All-quantum teleportation

TL;DR

This paper introduces an all-quantum teleportation protocol that replaces classical channels with quantum channels, reducing resource consumption and increasing efficiency compared to the original protocol.

Contribution

It proposes novel all-quantum teleportation schemes using either a single quantum channel or two nonlocal channels, improving resource efficiency over the original protocol.

Findings

Single-channel AQT uses one EPR pair for multiple runs.

Two nonlocal channels with superdense coding are more economical.

AQT reduces resource consumption compared to OP.

Abstract

I propose to replace the dual classical and nonlocal channels used for teleporting unknown quantum states in the original protocol (OP) [Bennett, C. H., et al. Phys. Rev. Lett. 70 1895 (1993)] by either (i) one single quantum channel or (ii) two nonlocal channels in order to turn the OP into an all-quantum teleportation (AQT) protocol. Ideally, N runs of single channel AQT can be achieved with a single Einstein-Podolsky-Rosen (EPR) pair, in contrast with the OP, which consumes N EPR pairs. In the two nonlocal channels proposal, Alice uses the superdense coding technique to send Bob her result, which makes AQT more economical than OP.

Equations

$${|{\psi }^{\pm }⟩}_{AB}=\sqrt{\frac{1}{2}}\left({|01⟩}_{AB}\pm {|10⟩}_{AB}\right),$$

$${|{\varphi }^{\pm }⟩}_{AB}=\sqrt{\frac{1}{2}}\left({|00⟩}_{AB}\pm {|11⟩}_{AB}\right).$$

$${|\psi ⟩}_{ABC}$$

$$+\frac{1}{2}{|{\psi }^{-}⟩}_{AC}\left(\alpha {|0⟩}_B+\beta {|1⟩}_B\right)$$

$$+\frac{1}{2}{|{\varphi }^{+}⟩}_{AC}\left(-\alpha {|1⟩}_B+\beta {|0⟩}_B\right)$$

$$-\frac{1}{2}{|{\varphi }^{-}⟩}_{AC}\left(\alpha {|1⟩}_B+\beta {|0⟩}_B\right),$$

$${|\psi ⟩}_{ABC}$$

$$-\frac{1}{2}{|{\psi }^{-}⟩}_{AC}\left(\alpha {|0⟩}_B-\beta {|1⟩}_B\right)$$

$$+\frac{1}{2}{|{\varphi }^{+}⟩}_{AC}\left(-\alpha {|1⟩}_B+\beta {|0⟩}_B\right)$$

$$-\frac{1}{2}{|{\varphi }^{-}⟩}_{AC}\left(\alpha {|1⟩}_B+\beta {|0⟩}_B\right),$$

$${|\psi ⟩}_{ABC}$$

$$-\frac{1}{2}{|{\psi }^{-}⟩}_{AC}\left(\alpha {|1⟩}_B+\beta {|0⟩}_B\right)$$

$$+\frac{1}{2}{|{\varphi }^{+}⟩}_{AC}\left(\alpha {|0⟩}_B-\beta {|1⟩}_B\right)$$

$$+\frac{1}{2}{|{\varphi }^{-}⟩}_{AC}\left(\alpha {|0⟩}_B+\beta {|1⟩}_B\right),$$

$${|\psi ⟩}_{ABC}$$

$$-\frac{1}{2}{|{\psi }^{-}⟩}_{AC}\left(\alpha {|1⟩}_B-\beta {|0⟩}_B\right)$$

$$+\frac{1}{2}{|{\varphi }^{+}⟩}_{AC}\left(\alpha {|0⟩}_B+\beta {|1⟩}_B\right)$$

$$+\frac{1}{2}{|{\varphi }^{-}⟩}_{AC}\left(\alpha {|0⟩}_B-\beta {|1⟩}_B\right),$$

$${|\psi ⟩}_{ABC}$$

$$+\frac{1}{2}{|{\psi }^{-}⟩}_{AC}\left(\alpha {|0⟩}_B+\beta {|1⟩}_B\right){|1⟩}_D{|1⟩}_E$$

$$+\frac{1}{2}{|{\varphi }^{+}⟩}_{AC}\left(-\alpha {|1⟩}_B+\beta {|0⟩}_B\right){|0⟩}_D{|0⟩}_E$$

$$-\frac{1}{2}{|{\varphi }^{-}⟩}_{AC}\left(\alpha {|1⟩}_B+\beta {|0⟩}_B\right){|0⟩}_D{|1⟩}_E.$$

