Weak Value Assisted Quantum Key Distribution
James E. Troupe
Jacob M. Farinholt
Applied Research Laboratories,
The University of Texas at Austin (ARL:UT)
Naval Surface Warfare Center,
Dahlgren Division (NSWCDD)
WEAK MEASUREMENTS & WEAK VALUES
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:
𝐻𝑖𝑛𝑡 = 𝑔 𝑡 𝑃𝑀𝐷 ⊗ 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
𝜓𝑓 𝐴 𝜓𝑖
𝐴𝑤 =
.
𝜓𝑓 𝜓𝑖
The conditional shift in the mean values of the weak measurement results for
each pre- and post-selected (PPS) ensemble are given by 𝜇 = 𝑔𝐴𝑤 [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.
EFFECTS OF DARK COUNTS ON WEAK VALUE ESTIMATES
Let 𝑑 denote the probability of a detector clicking when no photon
arrived. Since the weak measurement occurs immediately prior to postselection, 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 𝑔𝐴𝑤 .
ESTIMATING COUPLING STRENGTH
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.
RESEARCH POSTER PRESENTATION DESIGN © 2012
www.PosterPresentations.com
THE QKD PROTOCOL
EFFICACY AGAINST DETECTOR ATTACKS
The protocol is essentially the Bennett Brassard 84 protocol augmented with weak measurements
of four projectors:
1
1
𝐻± ≡ 𝐼 +
𝑋±𝑍
2
2
1
1
𝐻⊥± ≡ 𝐼 −
𝑋±𝑍
2
2
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.
|0
|𝐻 −
|𝐻 +
.
Alice prepares each photon in one of the states |0 , |1 , |+ , 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.
|+
|−
|𝐻⊥+
|𝐻⊥−
|1
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.
Alice announces the basis in which she prepared the photon. Bob notes whether or not it matches his choice.
WEAK MEASUREMENT INDUCED ERRORS
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.
Since the weak measurements have non-zero strength, the photons’
states will be very slightly altered by the interaction.
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.
The effect of the WMs is a depolarizing channel with a QBER given by
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.
𝑝𝑊𝑀 =
1
1 𝑔
1 − exp −
4
8 𝜎
2
,
where 𝜎 2 is the variance of the weak measurement pointer. Therefore,
the QBER due to the WMs can be less than 0.001 for realistic parameters.
CONCLUSION
WEAK VALUES AND CHANNEL NOISE
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
1
𝜌 = 1 − 𝑝 0 0 + 𝑝 1 1 + 1 − 𝑞 + + + 𝑞 − − − 𝐼,
2
for some 𝑝, 𝑞 ∈ 0, 1 .
Observe that 𝑇𝑟 𝜌 0 0 = (1 − 𝑝), 𝑇𝑟 𝜌 1 1 = 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
+
𝐻−
𝑤
𝑤
=
=
1
2 2
1
2 2
1
−𝑝
2+1 + 2
1−𝑞
1
−𝑝
2−1 + 2
1−𝑞
𝐻⊥+ 𝑤
𝐻⊥−
𝑤
=
=
1
2 2
1
2 2
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.
1
−𝑝
2−1 − 2
1−𝑞
1
−𝑝
2+1 − 2
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
1.
Y. Aharonov and L. Vaidman, "Properties of a quantum system during the time
interval between two measurements", Physical Review A 41, 11 (1990).
2.
J.E. Troupe and J.M. Farinholt, quant-ph/1512.02256 (2015).
ACKNOWLEDGMENTS
J.T. acknowledges support from the Office of Naval Research
(ONR) under Grant No. N00014-15-1-2225. J.F. acknowledges
support from ONR under Grant No. N0001416WX01474, as well
as an NSWCDD In-house Laboratory Independent Research (ILIR)
grant.