Quantum sampling of mixed states

Transcription

1 Quantum sampling of mixed states Philippe Lamontagne January 7th Philippe Lamontagne Quantum sampling of mixed states January 7th 1 / 9

2 The setup Philippe Lamontagne Quantum sampling of mixed states January 7th 2/9

3 The setup The goal of Alice is to pass while still being entangled with B 0. Philippe Lamontagne Quantum sampling of mixed states January 7th 2/9

4 What does sampling do? Bob has a reference n-qubit state ϕ and tests that ρ B ϕ ϕ by sampling the state, i.e., measuring random qubits and comparing results with expected ones. Philippe Lamontagne Quantum sampling of mixed states January 7th 3 / 9

5 What does sampling do? Bob has a reference n-qubit state ϕ and tests that ρ B ϕ ϕ by sampling the state, i.e., measuring random qubits and comparing results with expected ones. Example ϕ = : Bob measures random positions and check if result is 0. Philippe Lamontagne Quantum sampling of mixed states January 7th 3 / 9

6 What does sampling do? Bob has a reference n-qubit state ϕ and tests that ρ B ϕ ϕ by sampling the state, i.e., measuring random qubits and comparing results with expected ones. Example ϕ = : Bob measures random positions and check if result is 0. ϕ = H θ 1 x 1 H θn x n : Bob measures random positions i in basis θ i and check if result is x i. Philippe Lamontagne Quantum sampling of mixed states January 7th 3 / 9

7 What does sampling do? Bob has a reference n-qubit state ϕ and tests that ρ B ϕ ϕ by sampling the state, i.e., measuring random qubits and comparing results with expected ones. Example ϕ = : Bob measures random positions and check if result is 0. ϕ = H θ 1 x 1 H θn x n : Bob measures random positions i in basis θ i and check if result is x i. General form: ϕ = U 1 0 U n 0. Philippe Lamontagne Quantum sampling of mixed states January 7th 3 / 9

8 What does sampling do? Bob has a reference n-qubit state ϕ and tests that ρ B ϕ ϕ by sampling the state, i.e., measuring random qubits and comparing results with expected ones. Example ϕ = : Bob measures random positions and check if result is 0. ϕ = H θ 1 x 1 H θn x n : Bob measures random positions i in basis θ i and check if result is x i. General form: ϕ = U 1 0 U n 0. Bob tests that the state is pure. Philippe Lamontagne Quantum sampling of mixed states January 7th 3 / 9

9 The sampling theorem Theorem (Sampling, Bouman and Fehr) If Bob s measurement results coincide with the reference state, then the state resulting from the sampling is ɛ-close to the reference state. Philippe Lamontagne Quantum sampling of mixed states January 7th 4 / 9

10 The sampling theorem Theorem (Sampling, Bouman and Fehr) If Bob s measurement results coincide with the reference state, then the state resulting from the sampling is ɛ-close to the reference state. A consequence of this theorem is that if Alice passes in the experiment from slide 1, she has only approximately ɛn qubits entangled with B. Philippe Lamontagne Quantum sampling of mixed states January 7th 4 / 9

11 Why is this useful? In a cryptographic protocol, the reference state can be encoded by a classical cryptography: Bob can access the description of the state for the positions he wants to sample, but will have uncertainty on the state after sampling. Philippe Lamontagne Quantum sampling of mixed states January 7th 5 / 9

12 Why is this useful? In a cryptographic protocol, the reference state can be encoded by a classical cryptography: Bob can access the description of the state for the positions he wants to sample, but will have uncertainty on the state after sampling. Therefore, after sampling, Alice will have low entanglement with Bob s register and he will have high uncertainty on his register s state. This is a useful setup in quantum cryptography. Philippe Lamontagne Quantum sampling of mixed states January 7th 5 / 9

13 Sampling mixed states A restriction of the sampling theorem is that Bob needs a complete description of the reference state: it must be a pure state. Philippe Lamontagne Quantum sampling of mixed states January 7th 6 / 9

14 Sampling mixed states A restriction of the sampling theorem is that Bob needs a complete description of the reference state: it must be a pure state. If Bob only has a partial description of the reference string, Alice will have to fill the voids, but this leaves room to cheat. Philippe Lamontagne Quantum sampling of mixed states January 7th 6 / 9

15 Sampling mixed states A restriction of the sampling theorem is that Bob needs a complete description of the reference state: it must be a pure state. If Bob only has a partial description of the reference string, Alice will have to fill the voids, but this leaves room to cheat. Example Bob wants to check if Alice sent each qubit in either state 0 or + = 1 2 ( ), but does not know which. Philippe Lamontagne Quantum sampling of mixed states January 7th 6 / 9

16 Sampling mixed states A restriction of the sampling theorem is that Bob needs a complete description of the reference state: it must be a pure state. If Bob only has a partial description of the reference string, Alice will have to fill the voids, but this leaves room to cheat. Example Bob wants to check if Alice sent each qubit in either state 0 or + = 1 2 ( ), but does not know which. To sample a state of this form, he asks Alice to tell him in which basis (+ or ) he must measure to obtain 0. Philippe Lamontagne Quantum sampling of mixed states January 7th 6 / 9

17 Sampling mixed states A restriction of the sampling theorem is that Bob needs a complete description of the reference state: it must be a pure state. If Bob only has a partial description of the reference string, Alice will have to fill the voids, but this leaves room to cheat. Example Bob wants to check if Alice sent each qubit in either state 0 or + = 1 2 ( ), but does not know which. To sample a state of this form, he asks Alice to tell him in which basis (+ or ) he must measure to obtain 0. In the previous example, the reference state is a mixed state of the form θ α θh θ 0 0 H θ. Philippe Lamontagne Quantum sampling of mixed states January 7th 6 / 9

18 Alice can always cheat There is no sampling method that can prevent Alice from being entangled in the previous example. Alice can prepare n states of the form 1 2 ( ) and send the second qubit to Bob. Philippe Lamontagne Quantum sampling of mixed states January 7th 7 / 9

19 Alice can always cheat There is no sampling method that can prevent Alice from being entangled in the previous example. Alice can prepare n states of the form 1 2 ( ) and send the second qubit to Bob.When asked to announce the basis, Alice measures the qubit she kept. Therefore, Bob will always measure 0, and Alice will be entangled with each qubit not sampled. Philippe Lamontagne Quantum sampling of mixed states January 7th 7 / 9

20 Alice can always cheat There is no sampling method that can prevent Alice from being entangled in the previous example. Alice can prepare n states of the form 1 2 ( ) and send the second qubit to Bob.When asked to announce the basis, Alice measures the qubit she kept. Therefore, Bob will always measure 0, and Alice will be entangled with each qubit not sampled. We want a sampling theorem that shows that only states of the form θ α θ θ A H θ 0 B can pass the sampling. Philippe Lamontagne Quantum sampling of mixed states January 7th 7 / 9

21 How do you sample a mixed state? Philippe Lamontagne Quantum sampling of mixed states January 7th 8/9

22 Questions? Philippe Lamontagne Quantum sampling of mixed states January 7th 9 / 9