Quantum communication through devices with indefinite input-output direction

Let us start by reviewing the notion of indefinite input-output direction introduced in Ref. (36). This notion applies to bidirectional quantum processes, that is, quantum processes for which roles of the input and output ports are exchangeable. For example, the transmission of a single photon through a half-wave plate is a bidirectional process, because the exchange of the entrance point of the photon with its exit point gives rise to a valid single-photon evolution.

The set of bidirectional quantum processes on a given quantum system was characterised in Ref. (36). There, it was shown that a process is bidirectional if and only if it is described by a bistochastic channel (54; 55), that is, a completely positive linear map $\mathcal{C}$ with a Kraus representation $\mathcal{C}(\cdot)=\sum_{i=1}^{r}C_{i}\cdot C_{i}^{\dagger}$ where the Kraus operators $\{C_{i}\}_{i=1}^{r}$ satisfy the normalisation condition $\sum_{i=1}^{r}C_{i}^{\dagger}C_{i}=\sum_{i=1}^{r}C_{i}C_{i}^{\dagger}=I$, $I$ being the identity matrix on the system’s Hilbert space.

A given bidirectional device is associated to a forward channel $\mathcal{C}$ (describing the transformation of the system’s density matrix in a given direction, conventionally regarded as the “forward” one) and to a backward channel $\Theta(\mathcal{C})$ (describing the transformation of the system’s density matrix in the opposite direction). The map $\Theta$ transforming the forward channel into the backward channel is called an input-output inversion. Ref. (36) showed that the possible input-output inversions are of the form $\Theta(\mathcal{C})(\cdot)=\sum_{i}\theta(C_{i})\cdot\theta(C_{i}^{\dagger})$, where the matrices $\theta(C_{i})$ are either unitarily equivalent to the transpose, namely

where $U$ is a fixed unitary matrix and $C_{i}^{T}$ is the transpose of $C_{i}$ in a fixed basis, or unitarily equivalent to the adjoint, namely

where $C_{i}^{\dagger}$ is the adjoint (conjugate transpose) of $C_{i}$. A fundamental difference between the transpose and the adjoint is that the transpose can in principle be applied locally to part of a bipartite quantum channel, whereas the adjoint cannot, because it would generally give rise to maps that are not completely positive (36). Technically, the difference is that the transpose is an admissible transformation of quantum channels (an admissible quantum supermap, in the language of (56)), while the adjoint is not. In the following, we will focus our attention to bidirectional devices for which the input-output inversion is unitarily equivalent to the transpose. Example are photonic devices, such as half-wave plates and other polarisation rotators, or Stern-Gerlach devices.

Another example of input-output inversion is time reversal (57; 58). An extended discussion of this example is provided in (36), and here we only sketch the main ideas for the reader’s convenience. The standard approach to time reversal, dating back to Winger (57), is to define an anti-unitary operation $T$ acting on quantum states. The operator $T$ is generally taken to represent the change of description due to the inversion of time coordinate $t\mapsto-t$. Now, the time reversal of quantum states implicitly defines a time reversal of unitary evolutions: if a unitary evolution $U_{t_{0},t_{1}}$ transforms the state $|\psi\rangle$ at time $t_{1}$ into the state $U_{t_{1},t_{2}}|\psi\rangle$ at time $t_{2}>t_{1}$, then the time-reversed evolution should transform the state $TU_{t_{1},t_{2}}|\psi\rangle$ at time $-t_{2}$ into the state $T|\psi\rangle$ at time $-t_{1}$. Since this condition should hold for every state $|\psi\rangle$, setting $|\varphi\rangle:=TU_{t_{1},t_{2}}|\psi\rangle$ we obtain that the time-reversed evolution, denoted by $\theta(U_{t_{1},t_{2}})$, must transform the state $|\varphi\rangle$ into the state $TU_{t_{1},t_{2}}^{\dagger}T^{-1}|\varphi\rangle$. Hence, the time-reversed evolution is $\theta(U_{t_{1},t_{2}})=TU_{t_{1},t_{2}}^{\dagger}T^{-1}$. This expression is consistent with Eq. (1): by decomposing anti-unitary operator as $T=UK$, where $U$ is a unitary operator and $K$ is the complex conjugation, we obtain the expression $\theta(U_{t_{1},t_{2}})=U\,U_{t_{1},t_{2}}^{T}\,U^{\dagger}$, meaning that time reversal of the unitary evolution $U_{t_{1},t_{2}}$ is unitarily equivalent to its transpose $U_{t_{1},t_{2}}^{T}$. The above argument can be extended from unitary channels to general bistochastic channels by using the fact that bistochastic channels are linear combinations of unitary channels (see (36) for more details). In summary, the canonical time reversal in quantum theory is associated to an input-output inversion that is unitarily equivalent to the transpose. Furthermore, the combination of time reversal with other symmetries, such as charge conjugation $C$ and parity inversion $P$, gives rise to other examples of input-output inversions, such as those corresponding to the symmetries $CT$, $PT$, and $CPT$. More generally, an input-output inversion involves the exchange of two ports of a quantum device, without necessarily associating the two ports to two specific moments of time $t_{1}$ and $t_{2}$.

