1. with Weak Measurements
Quantum Cryptography
with Weak Measurements
APS March Meeting
16 March 2017
Jake Farinholt
Strategic & Computing Systems Department
Naval Surface Warfare Center, Dahlgren Div.
James Troupe
Applied Research Laboratories
The University of Texas at Austin
arXiv:

2. What are Weak Measurements?
Suppose the measuring system, called the “measuring device” (MD), has an initial quantum state that is a real-valued Gaussian with unit uncertainty.
Using the von Neumann representation of the measurement interaction between the MD and the system, we have
𝐻 𝑖𝑛𝑡 =𝑔 𝑡 𝑃 𝑀𝐷 ⊗ 𝐴 ,
where 𝑔(𝑡) is the coupling strength as a function of time.
Let 𝑔=∫𝑔 𝑡 𝑑𝑡 be the time-integrated coupling. In the limit of very weak coupling 𝑔 𝐴 ≪𝜎 between the measuring device and the system, the weak measurement of the observable 𝐴 results in a shift in the measuring device’s wavefunction according to
𝜓 𝑥 →𝜓 𝑥 −𝑔Re 𝐴 𝑤 .

3. What are Weak Measurements?
The outcome of a weak measurement of an observable on a single quantum system will provide essentially no information about the system, but it will also not significantly disturb the system by collapsing the initial state. If, however, a weak measurement of some observable 𝐴 is performed on a large ensemble of identically prepared systems in some initial state 𝜌 𝑖 , then the mean of the weak measurement results yields the expectation value: 𝜇=𝑔 𝐴 𝜌 𝑖 .

4. Including Weak Measurements in QKD
BB84:
Assumption:
Parameter estimation can only be performed on the subset of signals for which the bases agree.
Bob tries to enforce the assumed uniformity of Eve’s interactions by randomizing his basis choice.
Bases Agree
Eve
Alice
Bob
RNG
Bases Disagree
Visually emphasize Bob needs to randomly switch bases to “enforce” the uniformity of noise assumption.

5. Including Weak Measurements in QKD
BB84:
Reality:
Eve can correlate her measurements with Bob’s. Assumption is violated.
Eve can assure that her interactions when the bases agree introduce no noise.
INSECURE!!
Bases Agree
Eve
Alice
Bob
RNG
Eve
Bases Disagree
Visually emphasize Bob needs to randomly switch bases to “enforce” the uniformity of noise assumption.

6. Including Weak Measurements in QKD
Adding Weak Measurements:
Idea:
Using weak measurements for parameter estimation, Bob is able to use information from EACH signal, regardless of basis agreement.
Eliminates all measurement basis-dependent attacks.
Bob no longer needs to randomize (strong) measurement basis.
Eve
Alice
Bob WM
Visually emphasize Bob needs to randomly switch bases to “enforce” the uniformity of noise assumption.

7. Including Weak Measurements in QKD
THE INTUITION:
If we know the quantum system was initially prepared in the pure state | 𝜓 𝑖 〉, then the difference between the measured and ideal expectation values should provide some information about the evolution of the initial state.
𝐴 𝜓 𝑖 𝜓 𝑖 → 𝐴 𝜌 𝑖
𝜓 𝑖 𝜓 𝑖 → 𝜌 𝑖
The weak measurement observables for this protocol are
𝐻 ± = 1 2 𝐼 𝑍±𝑋

8. The Weak Measurement QKD Protocol
Alice generates a length 2𝑛 random string of bits, encoding each bit in either the 𝑋 or 𝑍 basis uniformly at random. Then she transmits each qubit to Bob.
Bob performs weak measurements of 𝐻 + or 𝐻 − chosen uniformly at random on each signal he receives, then strongly measures in the 𝒁 basis, recording both his weak and strong measurement result.
Bob publicly reveals the weak measurement results with Alice, who breaks them into subensembles conditioned on the transmitted state and averages each. She uses this information to calculate the 𝛿 𝑋 and 𝛿 𝑍 error rates.
If the error rates are low enough, Alice announces which signals were prepared in the 𝑍 basis. Alice and Bob perform classical post-processing on this subset to distill a smaller secure shared key.

