Operator Quantum Error Correcting Codes

Operator Quantum Error Correcting Codes Vlad Gheorghiu Department of Physics Carnegie Mellon University Pittsburgh, PA 15213, U.S.A. June 18, 2008 Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 1 / 16

Outline 1 References 2 Evolution of open quantum systems 3 The standard model of quantum error correction 4 Mathematical detour: structure of finite C algebras 5 Noiseless subsystems and operator quantum error correction Unital channels General channels Necessary and sufficient conditions for OQEC 6 Examples and discussions Examples Discussions Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 2 / 16

References References D.W. Kribbs, R. Laflamme and D. Poulin, Phys. Rev. Lett., 94, 180501 (2005) - Unified and Generalized Approach to Quantum Error Correction D. W. Kribs, R. Laflamme, D. Poulin and M. Lesosky, arxiv:quant-ph/0504189 - Operator Quantum Error Correction D. Kribbs, arxiv:math.oa/0506491 - A Brief Introduction to Operator Quantum Error Correction M. A. Nielsen and D. Poulin, Phys. Rev. A, 75, 064304 (2007) - Algebraic and information-theoretic conditions for operator quantum error-correction The present talk is posted online at http://quantum.phys.cmu.edu under the Quantum Information Seminar heading. Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 3 / 16

Evolution of open quantum systems Evolution of open quantum systems: Kraus representation The reversibility postulate of quantum mechanics implies that evolution in a closed quantum system occurs via unitary maps ρ UρU. An open quantum system is a part of a larger closed one H = H E H S where H E is the environment and H S is the open system. Let E describe the evolution of an open quantum system. Then E must be positive and trace preserving (maps density operators to density operators). Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 4 / 16

Evolution of open quantum systems Evolution of open quantum systems: Kraus representation The reversibility postulate of quantum mechanics implies that evolution in a closed quantum system occurs via unitary maps ρ UρU. An open quantum system is a part of a larger closed one H = H E H S where H E is the environment and H S is the open system. Let E describe the evolution of an open quantum system. Then E must be positive and trace preserving (maps density operators to density operators). This must hold for any environment, i.e. Identity E E : L(H E H S ) L(H E H S ) must be positive and trace-preserving for all E, or, equivalently, completely positive trace preserving (CPTP) map. Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 4 / 16

Evolution of open quantum systems A theorem of Choi and Kraus states that Kraus representation Every CPTP map E has an operator-sum representation of the form E(ρ) = a E a ρe a for some set of non-unique operators {E a } with E a E a = I. a Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 5 / 16

Evolution of open quantum systems A theorem of Choi and Kraus states that Kraus representation Every CPTP map E has an operator-sum representation of the form E(ρ) = a E a ρe a for some set of non-unique operators {E a } with E a E a = I. a The E a s are called noise operators or errors associated with E. Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 5 / 16

Evolution of open quantum systems A theorem of Choi and Kraus states that Kraus representation Every CPTP map E has an operator-sum representation of the form E(ρ) = a E a ρe a for some set of non-unique operators {E a } with E a E a = I. a The E a s are called noise operators or errors associated with E. Shorthand: E = {E a } when an error model for E is known. Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 5 / 16

The standard model of quantum error correction The standard model of QEC Triplet (R, E, C), where C is a subspace, a quantum code, of a Hilbert space H, and the error E and recovery R are quantum operations on L(H). In the trivial case when E = {U} is implemented by a single unitary error operator, the recovery is just the reversal operation R = {U } ρ E UρU R U (UρU )U = ρ. Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 6 / 16

The standard model of quantum error correction The standard model of QEC Triplet (R, E, C), where C is a subspace, a quantum code, of a Hilbert space H, and the error E and recovery R are quantum operations on L(H). In the trivial case when E = {U} is implemented by a single unitary error operator, the recovery is just the reversal operation R = {U } ρ E UρU R U (UρU )U = ρ. More generally, the set (R, E, C) forms an error triple if R undoes the effects of E on C Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 6 / 16

The standard model of quantum error correction The standard model of QEC Triplet (R, E, C), where C is a subspace, a quantum code, of a Hilbert space H, and the error E and recovery R are quantum operations on L(H). In the trivial case when E = {U} is implemented by a single unitary error operator, the recovery is just the reversal operation R = {U } ρ E UρU R U (UρU )U = ρ. More generally, the set (R, E, C) forms an error triple if R undoes the effects of E on C R(E(σ)) = σ, σ L(H). Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 6 / 16

The standard model of quantum error correction When there exists such an R for a given pair E, C, the subspace C is said to be correctable for E. Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 7 / 16

The standard model of quantum error correction When there exists such an R for a given pair E, C, the subspace C is said to be correctable for E. QEC conditions (Knill-Laflamme) Given E = {E a }, there exists a recovery operation R on C iff there exists a complex matrix Λ = (λ ab ) s.t. P C E a E b P C = λ ab P C, a, b, where P is the projector onto the coding space C. Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 7 / 16

