Secure Certification of Mixed Quantum States Frdric Dupuis, Serge - PowerPoint PPT Presentation

Secure Certification of Mixed Quantum States Frédéric Dupuis, Serge Fehr, Philippe Lamontagne and Louis Salvail

Quantum state certification H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H 1/8

Quantum state certification H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H 1/8

Quantum state certification H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H Certification • Measure H with {| ψ � � ψ | , I − | ψ � � ψ |} • If result is | ψ � for every H , then most of the remaining positions are in state | ψ � with overwhelming probability [BF10]. • The reference state | ψ � must be pure. 1/8

Quantum state certification H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H Certification • Measure H with {| ψ � � ψ | , I − | ψ � � ψ |} • If result is | ψ � for every H , then most of the remaining positions are in state | ψ � with overwhelming probability [BF10]. • The reference state | ψ � must be pure. 1/8

Quantum state certification H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H Certification • Measure H with {| ψ � � ψ | , I − | ψ � � ψ |} • If result is | ψ � for every H , then most of the remaining positions are in state | ψ � with overwhelming probability [BF10]. • The reference state | ψ � must be pure. 1/8

Quantum state certification H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H Certification • Measure H with {| ψ � � ψ | , I − | ψ � � ψ |} • If result is | ψ � for every H , then most of the remaining positions are in state | ψ � with overwhelming probability [BF10]. • The reference state | ψ � must be pure. 1/8

What about certifying mixed states ? Usual approach fail Notion of typical subspace not applicable 2/8

What about certifying mixed states ? Usual approach fail Notion of typical subspace not applicable X sample = 00 . . . 0 Pr ≈ 1 = ⇒ X rest ∈ { x : x has less than δ n 1s } 2/8

What about certifying mixed states ? Usual approach fail Notion of typical subspace not applicable X sample = 00 . . . 0 Pr ≈ 1 = ⇒ X rest ∈ { x : x has less than δ n 1s } 2/8

What about certifying mixed states ? Usual approach fail Notion of typical subspace not applicable X sample = 00 . . . 0 Pr ≈ 1 = ⇒ X rest ∈ { x : x has less than δ n 1s } 2/8

What about certifying mixed states ? Usual approach fail Notion of typical subspace not applicable X sample = 00 . . . 0 Pr ≈ 1 = ⇒ X rest ∈ { x : x has less than δ n 1s } • For pure states � � Pr ≈ 1 = | 0 � ⊗ k = ⇒ | ψ rest � ∈ span {| x � : x has less than δ n 1s } � ψ sample 2/8

What about certifying mixed states ? Usual approach fail Notion of typical subspace not applicable X sample = 00 . . . 0 Pr ≈ 1 = ⇒ X rest ∈ { x : x has less than δ n 1s } • For pure states � � Pr ≈ 1 = | 0 � ⊗ k = ⇒ | ψ rest � ∈ span {| x � : x has less than δ n 1s } � ψ sample 2/8

What about certifying mixed states ? Usual approach fail Notion of typical subspace not applicable X sample = 00 . . . 0 Pr ≈ 1 = ⇒ X rest ∈ { x : x has less than δ n 1s } • For pure states � � Pr ≈ 1 = | 0 � ⊗ k = ⇒ | ψ rest � ∈ span {| x � : x has less than δ n 1s } � ψ sample 2/8

What about certifying mixed states ? Usual approach fail Notion of typical subspace not applicable X sample = 00 . . . 0 Pr ≈ 1 = ⇒ X rest ∈ { x : x has less than δ n 1s } • For pure states � � Pr ≈ 1 = | 0 � ⊗ k = ⇒ | ψ rest � ∈ span {| x � : x has less than δ n 1s } � ψ sample • For some mixed states ϕ , supp ( ϕ ⊗ n ) = H ⊗ n 2/8

What about certifying mixed states ? Usual approach fail Notion of typical subspace not applicable X sample = 00 . . . 0 Pr ≈ 1 = ⇒ X rest ∈ { x : x has less than δ n 1s } • For pure states � � Pr ≈ 1 = | 0 � ⊗ k = ⇒ | ψ rest � ∈ span {| x � : x has less than δ n 1s } � ψ sample • For some mixed states ϕ , supp ( ϕ ⊗ n ) = H ⊗ n No local measurement for a discrete notion of errors for mixed states 2/8

