R of T with support in V. Thus, we consider i the linear L space of hermitian operators in V V T, constrained to be positive. To consider quantum operations, we add the additional constraint M(O) = 1l to obtain the desired SDP. 36 TOC

94 Appendix: Correction by Isomorphism Theorem 7. Let E( ˆB) the set of errors that are Â-information preserving via ˆB. Then  is a E( ˆB)-error-correcting subsystem. Proof sketch: Let m j=1 I j C j C 0 and m j=1 I j D j D 0 be the canonical decompositions of H into isotypical subspaces for  and ˆB, respectively. (We leave the isomorphisms implicit for objects whose membership is clear from context.) Pick fixed orthonormal bases for the C j, D j. Define operators R 1 (j; r, s) as the isometry from the r th basis element of D j to the s th basis element of C j. Then extend R 1 to H by defining R(j; r, s) = 1l Ij R 1 (j; r, s). (We define I 0 = C.) Then R : ρ j,r,s 1 dim(c j ) R(j; r, s)ρr(j; r, s) is a quantum operation that, loosely speaking, completely depolarizes ˆB -information and outputs completely depolarized  - information, all while preserving the stored A-dit. The transpose of R implements the isomorphism γ from  to ˆB. The result follows. Definition: A linear map γ :  ˆB (or on the abstract versions of these algebras) is multiplicative/p if γ(x)γ(y ) = γ(xy )P. Here, all such maps are also assumed to be - preserving. (The terminology is the best I could come up with for now, there is probably standard terminology, but I don t know what it is.) Lemma 1. If γ :  ˆB is multiplicative/p, then γ(1lâ)p = γ(1lâ) 2, γ is multiplicative/γ(1lâ) and γ(1lâ) γ(â). The map γ can be converted to a positivity preserving multiplicative/q map β by defining β(x) = γ(x)sgn(γ(1lâ) ) and Q = γ(1lâ) = β(1lâ). If γ is positivity preserving, then β : X γ(x)γ(1lâ) 1 is a homomorphism. Proof: The first part of the lemma follows from the identities γ(1lâ)p = γ(1lâ1lâ)p = γ(1lâ)γ(1lâ) = γ(1lâ) 2, γ(x)p = γ(x1lâ)p = γ(x)γ(1lâ), γ(x)p = γ(1lâx)p = γ(1lâ)γ(x), γ(xy )γ(1lâ) = γ(xy 1lÂ)P = γ(xy )P = γ(x)γ(y ). Theorem 8. Assume that for all ψ supp(1lâ), E ψ = {0}. If  is an E-error-correcting subsystem, then there exists an isomorphic ˆB such that E E( ˆB). Proof sketch: Let R : ρ j R j ρr j be a recovery operator for E, so that for all E E, R j E is Â-information preserving, that is, R j E1l  Â. Since  is -closed, also 1lÂE R j Â. Define ι E : X  EXE. For E, F E, X, Y Â, we find that ι E (X)ι F (Y ) = EXE F Y F = EXE R i R i F Y F i = EX(1lÂE R i )(R i F 1l  )Y F i = E1lÂ(E R i )(R i F )1l  XY F i = E1lÂE F XY F = ι E (1lÂ)ι F (XY ), and similarly, ι E (X)ι F (Y ) = ι E (XY )ι F (1lÂ). Choose {E j } j E such that E i ψ = 0 for all j implies E ψ = {0}. (A basis of E works.) Define γ : X  j E j XE j. The identities above imply that γ is multiplicative/γ(1lâ). It is manifestly positive, so we can define the homomorphism β as in Lemma 1. Define P = γ(1lâ). Homomorphisms of -algebras are either 0 or isomorphisms when restricted to minimal ideals. The assumption on E implies that the first case does not happen for γ on Â: Otherwise, consider the identity 1l l of a minimial ideal corresponding to summand I l C l of the decomposition of H with respect to Â. Then j E j 1l l E j = 0 and positivity of 1l l imply that there is a state of I l C l on which all the E j are 0. Thus β is an isomorphism. For F E and ρ a Â-pure state: F β(x)f ρ = i tr(p 1 E i XE i F ρf ) = tr(p 1 E i XE i R j R j F ρf ) i,j = tr(p 1 E i 1lÂE i R j XR j F ρf ) i,j = tr(r j XR j F ρf ) = tr(xr j F ρf R j ) j j X ρ. For the second part, the sign and absolute value of γ(1lâ) is well-defined because γ(1lâ) is hermitian. That β is multiplicative/q follows from sgn(h) H = H and sgn(h) 2 = 1l H, where 1l H is the projector onto the support of H. By definition, Q is positive and for any hermitian H, the support of β(h) is contained in the support of Q. Since Qβ(X X) = β(x X)Q = β(x )β(x), it follows that β(x X) is positive. The third part is by direct calculation. The technical assumption on E in Thm. 8 is satisfied if E is the span of a the operators in a quantum operations operator sum expression. One can weaken it, but then, the homomorphism β is not necessarily an isomorphism and one must redefine it on the minimal ideals on which it is zero. It would be simplest to just define it to be the identity on those ideals, but this fails because our definition allows for the range of an error to map over to the support of such an ideal. (The recovery operator can take care of this.) To fix this, one can allow for an extension of H outside the range of any error (when restricted to supp(1lâ) and map these ideals isomorphically into the extension. 37 TOC

95 Appendix: Detectability(E*E) Theorem 9. If E E is detectable, then  is an E-error-correcting subsystem. Proof sketch: We adopt the definitions in the proof of Thm. 8 and let ι E = γ. The properties of ι E and ι F can be proven in nearly the same way by shortcutting the step where i R i R i is inserted. The subsequent arguments in Thm. 8 apply to show that β(...) = P 1 ι E (...) is a -homorphism. Here, P = ι E (1lÂ) and P 1 is computed on its support, noting that ι E is positive. As written, β can readily be extended to an explicit, completely positive superoperator β(b) = i P 1/2 E i 1lÂB1lÂE i P 1/2. Its transpose is also well-defined, but trace-preserving only for states of supp(p ). It suffices to extend the transpose to supp(p ) by adding a projector onto supp(p ) to the explicit form of the superoperator. 38 TOC

96 Contents Title: A Bit of Algebra Lessons Physical Systems and Control top... 2 Accessible Observables top... 3 Information top... 4 Classification of -Algebras top... 5 Representations of -Algebras top... 6 Representation Examples top... 7 Interpretations States Pure States top Algebra Extensions top Algebra Extensions top Information Type Extensions Union and Concatenation top Error operators top Effects on Information top Example: The Even Bit top Loss and Error Detection top Example: The Even Bit Environmental Interactions top Zeno Error Control top Zeno Error Control top Implementing Zeno top Error-Correcting Subsystems top Example: The Majority Bit top Protecting Subsystems top Example: The Full Majority Bit top Protecting vs Correcting Subsystems Correction by Isomorphism top Correctability iff Detectability(E*E) Correctability iff Detectability(E*E) Detectability(E*E): Special Cases top Lessons Appendix: Accessible Observables Appendix: Information Preservation Appendix: SDPs Appendix: Correction by Isomorphism Appendix: Detectability(E*E) References TOC

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