1. Weak Value Assisted Quantum Key Distribution
James E. Troupe
Applied Research Laboratories,
The University of Texas at Austin (ARL:UT)
Jacob M. Farinholt
Naval Surface Warfare Center,
Dahlgren Division (NSWCDD)
WEAK MEASUREMENTS & WEAK VALUES
THE QKD PROTOCOL
EFFICACY AGAINST DETECTOR ATTACKS
A weak measurement is a special type of POVM that minimizes the measurement
disturbance of the system being measured. Using the von Neumann representation of the measurement interaction between the measuring device (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. If the time integrated coupling 𝑔 is very small compared to the MD’s position uncertainty, then the MD’s position will yield little information about the observable 𝐴 and the system will be almost undisturbed. However, with a large enough ensemble of weak measurement results from identically prepared systems, we can estimate the value of 𝐴 .
If we also condition the weak measurement results on identical post-selected states, then the resulting weak measurements yield access to the weak value given by
𝐴 𝑤 = 𝜓 𝑓 𝐴 𝜓 𝑖 𝜓 𝑓 𝜓 𝑖 .
The conditional shift in the mean values of the weak measurement results for each pre- and post-selected (PPS) ensemble are given by 𝜇=𝑔 𝐴 𝑤 [1].
The shift in the MD pointer is linear in the coupling strength while the probability of collapsing the system’s initial state into an orthogonal state is reduced quadratically in the coupling.
Note that the weak value in general can be complex valued and its magnitude can exceed the bounds of the eigenvalues of the observable. We will call real weak values that exceed the eigenvalue range anomalous weak values.
In the proposed QKD protocol, Bob will perform weak measurements of specially chosen sets of observables in order to test the degree to which the weak values exceed their eigenvalue range. If the estimated weak values exceed a threshold, then the quantum channel is secure.
An important class of attacks on QKD systems is based on Eve blinding
Bob’s detectors so that only photons that Bob has measured in the
same basis as Eve are detected. This allows Eve’s presence to be
completely hidden with respect to the usual QBER test of BB84, etc.
In the proposed QKD protocol, if Eve uses this attack to hide the errors
induce by her measurement of the photons, the weak measurements
performed by Bob must have the same pre-selected and post-selected
states since Eve’s and Bob’s bases now must agree. When this is the
case, the weak measurements will yield the expectation value of the
observable being measured. The expectation value must be within the observable’s eigenvalue bound, and so that all of the PPS ensembles will generate non-anomalous weak values. Thus, the protocol will make detector attacks such as this significantly more detectable than a simple intercept-resend attack.
The protocol is essentially the Bennett Brassard 84 protocol augmented with weak measurements
of four projectors:
Alice prepares each photon in one of the states , , , or uniformly at random.
Bob weakly measures the projector of one of the four H states given above chosen uniformly
at random.
Bob then randomly chooses the X or Z basis and performs a strong measurement. He records the
results of the post-selection (strong measurement) and of the weak measurement for each photon.
Alice announces the basis in which she prepared the photon. Bob notes whether or not it matches his choice.
The steps above are repeated for a large number of photons until Bob has recorded a sufficient number of detections.
Bob announces which detections were performed in the same basis as prepared by Alice. For the detections where their bases did not match, Alice broadcasts the initial state of each photon.
Bob separates his weak measurement data into lists labelled by identical initial and final photon states (PPS ensembles). These are used to calculate the mean weak measurement values for each of these PPS ensembles.
Bob calculates the estimated weak value for each PPS and uses this to obtain the QBER directly from the quantum channel. If the QBER is above the security threshold, Bob announces that the channel is secure, otherwise Alice and Bob abort the protocol.
Alice and Bob perform error reconciliation and privacy amplification to distill a secure key using the sifted key (i.e. for which their bases matched). Note: there is no need to distribute raw key data to estimate the QBER.
𝐻 ± ≡ 𝐼 𝑋 ± 𝑍
𝐻 ⊥ ± ≡ 𝐼 − 𝑋 ± 𝑍 .
| +
| −
| 0
| 1
| +
| −
| 0
| 1
| 𝐻 +
| 𝐻 ⊥ +
| 𝐻 −
| 𝐻 ⊥ −
WEAK MEASUREMENT INDUCED ERRORS
Since the weak measurements have non-zero strength, the photons’
states will be very slightly altered by the interaction.
The effect of the WMs is a depolarizing channel with a QBER given by
𝑝 𝑊𝑀 = −exp − 𝑔 𝜎 ,
where 𝜎 2 is the variance of the weak measurement pointer. Therefore,
the QBER due to the WMs can be less than for realistic parameters.