Full text

*Introduction. *Quantum teleportation was suggested by Bennett et al. Bennet93 and is recognized nowadays as one of the cornerstones of quantum computation and quantum communication Nielsen00 ; Gottesman99 ; Lo98 . Two important features characterizing this phenomenon is the transfer of information between noninteracting systems by means of a dual quantum and classical channels. Indeed, in the very tittle of the pioneering work Ref. Bennet93 , which henceforth will be called the *original protocol *(OP), the classical resource is highlighted, and in the abstract the authors says "Alice makes a joint measurement on her EPR particle and the unknown quantum system, and sends Bob the classical result of this measurement." Bennet93 . This classical result consists of two classical bits out of the combination $(00,01,10,11)$. **

Soon after the work by Bennett et al., several other theoretical schemes for teleportation differing from the OP were proposed. As for example, in Ref.Vaidman94 it was proposed a “cross measurement” method to achieve a two-way teleportation, and in Ref.moussa97 the author shows how to implement teleportation with identity interchange of quantum states. Yet, in Ref. deAlmeida98 , the authors used Greenberger–Horne–Zeilinger (GHZ) states as the nonlocal channel instead of the Einstein–Podolsky–Rosen (EPR) states used in the OP, and in Ref.Karlson98 , also using a GHZ channel, the authors show how to accomplish teleportation controlled by a third party. In Ref.Agrawal06 the authors show how to achieve perfect teleportation and superdense code using W states as the nonlocal channel. In Refs.Cardoso08 ; Cardoso09 the concept of controlled partial teleportation (CPT) was introduced**. **Recently, a protocol to send information in a secure way inspired by the OP was proposed, which used a classical correlated channel Costa16 .

Here I propose a modification in the OP to turn it into an all-quantum teleportation (AQT) protocol by either (i) abolishing the classical channel and using one single quantum channel, or (ii) replacing the classical channel by a quantum one and using two nonlocal channels. In (i), a joint quantum nondemolition (QND) measurement Guo01 is used, such that the particle whose unknown state is in Alice’s hands has a twofold usefulness: it is used both for Alice sending Bob her joint QND measurement result and also for reestablishing the nonlocal channel when Bob sends back Alice one particle to re-initiate the AQT protocol. To accomplish this, Alice and Bob must be able to perform nonlocal quantum nondemolition measurement and to store and send the information encoded in their particles. Next, I show how this can be theoretically accomplished.

Nonlocal channels for all-quantum teleportation. As in the OP, Alice and Bob share a nonlocal channel, which can be any of the four states composing the complete and orthonormal Bell basis for particles $A$ and $B$ $\left\{\left|\psi^{\pm}\right\rangle_{AB},\left|\phi^{\pm}\right\rangle_{AB}\right\}$, where

[TABLE]

[TABLE]

When Alice receives from Charles the state to be teleported to Bob, which can be cast in the form $\left|\chi\right\rangle_{C}=\alpha\left|0\right\rangle_{C}+\beta\left|1\right\rangle_{C}$, with $\left|\alpha\right|^{2}+\left|\beta\right|^{2}=1$, she performs a nonlocal or joint measurement on particles $A$ and $C$. Before Alice’s measurement, there will be four possibilities to the complete state of the three particles, depending on which of the four nonlocal channels, Eqs.(1) and (2), Alice and Bob are sharing. After knowing that Alice’s measurement was one of these four $\left|\psi^{\pm}\right\rangle$ or $\left|\phi^{\pm}\right\rangle$ states, Bob applies one of the four local unitary operation to complete teleportation: $\mathbf{1},\sigma_{z},\sigma_{x},\sigma_{x}\sigma_{z}$, depending on the nonlocal channel used. Explicitly, if the nonlocal channel is $\left|\psi^{-}\right\rangle_{AB}$, the complete state of the three particles $A$, $B$ and $C$ will be $\left|\psi\right\rangle_{ABC}=\left|\psi^{-}\right\rangle_{AB}\left|\chi\right\rangle_{C}$, which can be written in the following way

[TABLE]

where now the subscripts in Bell states is for $A$ and $C$. After the Alice’s joint measurement, particles state $A$ and $C$ gets entangled in one of the four Bell states, and the local unitary operation Bob must apply to convert his particle state into a perfect copy of $\left|\chi\right\rangle_{C}$ is either $\mathbf{1},$ or $\sigma_{z}$, or $\sigma_{x}$, or $\sigma_{x}\sigma_{z}$ if Alice’s result was either $\left|\psi^{-}\right\rangle_{AC}$ or $\left|\psi^{+}\right\rangle_{AC}$ or $\left|\phi^{-}\right\rangle_{AC}$, or $\left|\phi^{+}\right\rangle_{AC}$ , respectively.

