Protocol for Counterfactually Transporting an Unknown Qubit

Quantum teleportation circumvents the uncertainty principle using dual channels: a quantum one consisting of previously-shared entanglement, and a classical one, together allowing the disembodied transport of an unknown quantum state over distance. It has recently been shown that a classical bit can be counterfactually communicated between two parties in empty space,"Alice"and"Bob". Here, by using our"dual"version of the chained quantum Zeno effect to achieve a counterfactual CNOT gate, we propose the first protocol for transporting an unknown qubit counterfactually, that is without any physical particles travelling between Alice and Bob - no classical channel and no previously-shared entanglement.

Quantum teleportation [1] circumvents the uncertainty principle using dual channels: a quantum one consisting of previously-shared entanglement, and a classical one, together allowing the disembodied transport of an unknown quantum state over distance. It has recently been shown that a classical bit can be counterfactually communicated between two parties in empty space, "Alice" and "Bob" [2]. Here, using a novel "dual" version of the chained quantum Zeno effect [3], we propose a protocol for transporting an unknown qubit counterfactually, that is without any physical particles travelling between Alice and Bob-no classical channel and no previously-shared entanglement. In contrast to classical information, quantum states cannot be faithfully copied-as proven by the no-cloning theorem [4]. In fact doing so would not only violate the uncertainty principle, but would also violate the rule against faster-than-light signalling [5]. Teleporting an unknown qubit, whereby an identical version appears elsewhere, was therefore presumed to firmly belong to the realm of science fiction-until Bennett et al. [1] showed it possible using previously shared entanglement and a classical channel, such as a phone line. The original qubit, in accordance with no-cloning, is duly destroyed in the process. Quantum teleportation has since been demonstrated in countless experiments [6][7][8][9][10].
Here, we wonder: Is the disembodied transport of an unknown qubit over distance possible, even in principle, without recourse to previously-shared entanglement or a classical channel?-and intriguingly, without physical particles travelling between Alice and Bob? The answer, as we show, is surprisingly yes.
It has recently been shown that a classical bit can be counterfactually transferred between two parties in empty space [2]. The key ideas behind direct counterfactual quantum communication are interaction-free measurement [11,12] and the quantum Zeno effect [13][14][15][16][17]. The phenomenon of interaction-free measurement, or quantum interrogation, relies on the fact that the presence of an obstructing object, acting as a measuring device inside an interferometer setting, destroys interference even if no particle is absorbed by the object. This has the surprising consequence that sometimes the presence of such an object can be inferred without the object directly interacting with any (interrogating) particles. Noh [18] used this to design a counterfactual quantum key distribution protocol whereby, for the shared random bits, no photons travel between Alice and Bob. The quantum Zeno effect, on the other hand, refers to the fact that repeated measurement of an evolving quantum system can inhibit its evolution, leaving it in its initial state, an effect often paraphrased as "a watched kettle * salih.hatim@gmail.com never boils". Crucially, the Zeno effect can dramatically boost the efficiency of interaction-free measurements.
We now turn to our protocol for the counterfactually transporting an unknown qubit. The protocol uses a dual version of the chained quantum Zeno effect (CQZE) to construct a fully counterfactual quantum CNOT gate. Counterfactual transport then follows straightforwardly.
Let's first consider the Mach-Zehnder Zeno setup of FIG. 1(a) [2]. (Throughout, we will adopt similar notation and style to Salih et al. [2] for consistency.) The first concept we require here is Bob effecting a quantum superposition of blocking and not blocking the transmission channel [11,12,14]. Although this is easier to imagine from a practical point of view for the Michelson version we will discuss shortly, we will for now stick to the Mach-Zehnder one, which is easier to explain. Here, BS stands for beamsplitter. The action of BS on Alice's photon is the following, |10 → cos θ |10 + sin θ |01 and |01 → cos θ |01 − sin θ |10 , where the state |10 corresponds to the photon being on the left of BS, the state |01 corresponds to the photon being on the right of BS, and cos θ = √ R, with R being the reflectivity of BS. We set θ = π/2N , where N is the number of beamsplitters. Let the initial combined state of Bob's quantum object together with Alice's photon, impinging on the first beamsplitter BS from the top left, be (α |pass + β |block ) ⊗ |10 . After n beamsplitters, And after N beamsplitters, with N very large, the combined state of Bob's quantum object and Alice's photon becomes (α |pass |01 + β |block |10 ). The factor cos n−1 θ squared is the probability that Alice's photon is not lost due to measurement by Bob's object, which brings about the Zeno effect. We have implemented a CNOT gate, with Bob's the control bit, |block ≡ |0 , and |pass ≡ |1 , and Alice's the target bit, |10 ≡ |0 , and |01 ≡ |1 , albeit for only one of Alice's possible input states, namely |0 . Moreover, the scheme is only counter-factual for the part of the superposition corresponding to Bob blocking and is not counterfactual for the part of the superposition corresponding to Bob not blocking, where Alice's photon gradually "leaks" into the channel.
