On Complementarity In QEC And Quantum Cryptography David Kribs - PowerPoint PPT Presentation

Introduction Stinespring Dilation Theorem Private Quantum Codes Connection with QEC and Beyond Conclusion On Complementarity In QEC And Quantum Cryptography David Kribs Professor & Chair Department of Mathematics & Statistics University of Guelph Associate Member Institute for Quantum Computing University of Waterloo QEC II — USC — December 2011

Introduction Stinespring Dilation Theorem Private Quantum Codes Connection with QEC and Beyond Conclusion Outline 1 Introduction Notation 2 Stinespring Dilation Theorem Heisenberg & Schr¨ odinger Pictures Purification of Mixed States Conjugate/Complementary Channels 3 Private Quantum Codes Definition Single Qubit Private Channels 4 Connection with QEC and Beyond Complementarity of Quantum Codes From QEC to QCrypto? 5 Conclusion Summary

Introduction Stinespring Dilation Theorem Private Quantum Codes Connection with QEC and Beyond Conclusion Notation H A , H B will denote Hilbert spaces for systems A and B . B ( H ) will denote the set of (bounded) linear operators on H ; B ( H ) t will denote the trace class operators on H . In finite dimensions these sets coincide and so we’ll simply write L ( H ) B ( H A , H B ) will denote the set of linear transformations from H A to H B . We’ll write X , Y for operators in B ( H ), and ρ, σ for density operators in B ( H ) t . (And we’ll just refer to L ( H ) when appropriate.) Given a linear map Φ : B ( H A ) t → B ( H B ) t , its dual map Φ † : B ( H B ) → B ( H A ) is defined via the Hilbert-Schmidt inner product: Tr ( ρ Φ † ( X )) = Tr (Φ( ρ ) X ).

Introduction Stinespring Dilation Theorem Private Quantum Codes Connection with QEC and Beyond Conclusion Heisenberg Picture Suppose that Φ † : B ( H B ) → B ( H A ) is a completely positive (CP) unital (Φ † ( I B ) = I A ) linear map.

Introduction Stinespring Dilation Theorem Private Quantum Codes Connection with QEC and Beyond Conclusion Heisenberg Picture Suppose that Φ † : B ( H B ) → B ( H A ) is a completely positive (CP) unital (Φ † ( I B ) = I A ) linear map. Then there is a Hilbert space K (of dimension at most dim( A ) dim( B )) and a co-isometry V ∈ B ( H B ⊗ K , H A ) ( VV † = I A ) such that Φ † ( X ) = V ( X ⊗ I K ) V † ∀ X .

Introduction Stinespring Dilation Theorem Private Quantum Codes Connection with QEC and Beyond Conclusion Heisenberg Picture Suppose that Φ † : B ( H B ) → B ( H A ) is a completely positive (CP) unital (Φ † ( I B ) = I A ) linear map. Then there is a Hilbert space K (of dimension at most dim( A ) dim( B )) and a co-isometry V ∈ B ( H B ⊗ K , H A ) ( VV † = I A ) such that Φ † ( X ) = V ( X ⊗ I K ) V † ∀ X . V is unique up to a unitary on K . Here the Kraus operators for Φ † can be read off as the “coordinate operators” of V † :   V † 1  .  V † = .   . V † AB

Introduction Stinespring Dilation Theorem Private Quantum Codes Connection with QEC and Beyond Conclusion Schr¨ odinger Picture Suppose that Φ : B ( H A ) t → B ( H B ) t is a CP trace preserving (CPTP) linear map.

Introduction Stinespring Dilation Theorem Private Quantum Codes Connection with QEC and Beyond Conclusion Schr¨ odinger Picture Suppose that Φ : B ( H A ) t → B ( H B ) t is a CP trace preserving (CPTP) linear map. Then there is a Hilbert space K (of dimension at most dim( A ) dim( B )), an isometry U ∈ B ( H A ⊗ K , H B ⊗ K ), and a pure state | ψ � ∈ K such that Φ( ρ ) = Tr K ( U ( ρ ⊗ | ψ �� ψ | ) U † ) ∀ ρ.

Introduction Stinespring Dilation Theorem Private Quantum Codes Connection with QEC and Beyond Conclusion Schr¨ odinger Picture Suppose that Φ : B ( H A ) t → B ( H B ) t is a CP trace preserving (CPTP) linear map. Then there is a Hilbert space K (of dimension at most dim( A ) dim( B )), an isometry U ∈ B ( H A ⊗ K , H B ⊗ K ), and a pure state | ψ � ∈ K such that Φ( ρ ) = Tr K ( U ( ρ ⊗ | ψ �� ψ | ) U † ) ∀ ρ. The two pictures are connected via V † | φ � := U ( | φ � ⊗ | ψ � ), which gives Φ( ρ ) = Tr K ( V † ρ V ).