9. The Weak Measurement QKD Protocol: Parameter Estimation
Suppose Alice prepares the state 𝛼 𝛼 (𝛼∈{0,1,+,−}), but the mixed state
𝜌 𝛼 = 1 2 𝐼+ 𝑟 𝑥 𝛼 𝑋+ 𝑟 𝑦 𝛼 𝑌+ 𝑟 𝑧 𝛼 𝑍
arrives at Bob’s weak measurements.
We have
𝑟 𝑥 𝛼 = 𝐻 𝜌 𝛼 − 𝐻 − 𝜌 𝛼 ,
𝑟 𝑧 𝛼 = 𝐻 𝜌 𝛼 𝐻 − 𝜌 𝛼 −1 ,
and
𝛿 𝑋 = − 𝑟 𝑥 + + 𝑟 𝑥 − ,
𝛿 𝑍 = − 𝑟 𝑧 0 + 𝑟 𝑧 1 .

10. The Weak Measurement QKD Protocol: Parameter Estimation
Let 𝑔 ± denote the coupling constant associated with the weak measurement of observable 𝐻 ± .
Average of the weak measurement results of 𝐻 ± conditioned on initial state 𝛼 𝛼 is given by
𝜇 𝛼 ± = 𝑔 ± 𝐻 ± 𝜌 𝛼 .
Assuming unital noise, we have 𝐻 ± 𝜌 𝛼 𝐻 ± 𝜌 𝛼 ⊥ =1, so that
𝜇 𝛼 ± + 𝜇 𝛼 ⊥ ± = 𝑔 ± .
We can calculate the coupling constant independently from the error rates. We do not need to trust the coupling!

11. The Weak Measurement QKD Protocol: Device Imperfections
Imperfect, but unbiased weak measurement observables:
Expectation values stay the same, but variance of measurement results increases.
We place an upper bound on permissible variance, 𝜎 𝑈 2 .

12. The Weak Measurement QKD Protocol: Device Imperfections
Uniformly biased weak measurement observables:
Let 𝛿 𝑏 ≈ 𝛿 𝑋 + 𝛿 𝑍 . Then uniformly biasing the observables always increases the estimate of 𝛿 𝑏 .
In fact, this holds (within modest noise environments) even when the bias is not uniform (e.g. 𝐻 + and 𝐻 − can have separate biases applied).

13. The Weak Measurement QKD Protocol: Towards Weak Measurement Device Independence
Suppose weak measurements occur inside a black box in possession of Eve.
Eve receives the weak measurement result and can choose to either broadcast it or broadcast some other number.
Assumption 1: Eve cannot manipulate the path between weak measurement and post-selection. (She has not covertly placed any components inside of Bob’s measurement device.)
Assumption 2: Eve has limited knowledge of which observable is weakly measured on each signal. (She can correctly guess the observable choice with some probability 1 2 ≤ 𝑝 𝐻 <1.)

14. The Weak Measurement QKD Protocol: Towards Weak Measurement Device Independence
QUESTION: Can Eve fake the error estimates to her benefit?
Answer: Probably not.
We designed several weak measurement attack strategies we believe to be optimal given the constraints.
Weak Measurement Certification Step:
Verify 𝛿 𝑋 , 𝛿 𝑍 ≥0.
Verify estimated WM variance 𝜎 𝑀𝐷 2 is below some predefined threshold bound 𝜎 𝑈 2 .
These checks place very tight restrictions on what Eve can do given limited knowledge of the source and choice of WM observable
The threshold bound 𝜎 𝑈 2 determines the upper bound we place on Eve’s access to the choice of WM observable.

15. The Weak Measurement QKD Protocol: Expected Performance
MDI-QKD with decoys (blue) and WM-QKD with decoys (red) asymptotic performance using standard APDs and varying decoy parameters. As can be seen, WM-QKD is far more robust to varying decoy parameters than MDI-QKD.

16. Thank You! Acknowledgements
This research was funded in part by ONR Grants N and N WX01474, as well as an NSWCDD In-house Laboratory Independent Research (ILIR) grant.
Thank You!

17. Backup Slides

18. The Weak Measurement QKD Protocol: An Implementation

19. The Weak Measurement QKD Protocol: Expected Performance
WM-QKD with decoy states (red) and BB84 with decoy states (blue), using standard APDs (solid lines) and High Efficiency APDs (dotted lines). The asymptotic performance of WM-QKD is almost indistinguishable from that of BB84.

20. The Weak Measurement QKD Protocol: Expected Performance
MDI-QKD with decoys (blue) and WM-QKD with decoys (red) asymptotic performance using standard APDs and varying signal intensities.

21. The Weak Measurement QKD Protocol: Expected Performance
MDI-QKD with decoys (blue) and WM-QKD with decoys (red) asymptotic performance using SNSPDs and varying signal intensities.