We now show how to achieve complete CNOT counterfactuality, for Alice's input state |0 , using the chained quantum Zeno effect (CQZE) setup of FIG. 1(b) [2]. Here, Alice's photon goes through M beamsplitters BS M , with θ M = π/2M . Between successive BS M s the photon goes through N beamsplitters BS N , with θ N = π/2N . The state |100 corresponds to Alice's photon being on the left of BS M , the state |010 corresponds to the photon being on the right of BS M and on the left of BS N , and the state |001 corresponds to the photon being on the right of BS N . For the m-th cycle, And after N beamsplitters BS N , with N very large, the combined state of Bob's quantum object and Alice's photon becomes (α |pass |001 + β |block |010 ). The factor cos n−1 θ N squared is the probability that the photon is not lost due to measurement by Bob's object, which brings about the Zeno effect. But Alice's single photon is initially in the state |100 , as shown in FIG. 1(b), with all unused ports in the vacuum state. After the m-th BS M , And after the M -th BS M , with M very large, the combined state of Bob's quantum object and Alice's photon ≃ (α |pass |100 + β |block |010 ). The factor cos m−1 θ M squared is the probability that Alice's photon is not lost through detection by one of her D 3 s, which bring about the Zeno effect for the part of the superposition corresponding to Bob not blocking. We have thus implemented a fully counterfactual CNOT gate, with Bob's the control bit |pass ≡ |0 , and |block ≡ |1 , and Alice's the target bit |100 ≡ |0 , and |010 ≡ |1 , again for only one of Alice's possible input states; |0 .
The probability that the photon successfully avoids detection by all D 3 s for the |pass part of the superposition is cos 2(M−1) θ M . This part has a probability amplitude α. While for the |block part of the superposition, the probability that the photon avoids being lost due to measurement by Bob's object (or D 3 ) is This part has a probability amplitude β. Thus the maximum efficiency of this counterfactual CNOT gate is |α| 2 plots this success probability for different M s and N s, for α = β = 1/ √ 2. We can see that efficiency approaches unity for large M and N , given ideal implementation. For instance for M = 50 and N = 1250, efficiency is already 95%. (Unlike Mach-Zehnder, where for Bob passing, the last BS M produces an undesired rotation-tiny for large Min the Michelson implementation we will discuss next, a small number of cycles does not lead to output errors from the counterfactual CNOT gate, but would instead lead to more instances of the gate failing through photon loss. Imperfect implementation, however, would [2].) Complete counterfactuality is ensured: Any photon going into the channel would either be lost due to measurement by Bob's object or else end up at one of the detectors D 3 . Moreover, the probability amplitude of the photonic state |001 corresponding to the photon being in the channel is virtually zero for large enough M and N . Nevertheless, the scheme is not practical, and more fundamentally, it only works for one of Alice's input states.
We now switch to a much more practical and versatile Michelson implementation, with a massive saving of physical resources, where the function of BS in the CQZE setup of FIG. 1(b) is achieved by the combined action of switchable polarisation rotator SP R and polarising beamsplitter P BS, FIG. 3 [2]. The action of SP R , with i = 1, 2 corresponding to SP Rs with different rotation angles. We set the rotation angle θ 1(2) = π/2M (N ) with SP R 1(2) switched on once per cycle, when the photon, or part of it rather, is moving in the direction from SM 1(2) towards P BS 1 (2) . Initially, switchable mirror SM 1(2) is turned off allowing the photon in, but is then turned on for M (N ) cycles before it is turned off again, allowing the photon out.
Note This means that Alice can encode her bit using polarisation. She encodes a "0"("1") by sending a H(V ) photon into the corresponding H(V )-input CQZE setup. But can Alice encode a quantum superposition of "0" and "1"? Crucially, the answer is yes. She first passes her photon through P BS L in order to separate it into H and V components as shown in FIG. 4(a). The H(V ) component is then fed into the corresponding H(V )-input CQZE setup. Bob can block or not block the transmission channel-or a quantum superposition of blocking and not blocking-for both H and V components which are first recombined using P BS R , FIG. 4(a). The polarisation of Alice's exiting photon is determined by Bob's bit choice. This is our dual chained quantum Zeno effect.
All we need now is to combine the two photonic states from the upper and lower paths. This is done by replacing P BS L in FIG. 4(a) by a 50:50 beamsplitter BS, as shown in 4(b). We define the upper-path as above or to the right of BS, and the lower-path as below or to the left of BS. Let's rename the states (α |pass |H + β |block |V ) and (α |pass |V +β |block |H ) as |տ and |ւ respectively. We can rewrite the exiting state, Eq. 4, as (λ |տ |upper-path + µ |ւ |lower-path ). Feeding this state into BS, which applies a π/2-rotation to the path qubit, gives, which means we can obtain the desired state λ |տ + µ |ւ with 50% probability upon measuring the path qubit. This measurement is carried out at D 0 , FIG. 4(b), without destroying the photon when ideal [19,20]. If the photon is not detected there we know it is in the other path travelling towards the left, in the correct state. So starting with the most general input states we got, (α |pass + β |block ) ⊗ (λ |H + µ |V ) → λ(α |pass |H + β |block |V )+ µ(α |pass |V + β |block |H ).