Introduction Stinespring Dilation Theorem Private Quantum Codes Connection with QEC and Beyond Conclusion Schr¨ odinger Picture Suppose that Φ : B ( H A ) t → B ( H B ) t is a CP trace preserving (CPTP) linear map. Then there is a Hilbert space K (of dimension at most dim( A ) dim( B )), an isometry U ∈ B ( H A ⊗ K , H B ⊗ K ), and a pure state | ψ � ∈ K such that Φ( ρ ) = Tr K ( U ( ρ ⊗ | ψ �� ψ | ) U † ) ∀ ρ. The two pictures are connected via V † | φ � := U ( | φ � ⊗ | ψ � ), which gives Φ( ρ ) = Tr K ( V † ρ V ). The Kraus operators for Φ are the coordinate operators V i from above, and the general form for U is � � V † U = ∗

Introduction Stinespring Dilation Theorem Private Quantum Codes Connection with QEC and Beyond Conclusion Purification of Mixed States Fix a density operator ρ 0 ∈ B ( H ) t , and consider the CPTP map Φ : C → B ( H ) t defined by Φ( c · 1) = c ρ 0 ∀ c ∈ C .

Introduction Stinespring Dilation Theorem Private Quantum Codes Connection with QEC and Beyond Conclusion Purification of Mixed States Fix a density operator ρ 0 ∈ B ( H ) t , and consider the CPTP map Φ : C → B ( H ) t defined by Φ( c · 1) = c ρ 0 ∀ c ∈ C . Then the Stinespring Theorem gives (here K = C ⊗ H = H ): ρ 0 = Φ(1) = Tr K ( U (1 ⊗ | ψ �� ψ | ) U † ) = Tr K ( | ψ ′ �� ψ ′ | ) , where | ψ ′ � ∈ H ⊗ H is a purification of ρ 0 – and the unitary freedom in the theorem captures all purifications.

Introduction Stinespring Dilation Theorem Private Quantum Codes Connection with QEC and Beyond Conclusion Conjugate/Complementary Channels Definition (King, et al.; Holevo) Given a CPTP map Φ : L ( H A ) → L ( H B ) , consider V ∈ L ( H B ⊗ K , H A ) and K above for which Φ( ρ ) = Tr K ( V † ρ V ) . Then the corresponding conjugate (or complementary ) channel is the CPTP map � Φ : L ( H A ) → L ( K ) given by Φ( ρ ) = Tr B ( V † ρ V ) . � Φ, Φ ′ obtained in this way are related Fact: Any two conjugates � by a partial isometry W such that � Φ( · ) = W Φ ′ ( · ) W † . We talk of “the” conjugate channel for Φ with this understanding.

Introduction Stinespring Dilation Theorem Private Quantum Codes Connection with QEC and Beyond Conclusion Computing Kraus Operators for Conjugates Suppose that V i ∈ L ( H A , H B ) are the Kraus operators for Φ : L ( H A ) → L ( H B ). Then we can obtain Kraus operators { R µ } for � Φ as follows.

Introduction Stinespring Dilation Theorem Private Quantum Codes Connection with QEC and Beyond Conclusion Computing Kraus Operators for Conjugates Suppose that V i ∈ L ( H A , H B ) are the Kraus operators for Φ : L ( H A ) → L ( H B ). Then we can obtain Kraus operators { R µ } for � Φ as follows. Fix a basis {| e i �} for K and define for ρ ∈ L ( H A ), � | e i �� e j | ⊗ V i ρ V † F ( ρ ) = i ∈ L ( K ⊗ H B ) . i , j

Introduction Stinespring Dilation Theorem Private Quantum Codes Connection with QEC and Beyond Conclusion Computing Kraus Operators for Conjugates Suppose that V i ∈ L ( H A , H B ) are the Kraus operators for Φ : L ( H A ) → L ( H B ). Then we can obtain Kraus operators { R µ } for � Φ as follows. Fix a basis {| e i �} for K and define for ρ ∈ L ( H A ), � | e i �� e j | ⊗ V i ρ V † F ( ρ ) = i ∈ L ( K ⊗ H B ) . i , j Then Φ( ρ ) = Tr K F ( ρ ) and � � � Tr ( V i ρ V † R µ ρ R † Φ( ρ ) = Tr B F ( ρ ) = j ) | e i �� e j | = µ , i , j µ where R † µ = [ V † 1 | f µ � V † 2 | f µ � · · · ] and {| f µ �} is a basis for H B .

Introduction Stinespring Dilation Theorem Private Quantum Codes Connection with QEC and Beyond Conclusion Private Quantum Codes Definition (Ambainis, et al.) Let S ⊆ H be a set of pure states, let Φ : L ( H A ) → L ( H B ) be a CPTP map, and let ρ 0 ∈ L ( H ) . Then [ S , Φ , ρ 0 ] is a private quantum channel if we have Φ( | ψ �� ψ | ) = ρ 0 ∀| ψ � ∈ S . Motivating class of examples: random unitary channels, where Φ( ρ ) = � i p i U i ρ U † i .

Introduction Stinespring Dilation Theorem Private Quantum Codes Connection with QEC and Beyond Conclusion Single Qubit Private Codes Recall a single qubit pure state | ψ � can be written � 1 � � 0 � | ψ � = cos θ + e i ϕ sin θ . 0 1 2 2 We associate | ψ � with the point ( θ, ϕ ), in spherical coordinates, on � θ � � θ � and β = e i ϕ sin the Bloch sphere via α = cos . The 2 2 associated Bloch vector is � r = (cos ϕ sin θ, sin ϕ sin θ, cos θ ). Using the Bloch sphere representation, we can associate to any r ∈ R 3 satisfying single qubit density operator ρ a Bloch vector � � � r � ≤ 1, where ρ = I + � r · � σ . 2 σ = ( σ x , σ y , σ z ) T . We use � σ to denote the Pauli vector; that is, �