A mixed state certification protocol Possible to verify that a qubit is in state ϕ if we have access to its purifying register. 3/8

A mixed state certification protocol Possible to verify that a qubit is in state ϕ if we have access to its purifying register. Two-player «Game» Verifier wants to certify that his state is close to ϕ ⊗ n . Prover wants to fool the verifier into thinking he has the right state even though it’s not the case. 3/8

A mixed state certification protocol Possible to verify that a qubit is in state ϕ if we have access to its purifying register. Two-player «Game» Verifier wants to certify that his state is close to ϕ ⊗ n . Prover wants to fool the verifier into thinking he has the right state even though it’s not the case. AR , send A n to verifier. P. Prepare | ϕ � ⊗ n V. Choose a random sample, announce it to prover. P. Send R for each position in sample. V. Measure {| ϕ � � ϕ | AR , I − | ϕ � � ϕ | AR } for each joint system AR in sample. V. Accept if no errors, reject otherwise. 3/8

A mixed state certification protocol Possible to verify that a qubit is in state ϕ if we have access to its purifying register. Two-player «Game» Verifier wants to certify that his state is close to ϕ ⊗ n . Prover wants to fool the verifier into thinking he has the right state even though it’s not the case. AR , send A n to verifier. P. Prepare | ϕ � ⊗ n V. Choose a random sample, announce it to prover. P. Send R for each position in sample. V. Measure {| ϕ � � ϕ | AR , I − | ϕ � � ϕ | AR } for each joint system AR in sample. V. Accept if no errors, reject otherwise. 3/8

A mixed state certification protocol Possible to verify that a qubit is in state ϕ if we have access to its purifying register. Two-player «Game» Verifier wants to certify that his state is close to ϕ ⊗ n . Prover wants to fool the verifier into thinking he has the right state even though it’s not the case. AR , send A n to verifier. P. Prepare | ϕ � ⊗ n V. Choose a random sample, announce it to prover. P. Send R for each position in sample. V. Measure {| ϕ � � ϕ | AR , I − | ϕ � � ϕ | AR } for each joint system AR in sample. V. Accept if no errors, reject otherwise. 3/8

A mixed state certification protocol Possible to verify that a qubit is in state ϕ if we have access to its purifying register. Two-player «Game» Verifier wants to certify that his state is close to ϕ ⊗ n . Prover wants to fool the verifier into thinking he has the right state even though it’s not the case. AR , send A n to verifier. P. Prepare | ϕ � ⊗ n V. Choose a random sample, announce it to prover. P. Send R for each position in sample. V. Measure {| ϕ � � ϕ | AR , I − | ϕ � � ϕ | AR } for each joint system AR in sample. V. Accept if no errors, reject otherwise. 3/8

A mixed state certification protocol Possible to verify that a qubit is in state ϕ if we have access to its purifying register. Two-player «Game» Verifier wants to certify that his state is close to ϕ ⊗ n . Prover wants to fool the verifier into thinking he has the right state even though it’s not the case. AR , send A n to verifier. P. Prepare | ϕ � ⊗ n V. Choose a random sample, announce it to prover. P. Send R for each position in sample. V. Measure {| ϕ � � ϕ | AR , I − | ϕ � � ϕ | AR } for each joint system AR in sample. V. Accept if no errors, reject otherwise. 3/8

A few observations about the protocol Interaction is necessary How can you distinguish ≈ n / 2 times ≈ n / 2 times � | 0 � � ⊗ n � 0 | + | 1 � � 1 | � �� � � �� � from | 0 �| 0 � . . . | 0 � | 1 �| 1 � . . . | 1 � 2 2 4/8

A few observations about the protocol Interaction is necessary How can you distinguish ≈ n / 2 times ≈ n / 2 times � | 0 � � ⊗ n � 0 | + | 1 � � 1 | � �� � � �� � from | 0 �| 0 � . . . | 0 � | 1 �| 1 � . . . | 1 � 2 2 Interaction gives more power to prover P. V. 4/8

A few observations about the protocol Interaction is necessary How can you distinguish ≈ n / 2 times ≈ n / 2 times � | 0 � � ⊗ n � 0 | + | 1 � � 1 | � �� � � �� � from | 0 �| 0 � . . . | 0 � | 1 �| 1 � . . . | 1 � 2 2 Interaction gives more power to prover P. V. 1. Learns sample 4/8