If the nonlocal channel is $\left|\psi^{+}\right\rangle$, the complete state can be written as

[TABLE]

and the local unitary operation that Bob must apply is either $\mathbf{1},$ or $\sigma_{z}$, or $\sigma_{x}$, or $\sigma_{x}\sigma_{z}$ if Alice’s result was either $\left|\psi^{+}\right\rangle_{AC}$ or $\left|\psi^{-}\right\rangle_{AC}$ or $\left|\phi^{-}\right\rangle_{AC}$, or $\left|\phi^{+}\right\rangle_{AC}$ , respectively.

If If the nonlocal channel is $\left|\phi^{-}\right\rangle$, the complete $\left|\psi\right\rangle_{ABC}$ can be written as

[TABLE]

and the local unitary operation that Bob must apply is either $\mathbf{1},$ or $\sigma_{z}$, or $\sigma_{x}$, or $\sigma_{x}\sigma_{z}$ if Alice’s result was either $\left|\phi^{-}\right\rangle_{AC}$ or $\left|\phi^{+}\right\rangle_{AC}$ or $\left|\psi^{-}\right\rangle_{AC}$, or $\left|\psi^{+}\right\rangle_{AC}$ , respectively.

Finally, if If the nonlocal channel is $\left|\phi^{+}\right\rangle$ , $\left|\psi\right\rangle_{ABC}$ can be written as

[TABLE]

and the local unitary operation Bob must apply is either $\mathbf{1},$ or $\sigma_{z}$, or $\sigma_{x}$, or $\sigma_{x}\sigma_{z}$ if Alice’s result was either $\left|\phi^{+}\right\rangle_{AC}$ or $\left|\phi^{-}\right\rangle_{AC}$ or $\left|\psi^{+}\right\rangle_{AC}$, or $\left|\psi^{-}\right\rangle_{AC}$ , respectively. In the original protocol, right after Alice did the measurement, she sends two classical bits out of $\left\{00,01,10,11\right\}$ representing one of the four possible results $\left|\psi^{\pm}\right\rangle$ and $\left|\phi^{\pm}\right\rangle$ above. In AQT, Bob becomes aware of Alice’s measurement when he receives not the classical bits, but the quantum ones, either $\left|\psi^{\pm}\right\rangle$ or $\left|\phi^{\pm}\right\rangle$, sent by Alice. This can be done in two different ways: (i) in the single channel AQT, after Alice’s performs a joint QND measurement, which can be accomplished using two ancillary states of additional particles $D$ and $E$ interacting with particles $A$ and $C$ according to the circuit in Fig.1.

After the QND joint measurement, Alice sends Bob the two particles $A$ and $C$ whose entangled state contains the message on the unitary operation Bob must apply to complete AQT. (ii) In the two nonlocal channels, Alice uses one channel to teleport to Bob the state of particle $C$ and the second channel two send quantum information containing the message on the unitary operation Bob must apply to complete AQT. Next, I discuss this two ways for AQT in more details.

Single channel all-quantum teleportation. Let the nonlocal channel initially shared by Alice and Bob be $\left|\psi^{-}\right\rangle_{AB}$, such that the complete state before Alice’s measurement is given by Eq.(3).

