An Optical Implementation of Quantum Bit Commitment Using Infinite-Dimensional Systems

Q: How can quantum coin tossing be applied?

Quantum coin tossing (QCT) can be applied in scenarios where two parties, Alice and Bob, need to generate a random bit value remotely without trusting each other. It can be used for property division between divorced couples or in online gambling. Strong QCT ensures equal probabilities of 0 and 1, making it a secure method for generating random outcomes. The QCT protocol can be built upon the Quantum Bit Commitment (QBC) protocol, which ensures the randomness and security of the coin tossing result. The commit and unveil phases of the QBC protocol are separated by a single classical bit transfer, allowing for quick and practical implementation of QCT in real-world applications.

Question

1. What is the significance of the QBC protocol proposed in ref. [56]?

2. Is the theoretical description of the Protocol in ref. [56] able to prevent dishonest Alice from cheating in the unveil phase?

3. How to implement infinite-dimensional systems?

4. What is the commit phase in the experimental QBC protocol?

Accepted Answer

The QBC protocol proposed in ref. [56] is significant because it remains secure against the cheating strategy in the no-go proofs, re-opening the potential for many useful multi-party secure computation protocols that were once closed by the MLC no-go theorem. The protocol requires infinite-dimensional systems, but it can be implemented using simple optical devices by using the arrival time of photons as a trick. This allows for the realization of infinite-dimensional systems using a Mach-Zehnder (MZ) interferometer, which is within the capability of currently available technology. Implementing this protocol in practice can have great significance in the field of quantum cryptography.

Accepted Answer

No, the cheating strategy in the no-go proofs fails in our protocol. In ref. [56], it is shown that dishonest Alice can prepare a bipartite system a b in the state: EQUATION, which satisfies Equation (3) and allows her to unveil b as whatever value she likes in the unveil phase by choosing between the two measurements M 0 and M 1. However, in our protocol, the two sets of states take the forms in Equations (1) and (2), and the cheating strategy fails. The HJW theorem applies, but the measurements M 0 and M 1 are actually the same, leaving Alice no freedom to choose between them to alter the value of her committed bit b. Therefore, the protocol is unconditionally secure against dishonest Alice in the unveil phase.

Accepted Answer

To implement infinite-dimensional systems, a trick is proposed using a single photon in each round of the protocol. Alice sends a single photon either from source S 0 or S 1, then splits it into two wave packets |x and |y using a 50:50 non-polarizing beam splitter BS A. |x is sent directly to Bob via path X, while |y is delayed by the storage ring SR A before sending via path Y. At Bob's site, |y is delayed by the storage rings SR y and SR B, then meets |x at the 50:50 beam splitter BS B and interferes. By tuning the delay times caused by SR A and SR B, the complete apparatus forms a balanced MZ interferometer, ensuring that ps 0 i (ps 1 i ) will make the detector D 0 (D 1 ) click with certainty. Optical fibers should be used for paths X and Y for better performance, and Bob's reference set should have optical fibers of the same length in a similar environment to ensure accurate error rate estimation.

Accepted Answer

The commit phase in the experimental QBC protocol involves Alice and Bob agreeing on a maximum delay time (t max) and sending times (t j) for storing ring SR A. Alice randomly picks a delay time (t j) within the range [0, t max] and sends Bob a photon (Ps j) from the source S b at time t j. The value of b (0 or 1) that Alice commits to is kept constant for all j. This phase ensures that Alice commits to a specific value of b and determines the delay times for storing ring SR A.

Accepted Answer

The experimental protocol is a faithful implementation of the theoretical protocol. In the experimental protocol, the state of a photon on path X and Y is denoted by |1 X |0 Y or |0 X |1 Y, depending on the presence of a wave packet. The time t represents the time that the wave packet is present in the path, and |0 P denotes the absence of a wave packet in the path. The initial state of a photon after passing through devices is represented by EQUATION. In the unveil phase, the final state of the photon arriving at BS B is represented by EQUATION. The detectors D0 and D1 serve as projective operators, and if Bob takes tB = tj, the detector Db should click with certainty. The experimental protocol is equivalent to the theoretical protocol when time is treated as a continuous variable, allowing for an infinite number of choices for time slots T1, T2, ..., Tn-1. This equivalence is demonstrated by rewriting Equation (15) in the experimental protocol as the state that Alice sends in step (i) of the theoretical protocol. Therefore, the experimental protocol is secure as long as time can be treated as a continuous variable, satisfying the condition n-.