Rewriting using the equivalent binary states we get, We have thus built a fully counterfactual quantum CNOT gate using our dual CQZE setup. We now show how counterfactual transportation follows. Using two CNOT gates, the circuit of FIG. 5(a) swaps the input states (α |0 + β |1 ) and |0 , effectively transferring α |0 + β |1 from one side (top) to the other (bottom) [21]. Can we use our counterfactual CNOT gate in this circuit-with Bob's quantum object as the control qubit-to counterfactually transport an unknown state from Bob to Alice? The problem with the circuit is that the control qubits of the two CNOT gates are on opposite sides. But there is a way around it. By means of four Hadamard gates, the circuit of FIG. 5(b) interchanges the control and target qubits of a CNOT gate [22]. Applying this to the circuit of FIG. 5(a) we get the circuit of FIG. 5(c) which forms the basis of our protocol.
What about the case of Alice's exiting photon ending up on path D 0 ? From Eq. 5, the combined state of Bob's object and Alice's photon in this path is, λ(α |pass |H + β |block |V ) − µ(α |pass |V + β |block |H ), which in binary is, λ(α |0 |0 + β |1 |1 ) − µ(α |0 |1 + β |1 |0 ). This is equivalent to the output of a CNOT gate but with Alice applying a Z-gate to her input qubit. Incorporating this in the circuit of FIG. 5(c) we get the circuit of FIG .  5(d). It turns out that an X-gate is needed at the end for the state α |0 +β |1 to be transferred from one side (top) to the other (bottom). We finally arrive at our protocol for counterfactually transporting an unknown qubit.
Protocol for counterfactual quantum transportation-Alice starts by sending a H photon into the dual CQZE setup (FIG. 4) with Bob's quantum object, his qubit to be counterfactually transported, placed in a superposition of blocking and not blocking the channel: α |pass + β |block . Alice then applies a Hadamard transformation to (the polarisation of) her exiting photon, as does Bob to his qubit. Alice sends her photon back into the dual CQZE setup. If her exiting photon is not found in path D 0 , she knows it is in the other path travelling towards the left. She applies a Hadamard transformation to (the polarisation of) her photon, as does Bob to his qubit. The photon is now in the state α |H + β |V . If Alice's exiting photon is found instead in path D 0 , she first applies a Hadamard transformation to (the polarisation of) her photon, as does Bob to his qubit. She finally applies an X-transformation to her qubit. The photon is now in the state α |H + β |V . Bob's qubit has been counterfactually transported to her. His original qubit ends up in the state |0 or |1 randomly; in other words destroyed.
We have so far not said anything about how Bob may practically implement his qubit. Tremendous recent advances mean that there are several candidate technologies. Perhaps most promising for our purpose here are trapped-ion techniques [23,24], whereby a carefully shielded and controlled ion can be placed in a quantum superposition of two spatially separated states-one of which in our case blocks the channel. Trapped ions offer relatively long decay times, needed for a large-numberof-cycles implementation of the protocol. Moreover, the Hadamard transformation, key to this protocol, can be directly applied. What this means is that all elements of our protocol are implementable using current technology. The protocol could well be useful for transferring atomic qubits to photonic ones inside a future quantum computer-with the key advantage that no previously shared entanglement is required. (If, for practicality, counterfactuality is relaxed, only the inner cycles of the counterfactual CNOT gate would be required, quadratically reducing the number of cycles.) On the flip side, the counterfactual CNOT gate at the heart of this protocol can itself be used to entangle atomic and photonic qubits from scratch.
We have proposed a protocol for the counterfactual, disembodied transport of an unknown qubit-much like in quantum teleportation except that Alice and Bob do not require previously-shared entanglement nor a classical channel. No physical particles travel between them either. Here, Bob's qubit is gradually "beamed up" to Alice. In the ideal asymptotic limit, efficiency approaches unity as the probability amplitude of the photon being in the channel approaches zero. This brings into sharp focus both the promise and mystery of quantum information.  is turned off allowing the photon in, but is then turned on for M (N ) outer(inner) cycles before it is turned off again, allowing the photon out. The combined action of switchable polarisation rotators SP R and polarising beamsplitters P BS achieves the function of beamsplitters BS in the Mach-Zehnder version of FIG. 1(b). M R stands for mirror, and OD for optical delay. Again, complete counterfactuality is ensured as any photon going into the channel would either be lost due to measurement by Bob's object QOB or else end up at detector D3: the chained quantum Zeno effect. For large enough M and N , the probability amplitude of the photon being in the channel is virtually zero.