Mathematical detour: structure of finite C algebras The structure of finite dimensional C algebras Let E = {E a } be a quantum operation. Let A be the C -algebra generated by the E a, i.e. the set of polynomials in the E a and E a. Furthermore, A = (M mj I nj ). J This means that there is an orthonormal basis such that the matrix representation of operators in A w.r.t. this basis have the form given above. A is called the interaction algebra associated with E. Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 8 / 16

Mathematical detour: structure of finite C algebras The structure of finite dimensional C algebras Let E = {E a } be a quantum operation. Let A be the C -algebra generated by the E a, i.e. the set of polynomials in the E a and E a. Furthermore, A = (M mj I nj ). J This means that there is an orthonormal basis such that the matrix representation of operators in A w.r.t. this basis have the form given above. A is called the interaction algebra associated with E. The noise commutant associated with E { } A = σ : [E, σ] = 0, E {E a, E a }. Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 8 / 16

Mathematical detour: structure of finite C algebras The structure of finite dimensional C algebras Let E = {E a } be a quantum operation. Let A be the C -algebra generated by the E a, i.e. the set of polynomials in the E a and E a. Furthermore, A = (M mj I nj ). J This means that there is an orthonormal basis such that the matrix representation of operators in A w.r.t. this basis have the form given above. A is called the interaction algebra associated with E. The noise commutant associated with E { } A = σ : [E, σ] = 0, E {E a, E a }. The structure of A implies that the noise commutant is unitarily equivalent to A = (I mj M nj ). J Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 8 / 16

Noiseless subsystems and operator quantum error correction Unital channels Noiseless subsystems for unital channels When E is unital (i.e. a E ae a = I), then all states encoded in A are immune to the errors of E. Furthermore, it has been shown that when E is unital, the noise commutant coincides with the set of fixed points for E, i.e. { } A = Fix(E) = σ : E(σ) = a E a σe a = σ. Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 9 / 16

Noiseless subsystems and operator quantum error correction Unital channels Noiseless subsystems for unital channels When E is unital (i.e. a E ae a = I), then all states encoded in A are immune to the errors of E. Furthermore, it has been shown that when E is unital, the noise commutant coincides with the set of fixed points for E, i.e. { } A = Fix(E) = σ : E(σ) = a E a σe a = σ. Note that decoherence-free subspaces arise as a special case, when m J = 1. Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 9 / 16

Noiseless subsystems and operator quantum error correction General channels Noiseless subsystems for general channels The structure of the algebra A induces a natural decomposition of the Hilbert space as H = HJ A HB J, J where the noisy subsystems H A J have dimension m J and the noisless subsystems H B J have dimension n J. Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 10 / 16

Noiseless subsystems and operator quantum error correction General channels Noiseless subsystems for general channels The structure of the algebra A induces a natural decomposition of the Hilbert space as H = HJ A HB J, J where the noisy subsystems H A J have dimension m J and the noisless subsystems H B J have dimension n J. For simplicity, assume that the information is encoded into a single noiseless sector of L(H), and hence H = (H A H B ) K, with dim(h A )=m, dim(h B )=n and dim(k)=dim(h)-mn. Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 10 / 16

Noiseless subsystems and operator quantum error correction General channels Let { α k : 1 k m} be an orthonormal basis for H A and let {P kl = α k α l I n : 1 k, k m}. Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 11 / 16

Noiseless subsystems and operator quantum error correction General channels Let { α k : 1 k m} be an orthonormal basis for H A and let {P kl = α k α l I n : 1 k, k m}. Define the semigroup U = {σ L(H) : σ = σ A σ B, for some σ A and σ B }. Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 11 / 16

Noiseless subsystems and operator quantum error correction General channels Let { α k : 1 k m} be an orthonormal basis for H A and let {P kl = α k α l I n : 1 k, k m}. Define the semigroup U = {σ L(H) : σ = σ A σ B, for some σ A and σ B }. Let P k = P kk, P U = P 1 + P 2 +... + P m and P U H = H A H B. Define a map P U ( ) = P U ( )P U. Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 11 / 16

Noiseless subsystems and operator quantum error correction General channels The following three conditions are equivalent for the H B sector of the semigroup U to encode a noiseless subsystem: Equivalent conditions for noiseless subsytems 1 σ A, σ B, τ A : E(σ A σ B ) = τ A σ B 2 σ B, τ A : E(I A σ B ) = τ A σ B 3 σ U : (Tr A P U E)(σ) = Tr A (σ). Note that decoding is not necessary. Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 12 / 16