The sequence of unitary operations described in Fig. 1 are the following: The ancilla particle state $D$ flips controlled by the particle states $B$ and $A$, respectively. After that, a Hadarmard gate is applied on particles $A$ and $B$. Finally, the ancilla particle state $E$ flips controlled by particle states $B$ and $A$, respectively. Starting from Eq.(3), it is straightforward to calculate the complete state after interacting with ancillas $E$ and $D$ as given by the circuit of Fig.1:

[TABLE]

From Eq.(7) we can clearly see that Alice’s measurement on ancilla states $D$ and $E$ allows her to discern in which of the four states of Bell basis particles $A$ and $C$ collapse. For example, assume the Alice’s result was $\left|1\right\rangle_{D}\left|0\right\rangle_{E}$. Then, the collapsed state is the tensorial product $\left|\psi\right\rangle_{ABC}\rightarrow\left|\psi^{+}\right\rangle_{AC}\left(\alpha\left|0\right\rangle_{B}-\beta\left|1\right\rangle_{B}\right)\left|1\right\rangle_{D}\left|0\right\rangle_{E}$, and Bob is left with state $\alpha\left|0\right\rangle_{B}-\beta\left|1\right\rangle_{B}$. Alice thus sends Bob particles $A$ and $C$ in state $\left|\psi^{+}\right\rangle_{AC}$ altogether, and Bob applies the same operations needed to make the QND measurement described in Fig.1, with the label $B$ replaced by $C$ and keeping the same labels $D$ and $E$ to his ancilla states. Bob’s result is, of course, $\left|\psi^{+}\right\rangle_{AC}\left|1\right\rangle_{D}\left|0\right\rangle_{E}$. Thus, after measuring the ancilla in states $\left|1\right\rangle_{D}\left|0\right\rangle_{E}$ , Bob applies the local unitary operation $\sigma_{z}$ on the particle $B$ to convert its state in a perfect replica of particle $C$ and completing one run of single-channel AQT.

To complete subsequent runs of AQT, Bob and Alice have two alternatives which, of course, needs to be previously combined between them: (i) According to Fig.2, Bob applies a local unitary operation $U_{A}$ to reestablish the original nonlocal state, which, in the case exemplified above, can be $\sigma_{z}^{(A)}\left|\psi^{+}\right\rangle_{AC}\rightarrow\left|\psi^{-}\right\rangle_{AC}$, and sends back particle $A$ to Alice, keeping particle $C$ with him relabeling $C\rightarrow B$. Once reestablished the nonlocal channel $\left|\psi^{+}\right\rangle_{AC}\rightarrow\left|\psi^{-}\right\rangle_{AB}$, Alice then repeats the nonlocal QND measurement on particles $A$ and $C$ and the protocol continues unchanged. (ii) Bob sends Alice exactly the state he has measured, in this example $\left|\psi^{+}\right\rangle_{AC}$. Alice, knowing that Bob has just send back particle $A$, makes a nonlocal measurement having in mind the nonlocal channel as given by Eq.(4). Alice performs the nonlocal QND measurement and sends her result to Bob who, after performing a nonlocal QND measurement, applies one of the following local operation to complete the second run of AQT: $\mathbf{1},$ $\sigma_{z}$, $\sigma_{x}$, or $\sigma_{x}\sigma_{z}$ if Alice’s result was either $\left|\psi^{+}\right\rangle_{AC}$ or $\left|\psi^{-}\right\rangle_{AC}$ or $\left|\phi^{-}\right\rangle_{AC}$, or $\left|\phi^{+}\right\rangle_{AC}$, respectively - see Eq.(4). A third run will follow a similar procedure, with the subsequent nonlocal channel being one of those four states Fig.(3)-(6). Fig.2 displays the circuit for the one single channel AQT protocol using approach (i).

We note that the second approach will be useful, and even necessary, if any of the unitary local operations is unavailable. This is the case, for example, if we are working with Fock states, for which no Hadamard gate exists. Although conceptually simple, AQT may pose some challenge in its accomplishment, mainly due to the maintenance of such a fragile resource as entanglement. Notwithstanding, it would be interesting to known how many runs of single channel AQT can be achieved based on existing technology.