CONCLUSION
EFFECTS OF DARK COUNTS ON WEAK VALUE ESTIMATES
WEAK VALUES AND CHANNEL NOISE
We proposed a new QKD protocol that uses weak measurements to obtain better quantum channel estimates, and which is particularly resilient against detector blinding attacks.
The protocol uses the usually discarded detection events in which Alice’s
and Bob’s bases disagree in order to detect Eve. This increases the amount
of sifted raw key available for distillation by almost a factor of two.
Let 𝑑 denote the probability of a detector clicking when no photon arrived. Since the weak measurement occurs immediately prior to post-selection, the probability of the photon getting lost between weak measurement and post-selection is negligible. In the event of a dark count, the weak measurement result is completely uncorrelated with the quantum system. This affects the mean value 𝜇 of the conditional weak measurement results according to
𝜇= 1−𝑑 𝑔 𝐴 𝑤 .
Since all measurements (strong and weak) are performed along the X-Z plane, without loss of generality, we may assume that all noise maps the initial state to some other density operator on this plane. Such a density operator 𝜌 can be expanded as
𝜌= 1−𝑝 𝑝 −𝑞 𝑞 − − − 1 2 𝐼,
for some 𝑝, 𝑞∈ 0, 1 .
Observe that 𝑇𝑟 𝜌 =(1−𝑝), 𝑇𝑟 𝜌 =𝑝, and likewise for the other post-selections. If 0 ( 1 ) was transmitted and mapped to the state 𝜌, then the value 𝑝 (resp. 1−𝑝) corresponds to the probability of error if the state were measured in the computational basis. Similarly, if + ( − ) was transmitted and mapped to the state 𝜌, then the value 𝑞 (resp. 1−𝑞) corresponds to the probability of error if the state were measured in the +/- basis.
When the pre- and post-selection bases disagree, we can obtain the parameters 𝑝 and 𝑞 for each of the 4 states Alice sends by looking at the weak measurement statistics. This provides far more information about the noise in the quantum channel than the standard method of calculating the QBER, and does not require the public disclosure of any of the distilled key.
EXAMPLE: Suppose Alice transmitted the state |0〉, which was mapped to the state 𝜌 immediately prior to weak measurement. Suppose Bob measured in the +/- basis, obtaining the state |+〉. Then the averages of the weak measurement results for each of the four observables in this case are given below:
𝐻 ⊥ + 𝑤 = −1 − −𝑝 1−𝑞
𝐻 ⊥ − 𝑤 = − −𝑝 1−𝑞
𝐻 + 𝑤 = −𝑝 1−𝑞
𝐻 − 𝑤 = − −𝑝 1−𝑞
Repeating these calculations for the same initial state, but conditioned on the post-selected result being |−〉 allows us to obtain accurate estimates of both 𝑝 and 𝑞.
The parameter 𝑝 is the probability that Bob would have measured |1〉 given that Alice sent |0〉 if he had measured in the computational basis. If the channel acts uniformly on states from the same basis (as is generally assumed) then 𝑝 is precisely the BER associated with the computational basis.
REFERENCES
ESTIMATING COUPLING STRENGTH
Y. Aharonov and L. Vaidman, "Properties of a quantum system during the time interval between two measurements", Physical Review A 41, 11 (1990).
J.E. Troupe and J.M. Farinholt, quant-ph/ (2015).
Because we will be using the weak measurement results to infer information about the quantum channel, a clever Eve could attempt to manipulate the coupling strength 𝑔 to alter our noise estimates.
To counter this, we design the QKD system such that the same interaction strength is used for the weak measurement of both 𝐻 ± and 𝐻 ⊥ ± . To estimate 𝑔 in situ, we use the sum of the mean weak measurement results for the orthogonal pair:
𝜇 ± + 𝜇 ⊥ ± =𝑔 1−𝑑 𝐻 ± + 𝐻 ⊥ ± 𝑤 =𝑔 1−𝑑 𝐼𝑑 𝑤 = 𝑔 ± 1−𝑑 .
As can be seen above, in order to obtain accurate estimates of the coupling strength, it is important that we have very good estimates of the detector dark count rates.
ACKNOWLEDGMENTS
J.T. acknowledges support from the Office of Naval Research (ONR) under Grant No. N J.F. acknowledges support from ONR under Grant No. N WX01474, as well as an NSWCDD In-house Laboratory Independent Research (ILIR) grant.