Noiseless subsystems and operator quantum error correction General channels The following three conditions are equivalent for the H B sector of the semigroup U to encode a noiseless subsystem: Equivalent conditions for noiseless subsytems 1 σ A, σ B, τ A : E(σ A σ B ) = τ A σ B 2 σ B, τ A : E(I A σ B ) = τ A σ B 3 σ U : (Tr A P U E)(σ) = Tr A (σ). Note that decoding is not necessary. Operator quantum error correction: same conditions as for the noiseless subsystems, but decoding (or recovery) is included, i.e. σ A, σ B, τ A : R(E(σ A σ B )) = τ A σ B. Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 12 / 16

Noiseless subsystems and operator quantum error correction Necessary and sufficient conditions for OQEC Necessary and sufficient conditions for OQEC To be use in practical applications, one needs easily testable conditions for a map E = {E a } to admit a noiseless subsystem. Compact algebraic characterization of OQEC The following conditions are equivalent: 1 H B is an E-correcting subsystem with respect to the decomposition H = (H A H B ) K. 2 PE a E b P = A ab I B, a, b, where P is the projector onto H A H B and the A ab are operators on A. Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 13 / 16

Examples Examples and discussions Examples Let { a, b, a, b } and { a 1, b 1, a 2, b 2 } be two orthonormal bases for C 4. Let P 1 be the projection onto span{ a, b } and P 2 the projection onto span{ a, b }. Let Q i, i = 1, 2, be the projection onto span{ a i, b i }. Define operators U 1, U 1, U 2, U 2 on C4 as U 1 a = a 1, U 1 b = b 1 ; U 1 a = a 1, U 1 b = b 1 ; U 2 a = a 2, U 2 b = b 2 ; U 2 a = a 2, U 2 b = b 2 and put U 1 P 2 U 1 P 1 U 2 P 2 U 2 P 1 0. Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 14 / 16

Examples Examples and discussions Examples Let { a, b, a, b } and { a 1, b 1, a 2, b 2 } be two orthonormal bases for C 4. Let P 1 be the projection onto span{ a, b } and P 2 the projection onto span{ a, b }. Let Q i, i = 1, 2, be the projection onto span{ a i, b i }. Define operators U 1, U 1, U 2, U 2 on C4 as U 1 a = a 1, U 1 b = b 1 ; U 1 a = a 1, U 1 b = b 1 ; U 2 a = a 2, U 2 b = b 2 ; U 2 a = a 2, U 2 b = b 2 and put U 1 P 2 U 1 P 1 U 2 P 2 U 2 P 1 0. These operators are partial isometries and satisfy U 1 = U 1 P 1, U 1 = U 1 P 2, U 2 = U 2 P 1, U 2 = U 2 P 2. Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 14 / 16

Examples Examples and discussions Examples Let { a, b, a, b } and { a 1, b 1, a 2, b 2 } be two orthonormal bases for C 4. Let P 1 be the projection onto span{ a, b } and P 2 the projection onto span{ a, b }. Let Q i, i = 1, 2, be the projection onto span{ a i, b i }. Define operators U 1, U 1, U 2, U 2 on C4 as U 1 a = a 1, U 1 b = b 1 ; U 1 a = a 1, U 1 b = b 1 ; U 2 a = a 2, U 2 b = b 2 ; U 2 a = a 2, U 2 b = b 2 and put U 1 P 2 U 1 P 1 U 2 P 2 U 2 P 1 0. These operators are partial isometries and satisfy U 1 = U 1 P 1, U 1 = U 1 P 2, U 2 = U 2 P 1, U 2 = U 2 P 2. The operators E = {E 1, E 2 } define a quantum channel where E 1 = 1 2 (U 1 P 1 + U 1P 2 ) E 2 = 1 2 (U 2 P 1 U 2P 2 ). Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 14 / 16

Examples and discussions Examples The action of E 1 and E 2 is indicated below. Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 15 / 16

Examples and discussions Examples The action of E 1 and E 2 is indicated below. Recovery: define V 11 = U 1 P 1, V 12 = U 1 P 2, V 21 = U 2 P 1, V 22 = U 2 P2 and let the recovery operators be { } 1 R = 2 V jk Q j : 1 j, k 2. Then all errors induced by E on U 0 = I2 M 2 can be corrected, i.e. R(E(σ)) = σ, σ L(C 4 ) that have a matrix representation of the form σ = σ 1 σ 1, σ 1 M 2 with respect to the ordered basis { a, b, a, b }. Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 15 / 16

Discussions Examples and discussions Discussions QEC is a subset of OQEC (set dim(h A )=1). Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 16 / 16

Discussions Examples and discussions Discussions QEC is a subset of OQEC (set dim(h A )=1). As seen in previous examples, there are OQECC that cannot be realized as QECC, so QEC is strictly included in OQEC. Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 16 / 16

Discussions Examples and discussions Discussions QEC is a subset of OQEC (set dim(h A )=1). As seen in previous examples, there are OQECC that cannot be realized as QECC, so QEC is strictly included in OQEC. What about fault-tolerant OQECC? Vlad Gheorghiu (CMU) Operator Quantum Error Correcting Codes June 18, 2008 16 / 16