*Two nonlocal channels all-quantum teleportation. *Let us now show that AQT can be achieved replacing the classical channel by a nonlocal channel and thus using two nonlocal channels instead of a dual classical and nonlocal channels as in Ref.Bennet93 . Indeed, one of the two channels can be used by Alice to communicate, e.g., via superdense coding Nielsen00 ; dense1 ; dense2 , her result to Bob. This, besides turning the OP all-quantum, would be more economical since Alice can use just one qubit to send Bob two classical bits. In the two nonlocal channels AQT, after Alice’s joint measurement on the first channel, she only needs to apply local operations on the state of the particle composing the second quantum channel to convert one state of the Bell basis into another, as for example, $\sigma_{z}\left|\psi^{\pm}\right\rangle=\left|\psi^{\mp}\right\rangle$; $\sigma_{x}\left|\phi^{\pm}\right\rangle=\left|\psi^{\pm}\right\rangle$; $\sigma_{x}\sigma_{z}\left|\phi^{\pm}\right\rangle=\left|\psi^{\mp}\right\rangle$, thus encoding the message $\left\{00,01,10,11\right\}$ into one of the four Bell states $\left\{\left|\psi^{\pm}\right\rangle,\left|\phi^{\pm}\right\rangle\right\}$, and then sending her qubit to Bob who, in turn, must read which state he is receiving. Bob can perform the measurement at least in two ways. (i) He can perform a nonlocal QND measurement, see Fig.1, on the particles composing the second quantum channel to read in which of the four Bell state these particles are in, or (ii) he can perform a CNOT unitary operation followed by a Hadamard gate on the entangled qubit, leading the entangled state to one of these four states: $\left|00\right\rangle_{AC}$, $\left|01\right\rangle_{AC}$, $\left|10\right\rangle_{AC}$, or $\left|11\right\rangle_{AC}$, depending on which Bell state it was. Therefore, by performing a projective measurement on the two-particle system composing the second channel, Bob decodes the Alice’s message encoded into the state of particles $A$ and $C$ and applies one of the four local unitary operation $\mathbf{1},$ or $\sigma_{z}$, or $\sigma_{x}$, or $\sigma_{x}\sigma_{z}$ on the state of particle $B$ to complete the two channels AQT process.

It is interesting to note that superdense coding, a technique used to send two bits of classical information using only one qubit, is the inverse of quantum teleportation, which sends one qubit with two classical bits. The present proposal thus combines two quantum protocols, which one being the inverse of the other, to transmit information between two remote places using an all-quantum teleportation protocol.

Security. In the single channel AQT, Alice sends Bob an entangled state of particles $A$ and $C$. If an eavesdropper (Eve) has access to a joint QND measurement device, she can intercept particles $A$ and $C$ to access the Bell states sent by Alice to Bob. Since the state of particle C was destroyed by the Alice’s joint QND measurement, she cannot extract useful information from the Bell state only. This is similar to what occurs within the OP, where Eve can intercept the classical message sent by Alice but cannot decode the message. Regarding AQT with superdense coding, if Eve intercepts Alice’s qubit during its flight to Bob, what is obtained is just a part of an entangled state, with no useful information as well. Therefore, AQT is as secure as the OP.

Conclusion. I have proposed an all-quantum protocol for teleporting unknown quantum states, which can be done in two different ways. In the first way, the classical channel is abolished altogether and Alice and Bob share one single quantum channel. It is important to have in mind that by one single quantum channel is meant just one pair of entangled particles with no extra-particles. Alice performs nonlocal QND measurement on the particle to be teleported and on that composing the nonlocal state shared with Bob. Due to the nonlocal Alice’s measurement, these two particles is now entangled, and since Alice makes a QND measurement, she is able to perform local unitary operation to convert one state of the Bell basis into another, and then to encode a two-bit message to Bob by sending him the two-particle entangled states. Finally, bob performs another nonlocal QND measurement to know what local operation he must perform to complete the teleportation process and sends back Alice one particle of the pair to retrieve the nonlocal channel. The single channel AQT protocol is thus prompt to be reinitiated. Ideally, just one single Einstein-Podolsky-Rosen (EPR) pair is consumed to achieve N runs of single channel AQT, in contrast with the OP that consumes N EPR pairs besides needing a classical channel. In the second way, the original protocol for teleporting unknown quantum states via dual classical and nonlocal channels is modified by replacing the classical channel by a quantum one, such that Alice and Bob share two quantum channels. Then, after Alice’s nonlocal QND measurement, she makes a local unitary operation on the particle of the second nonlocal channel shared with Bob and sends him two bits of information through a single qubit using the superdense coding protocol, which is the most economical way to transmit quantum information. By performing another joint measurement, Bob decodes the information to know which of four local operation he must apply to complete the two nonlocal channels AQT.

It is worthwhile to stress that, differently from classical information, which can be stored, copied and transmitted easily, single channel AQT, although theoretically simple, may pose some experimental challenges related to the storage and transmission of the quantum information by both Alice and Bob. However, AQT, in its two versions presented here, is by far more economical then the conventional teleportation protocol.