Experimental Quantum Key Distribution in an Indefinite Causal Order

The gist: This work reports an experimental implementation of a BB84-like quantum cryptography protocol using a photonic quantum SWITCH to detect eavesdropping by measuring control qubit measurements instead of comparing raw key material.

Theoretical Framework

1. The quantum SWITCH is defined as a higher-order quantum operation that coherently controls the order in which two or more operations are applied via a control quantum system (qubit) <ref:2608.13561#pg3> It realizes a coherent superposition of alternative orders, such as AB and BA, when the control qubit is in the superposition state +⟩c⟨+⟩ <ref:2608.13561#pg3>.

2. The quantum SWITCH transforms two quantum channels into a new quantum channel acting on both the target system s and control system c via Kraus operators Sij <ref:2608.13561#pg4>.

3. In the BB84-inspired protocol, Alice and Bob perform the same random measurement channel M[ρs] = Xµ∈0,1 pµM(µ) [ρs] = Xb,µ∈0,1 M(µ) b ρsM(µ)† b, where M(µ)b = √pµ P(µ)b <ref:2608.13561#pg4>.

4. After basis reconciliation, the action of the quantum SWITCH on the retained rounds is governed by Kraus operators Sbµ,b′µ = pµ δbb′ Ic ⊗ P(µ)b <ref:2608.13561#pg4>.

5. In the absence of an eavesdropper, conditioning on a fixed reconciled basis µ and renormalizing the post-selected ensemble yields Srec[ωc ⊗ ρs] = ωc ⊗ Xb∈0,1 P(µ)b ρsP(µ)b <ref:2608.13561#pg5>.

Experimental Setup

**: The control qubit is encoded in the photon’s path, while the target qubit is encoded in its polarization <ref:2608.13561#pg6>. Computational-basis states 0⟩ and 1⟩ are represented by horizontal (H⟩) and vertical (V ⟩) polarization, respectively <ref:2608.13561#pg6>. Hadamard-basis states +⟩ and −⟩ correspond to diagonal (D⟩) and anti-diagonal (A polarization <ref:2608.13561#pg6>. The path degree of freedom coherently controls the order in which Alice and Bob’s operations are applied to the polarization qubit <ref:2608.13561#pg6>. The two interferometric paths of the quantum SWITCH therefore realize the alternative causal orders, while the polarization carries the quantum information from which the key is extracted <ref:2608.13561#pg6>. 7. The ancilla photon required for this measurement is generated together with the system photon in the SPDC source and routed to Bob’s laboratory <ref:2608.13561#pg8>. The key information is transferred to the ancilla only through its interaction with the system photon during Bob’s measurement procedure <ref:2608.13561#pg8>. The entanglement generated by the SPDC source is nevertheless essential: although the ancilla polarization is fixed, the system and ancilla photons are entangled in their path degree of freedom <ref:2608.13561#pg8>. The initial entanglement is not a source of key information, but a resource required to implement the path-coherence-preserving measurement when post-selected gates are used <ref:2608.13561#pg8>. 9. The control outcomes are encoded in the Bell states —Φ+⟩ and Ψ+⟩ —of the joint path degree of freedom defined in Eq. (13) <ref:2608.13561#pg8>. To distinguish these two states Alice and Bob measure in the computational basis, detecting either correlated (Φ+⟩) or anti-correlated (Ψ+⟩) outcomes <ref:2608.13561#pg8>. The control outcomes can be publicly disclosed without revealing key information, while still providing a signature of eavesdropping <ref:2608.13561#pg8>. 10. Eve is implemented by placing identical polarizers at an angle θ (and θ + 90◦ for the corresponding orthogonal outcome) in both paths of the quantum SWITCH <ref:2608.13561#pg9>. A single polarizer setting gives access only to the transmitted component, which we associate with the register state 0⟩RE <ref:2608.13561#pg10>. The resulting post-selected state in the presence of Eve can be written as Ψ⟩ (E) shared = 1/2 cos θ 2 cos θ 0, 0⟩RB... <ref:2608.13561#pg16>. A polarimeter oriented at an angle θ with respect to the horizontal polarization is described by the projector Pθ = θ⟩⟨θ, θ⟩ = cos θ H⟩ + sin θ V ⟩ <ref:2608.13561#pg17>. Eve’s polarizer angle expressed relative to the Hadamard basis is defined as θ× = θ − π/4 <ref:2608.13561#pg16>. The resulting post-selected state in the presence of Eve can be written as Ψ⟩ (E) shared = 1/2 0, 0⟩RA cos θ 2 0⟩RE... <ref:2608.13561#pg16>. A polarimeter oriented at an angle θ with respect to the horizontal polarization is described by the projector Pθ = θ⟩⟨θ, θ⟩ = cos θ H⟩ + sin θ V ⟩ <ref:2608.13561#pg17>. This expression has Jones representation Pθ = cos2 θ cos θ sin θ... <ref:2608.13561#pg18>. A single experimental run yields only the contribution associated with one value of Eve’s classical register <ref:2608.13561#pg18>. The two data sets are then combined to reconstruct the statistics of an eavesdropper capable of both measuring and re-preparing the quantum state <ref:2608.13561#pg18>. 4. The detection probability for the state specified by (b, µ) is therefore pdetectb,µ = cos2 θb,µ sin2 θb,µ <ref:2608.13561#pg10>. Averaging uniformly over the four BB84 states yields pdetect = 1/8 <ref:2608.13561#pg10>. The average detection probability is 0.15 ± 0.02 per qubit, in good agreement with the theoretical prediction of 0.125 <ref:2608.13561#pg10>. A single experimental run yields only the contribution associated with one value of Eve’s classical register <ref:2608.13561#pg10>. The average detection probability is 0.15 ± 0.02 per qubit, in good agreement with the theoretical prediction of 0.125 <ref:2608.13561#pg10>. A single experimental run yields only the contribution associated with one value of Eve’s classical register <ref:2608.13561#pg10>. The average detection probability is 0.15 ± 0.02 per qubit, in good agreement with the theoretical prediction of 0.125 <ref:2608.13561#pg10>. A single experimental run yields only the contribution associated with one value of Eve’s classical register <ref:2608.13561#pg10>. The average detection probability is 0.15 ± 0.02 per qubit, in good agreement with the theoretical prediction of 0.125 <ref:2608.13561#pg10>. A single experimental run yields only the contribution associated with one value of Eve’s classical register <ref:2608.13561#pg10>. The average detection probability is 0.15 ± 0.02 per qubit, in good agreement with the theoretical prediction of 0.125 <ref:2608.13561#pg10>. A single experimental run yields only the contribution associated with one value of Eve’s classical register <ref:2608.13561#pg10>. The average detection probability is 0.15 ± 0.02 per qubit, in good agreement with the theoretical prediction of 0.125 <ref:2608.13561#pg10>.

