Private quantum subsystems and error correction Private quantum subsystems and error Tomas Jochym- O’Connor correction Privacy & error correction Restrictions of operator privacy Tomas Jochym-O’Connor Generalization of subsystem privacy Extended duality Institute for Quantum Computing and Department of Physics and Astronomy University of Waterloo 15 December 2014 QEC 2014 – Zurich Work in collaboration with David Kribs, Raymond Laflamme, and Sarah Plosker
Last time at QEC... Introduction Stinespring Dilation Theorem Private Quantum Codes Connection with QEC and Beyond Conclusion Private quantum subsystems and error correction Tomas Jochym- O’Connor On Complementarity In QEC Privacy & error And Quantum Cryptography correction Restrictions of operator privacy Generalization of David Kribs subsystem privacy Extended duality Professor & Chair Department of Mathematics & Statistics University of Guelph Associate Member Institute for Quantum Computing University of Waterloo QEC II — USC — December 2011
Outline Private quantum subsystems and error correction Tomas Jochym- O’Connor Privacy & error correction Review definitions of operator quantum privacy and error Restrictions of operator privacy correction Generalization of Complementary between privacy and error correction subsystem privacy Extended duality Restrictions of operator quantum privacy Generalized notion of subsystem privacy Recovering the duality with quantum error correction
Notation Private quantum subsystems and error correction Tomas Jochym- O’Connor Subsystems: S = ( A ⊗ B ) ⊕ ( A ⊗ B ) ⊥ Privacy & error Density matrices: Bounded linear operators with trace 1, correction σ A ∈ A, σ B ∈ B, ρ ∈ A ⊗ B Restrictions of operator privacy Quantum channel: Completely positive trace preserving map Generalization of subsystem privacy between linear operators, Φ : B ( A ) → B ( C ) Extended duality Complementary channel: Given a quantum channel Φ , there always exists a unitary U Φ and ancillary state | φ �� φ | K such that U Φ ( ρ A ⊗ | φ �� φ | K ) U † � � Φ( ρ A ) = Tr K , ∀ ρ i . The Φ complementary channel is then defined as: Φ ♯ ( ρ ) = Tr C U Φ ( ρ ⊗ | φ �� φ | K ) U † � � . Φ
Operator QEC and privacy Private quantum subsystems and error correction Tomas Jochym- O’Connor S = ( A ⊗ B ) ⊕ ( A ⊗ B ) ⊥ Privacy & error correction Restrictions of operator privacy Generalization of A subsystem B is an operator private subsystem for Φ if there subsystem privacy exists ρ 0 such that Extended duality Φ( σ A ⊗ σ B ) = ρ 0 , ∀ σ A , σ B A subsystem B is operator quantum error correctable for E if there exist τ A ( σ A ) , R such that R ◦ E ( σ A ⊗ σ B ) = τ A ⊗ σ B
Random Unitary channels Private quantum subsystems and error correction Tomas Jochym- What type of channels are required to privatize quantum information? O’Connor Privacy & error correction In classical communication, messages can be encrypted using a Restrictions of one-time pad. operator privacy Generalization of subsystem privacy Message: 1 0 0 1 1 1 1 0 0 1 1 1 : Final Message Extended duality Encryption: X X X I X I X X X I X I : Decryption Sent message: 0 1 1 1 0 1 0 1 1 1 0 1 : Received Message !"#$%&'$()*+""$,( The key property of the one-time pad is the uniform randomization of each of the bits of the message.