Accepted Answer

The experimental apparatus, shown in Figure 1, is similar to QKD and quantum private query protocols in refs. [57] [58] [59]. The main difference is the inclusion of two additional storage rings, SR x and SR y. The QKD protocol in ref. [57] was previously implemented experimentally in ref. [60]. By comparing the apparatus with Figure 1 of ref. [60], it is evident that the technology used in ref. [60] is adequate for implementing the protocol. The actual experimental devices are described in Section III of ref. [60]. The implementation requires finding storage rings SR x and SR y with a long delay time, as they determine the holding time of the protocol. Using 150 km of optical fiber can generate approximately 500 us delay time, which is sufficient for practical applications like quantum coin tossing.

Accepted Answer

The two practical difficulties in implementing the protocol are the finite length of classical communications and the limited precision of storage ring delay times. The length of classical communications between Alice and Bob must remain finite, preventing Alice from announcing delay times with an unlimited number of digits. Additionally, the delay time of storage rings cannot be adjusted to an unlimited precision, limiting Alice and Bob's ability to set delay times precisely. These limitations affect the number of time slots available for choosing delay times, resulting in a finite n-dimensional system instead of an infinite-dimensional one. Despite these challenges, the protocol can still be used as a practically secure quantum bit commitment (QBC) scheme.

Accepted Answer

The probability of Alice's cheating strategy in the QBC protocol is determined by the inequality Pr(s - p >= d) <= 2e^-sd^2, where s denotes the actual number of photons being detected correctly, p is the probability of Alice's cheating strategy, and d is a small positive value. This inequality ensures that s should be within the range [(1 - e)s - ds, (1 - e)s + ds], with a probability of 2e^-sd^2 at most. For example, when s = 10,000, e = 10%, and d = 5%, the probability is extremely small (2.8 x 10^-11), indicating that the number of photons detected by Bob's D b should be within the range [8500, 9500] to confidently conclude Alice's dishonesty.

Accepted Answer

Quantum coin tossing (QCT) can be applied in scenarios where two parties, Alice and Bob, need to generate a random bit value remotely without trusting each other. It can be used for property division between divorced couples or in online gambling. Strong QCT ensures equal probabilities of 0 and 1, making it a secure method for generating random outcomes. The QCT protocol can be built upon the Quantum Bit Commitment (QBC) protocol, which ensures the randomness and security of the coin tossing result. The commit and unveil phases of the QBC protocol are separated by a single classical bit transfer, allowing for quick and practical implementation of QCT in real-world applications.

An Optical Implementation of Quantum Bit Commitment Using Infinite-Dimensional Systems

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Most frequently asked questions

1. What is the significance of the QBC protocol proposed in ref. [56]?

2. Is the theoretical description of the Protocol in ref. [56] able to prevent dishonest Alice from cheating in the unveil phase?

3. How to implement infinite-dimensional systems?

4. What is the commit phase in the experimental QBC protocol?

5. How does the experimental protocol relate to the theoretical protocol?

6. How does the experimental apparatus differ from QKD and quantum private query protocols?

7. What are the two practical difficulties in implementing the protocol?

8. What is the probability of Alice's cheating strategy in the QBC protocol?

9. How can quantum coin tossing be applied?

References

The “transition probability” in the state space of a ∗-algebra

Founding crytpography on oblivious transfer

Unconditionally Secure Quantum Bit Commitment is Impossible

A Complete Classification of Quantum Ensembles Having a Given Density Matrix

Is Quantum Bit Commitment Really Possible

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