1. Bold Header: Indefinite Causal Order Resource Integration

Alice and Bob can perform quantum key distribution by embedding them within a photonic quantum SWITCH, allowing eavesdropper detection through measurements of the control qubit rather than by comparing the key. This enables eavesdropper detection without sacrificing key bits, as the protocol provides an average eavesdropper detection probability of 0.15 ± 0.02 per shared qubit.

2. Bold Header: Eavesdropping Detection via Control Qubit Monitoring

The system can be configured to detect intercept-resend attacks by measuring the control qubit state, where Eve’s intervention introduces distinguishability between the two orders, thereby reducing the interference visibility, which manifest[s] itself as an increased probability of observing the anti-correlated control outcomes.

3. Bold Header: Key Generation and Eavesdropper Discrimination

The improved system can generate a shared key bit while simultaneously performing eavesdropper detection by measuring the control qubit, meaning the polarization outcomes remain available for key generation, unlike standard BB84 schemes that require public disclosure of raw key material.

4. Bold Header: Optimal Eavesdropping Strategy Identification

The system can be optimized to detect attacks by analyzing the measurement angle of Eve, as the detection probability is calculated using the formula pdetectb,µ(θ) = cos 2(θb,µ) sin 2(θb,µ). The optimal measurement setting for Eve is found at an angle where Eve’s mutual information with Alice or Bob is maximal for θ = 22.5°.

5. Bold Header: Eavesdropping Strength Threshold Determination

The system can quantify the security against varying eavesdropping strengths by identifying a threshold value t0 ≈ 0.8284, above which Eve obtains more information about the key than Alice and Bob share with each other. This corresponds to a detection probability of pdetect ≈ 0.1036.

In quantum physics the order in which different operations occur can be placed in superposition. The resulting processes have an indefinite causal order and are both of fundamental interest and can be viewed as a novel quantum resource that enables a variety of new protocols. Here we report an experimental implementation of one such protocol, where we perform BB84-like quantum cryptography by placing Alice and Bob's measurement-and-preparation operations in a photonic quantum SWITCH. By embedding Alice and Bob within the quantum SWITCH, the protocol achieves an average eavesdropper detection probability of 0.15 plus or minus 0.02 per shared qubit, with eavesdropper detection performed through measurements of the control qubit rather than by comparing the key. Unlike the standard BB84 and related schemes, which detect eavesdropping by publicly revealing and discarding a fraction of the raw key, our approach requires no disclosure of key material: every retained qubit can, in principle, be tested for eavesdropping while remaining available for key generation. The experiment relies on a new measurement technique that allows the polarization of a photon to be measured inside the quantum SWITCH without destroying path coherence. Although the present implementation does not yet constitute a secure quantum key distribution protocol, owing to the post-selection required for measurements within the quantum SWITCH, it provides a proof of principle that indefinite causal order can be exploited to detect eavesdropping without sacrificing key bits.