Random Unitary channels What type of channels are required to privatize quantum information? Private quantum subsystems and error correction Tomas Jochym- In classical communication, messages can be encrypted using a O’Connor one-time pad. Privacy & error correction Restrictions of Message: 1 0 0 1 1 1 1 0 0 1 1 1 : Final Message operator privacy Encryption: X X X I X I X X X I X I : Decryption Generalization of subsystem privacy Sent message: : Received Message Extended duality 0 1 1 1 0 1 0 1 1 1 0 1 !"#$%&'$()*+""$,( The key property of the one-time pad is the uniform randomization of each of the bits of the message. The state of any given bit of encrypted data x b is given by a classical probability distribution: Φ( x b ) = 1 2 x b + 1 2 x b
Random Unitary channels Private quantum What type of channels are required to privatize quantum information? subsystems and error correction Tomas Jochym- O’Connor In classical communication, messages can be encrypted using a one-time pad. Privacy & error correction Restrictions of operator privacy Message: 1 0 0 1 1 1 1 0 0 1 1 1 : Final Message Generalization of Encryption: X X X I X I X X X I X I : Decryption subsystem privacy Extended duality Sent message: 0 1 1 1 0 1 0 1 1 1 0 1 : Received Message !"#$%&'$()*+""$,( Random unitary channels provide the quantum analogue to the classical one-time pad, � p i U i ρU † Φ( ρ ) = i i
Operator duality Private quantum subsystems and error correction Tomas Jochym- Theorem (KKS08 1 ) O’Connor A subsystem B is an operator private subsystem for a channel Φ if Privacy & error correction and only if it is operator QEC for the complementary channel Φ ♯ . Restrictions of operator privacy Generalization of subsystem privacy Φ( ρ ) Extended duality ρ U Φ | a ih a | Φ ] ( ρ ) 1 D. Kretschmann, D. W. Kribs, R. Spekkens, (2008)
Quest for small private channels Private quantum subsystems and Inspiration from quantum error correction! error correction Tomas Jochym- O’Connor The dephasing channel is not private on a single qubit: Privacy & error correction Λ i ( ρ ) = 1 2( ρ + Z i ρZ i ) ∀ ρ ∈ S. Restrictions of operator privacy Generalization of subsystem privacy How about the same identical channel on multiple qubits? Extended duality Λ( ρ ) = Φ 2 ◦ Φ 1 ( ρ )
Quest for small private channels Private quantum subsystems and Inspiration from quantum error correction! error correction Tomas Jochym- O’Connor The dephasing channel is not private on a single qubit: Privacy & error correction Λ i ( ρ ) = 1 2( ρ + Z i ρZ i ) ∀ ρ ∈ S. Restrictions of operator privacy Generalization of subsystem privacy How about the same identical channel on multiple qubits? Extended duality Λ( ρ ) = Φ 2 ◦ Φ 1 ( ρ ) The resulting mapping yields: α 00 α 01 α 02 α 03 α 00 0 0 0 α 10 α 11 α 12 α 13 0 α 11 0 0 Λ − → α 20 α 21 α 22 α 23 0 0 α 22 0 α 30 α 31 α 32 α 33 0 0 0 α 33
No-go result for private subspaces Private quantum Theorem (JKLP13 2 ) subsystems and error correction i p i U i ρU † Tomas Jochym- Let Φ( ρ ) = � i be a random unitary channel with mutually O’Connor commuting Kraus operators. Then Φ has no private subspace. Privacy & error correction Restrictions of operator privacy Generalization of subsystem privacy Extended duality 2 TJ, D. W. Kribs, R. Laflamme, S. Plosker (2013)
No-go result for private subspaces Private quantum Theorem (JKLP13 2 ) subsystems and error correction i p i U i ρU † Tomas Jochym- Let Φ( ρ ) = � i be a random unitary channel with mutually O’Connor commuting Kraus operators. Then Φ has no private subspace. Privacy & error correction Restrictions of A subsystem B is an operator private subsystem for Φ if there operator privacy exists ρ 0 such that Generalization of subsystem privacy Extended duality Φ( σ A ⊗ σ B ) = ρ 0 , ∀ σ A , σ B 2 TJ, D. W. Kribs, R. Laflamme, S. Plosker (2013)
No-go result for private subspaces Private quantum Theorem (JKLP13 2 ) subsystems and error correction i p i U i ρU † Tomas Jochym- Let Φ( ρ ) = � i be a random unitary channel with mutually O’Connor commuting Kraus operators. Then Φ has no private subspace. Privacy & error correction Restrictions of operator privacy Generalization of subsystem privacy Extended duality A subsystem B is an operator private subsystem for Φ if there exists ρ 0 such that Φ( σ A ⊗ | ψ �� ψ | ) = ρ 0 , ∀ σ A , | ψ �� ψ | Therefore, the channel Λ = Λ 2 ◦ Λ 1 cannot be operator quantum private 2 TJ, D. W. Kribs, R. Laflamme, S. Plosker (2013)
However... Private quantum subsystems and error correction Tomas Jochym- O’Connor Privacy & error Consider the following encoding of a quantum state: correction Restrictions of operator privacy ρ L = 1 2( I + αXX + βY I + γZX ) . Generalization of subsystem privacy Extended duality ρ L is privatized by the channel Λ = Λ 2 ◦ Λ 1 . A contradiction? It can be shown that the state space defined by the parameters α, β, γ is unitarily equivalent to I 2 ⊗ D 2 , where D 2 is the space of 2-dimensional density matrices.
Where is the loophole? Private quantum subsystems and error correction Tomas Jochym- O’Connor Λ privatizes the state space I 2 ⊗ D 2 , why is this not equivalent to Privacy & error correction operator privacy? Restrictions of A subsystem B is an operator private subsystem for Φ if there operator privacy Generalization of exists ρ 0 such that subsystem privacy Extended duality Φ( σ A ⊗ σ B ) = ρ 0 , ∀ σ A , σ B
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