Traditionally, the roles of the input and output ports of quantum devices have been taken to be fixed. In general, however, the user is free to choose which port serves as the input and which one serves as the output, and the choice can even be controlled by the state of a quantum system. The ability to control the input-output direction of the information flow within a given device is described by a higher order transformation, called the quantum time flip (36). The quantum time flip takes as input a bistochastic quantum channel $\mathcal{C}$, and produces as output another bistochastic quantum channel $\mathcal{F}(\mathcal{C})$, acting on the original quantum system (hereafter called the target) and on a control qubit, which determines the input-output direction. The channel $\mathcal{F}(\mathcal{C})$ can be specified via its Kraus representation, given by $\mathcal{F}(\mathcal{C})(\cdot)=\sum_{i}F_{i}\cdot F_{i}^{\dagger}$ with

If the control system is set to $|0\rangle$, then the action of the channel $\mathcal{F}(\mathcal{C})$ on the target system is given by the forward channel $\mathcal{C}$ (Fig. 1(a)); if instead the control system is set to $|1\rangle$, then the action on the target is given by the backward channel $\mathcal{C}^{T}$ (Fig. 1(b)). Indefiniteness of the input-output direction arises when the control system is set to a coherent superposition of $|0\rangle$ and $|1\rangle$, such as e.g. the maximally coherent state $|+\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$. In this case, the channel $\mathcal{F}(\mathcal{C})$ can be regarded as a coherent superposition of the forward and backward channels $\mathcal{C}$ and $\mathcal{C}^{T}$, in the sense of Ref. (4; 59; 60; 5; 5; 61; 62; 63; 64; 65; 66; 9; 6; 7; 8; 10; 52; 22; 26) (Fig. 1(c)).

We now develop a model of quantum communication using bidirectional devices in an indefinite input-output direction. The model is illustrated in Fig. 2. A sender (Alice) wants to transmit a quantum state $\rho$ to a receiver (Bob). To this purpose, Alice uses a bidirectional device that acts as channel $\mathcal{C}$ when traversed in a given direction and as channel $\mathcal{C}^{T}$ when traversed in the opposite direction. A third party, Charlie, controls the direction in which Alice’s particle traverses the device, thereby realising the quantum time flip $\mathcal{F}$ operation described in the previous subsection. At the beginning of the protocol, Charlie initialises the control qubit in a quantum state $\omega$. Then, the target and control, initially in the product state $\rho\otimes\omega$, undergo the joint evolution described by the quantum channel $\mathcal{F}(\mathcal{C})$, thereby evolving into the state $[\mathcal{F}(\mathcal{C})](\rho\otimes\omega)$. At this point, Charlie performs a measurement $\mathcal{M}$ on the control system and communicates the outcome to Bob via a classical communication channel. Finally, Bob tries to retrieve Alice’s input state $\rho$, using the classical information provided by Charlie.

The above communication model can be applied to different communication tasks, including the communication of classical data (in which case the input state $\rho$ is chosen from a set of states $\{\rho_{x}\}_{x\in\mathsf{X}}$ used to encoded a message $x$ from some finite alphabet $\mathsf{X}$) and the communication of quantum data (in which case the input state $\rho$ is an arbitrary density matrix with support contained in a suitable subspace). In turn, the communication of quantum data can be used for a number of applications, including for example quantum key distribution (1; 2) or quantum secure direct communication (67; 68; 69)

Let us now analyze explicitly the steps of our communication protocol. At the beginning, the target system is in the state $\rho$ and the control system is in the state $\omega$. Then, the two states are sent through the channel $\mathcal{F}(\mathcal{C})$. The effective evolution from Alice’s input state $\rho$ to the joint output state of the target and control is given by a channel $\mathcal{F}_{\omega}(\rho)$, defined as

where $(C_{i})_{i=1}^{r}$ is a Kraus representation of the channel $\mathcal{C}$, $\overline{C}_{i}:=\left(C_{i}^{T}\right)^{\dagger}$ is the complex conjugate of the matrix $C_{i}$, and $Z:=|0\rangle\langle 0|-|1\rangle\langle 1|$ is the Pauli $z$ matrix. Here, the second equality follows by substituting the expressions $|0\rangle\langle 0|=(I+Z)/2$ and $|1\rangle\langle 1|=(I-Z)/2$ into Eq. (3). The channel $\mathcal{F}_{\omega}(\mathcal{C})$ describes the use of a bidirectional device in a combination of the forward and backward directions determined by the quantum state $\omega$. In the following, we will call $\mathcal{F}_{\omega}(\mathcal{C})$ the flipped channel.

The next step in the protocol is Charlie’s measurement on the control qubit. The measurement can be described by a quantum-to-classical channel $\mathcal{M}$, of the form

where $(P_{j})_{j=1}^{N}$ is a positive operator-valued measure (POVM) and $\{|j\rangle\}_{j=1}^{N}$ are orthogonal states of a classical system, on which the measurement outcome is recorded.

After Charlie’s measurement, the target system and the classical system end up in the quantum-classical state

where $\mathcal{I}_{\rm target}$ is the identity map acting on the target system. Finally, Charlie communicates to Bob the outcome of his measurement via a classical communication channel. Hence, the classical system at the output of channel $\mathcal{M}$ is eventually in Bob’s laboratory.

Overall, the flow of information from Alice to Bob is described by the effective channel

The goal of this paper is the study of the classical and quantum capacities of the effective channel $\mathcal{C}_{\rm eff}$, optimised over all possible state preparations $\omega$ and over all possible measurements $\mathcal{M}$.

To better understand our communication model, it is useful to consider its similarities and differences with other models considered in the literature. First, it is important to observe that our model is different from two-way communication models in which information travels simultaneously from Alice to Bob and from Bob to Alice (70; 71; 72), or from related models of multiple access quantum communication channels (73; 74). Unlike these models, our model does not exhibit any indefiniteness in the roles of sender and receiver, but only in the direction in which the sender’s messages traverse a given quantum device.

Second, it is important to clarify the role of the third party, Charlie. In our model, one can think of Charlie as a communication provider, who assists Alice and Bob, without interfering with Alice’s choice of input states. Accordingly, Charlie initialises the control system in fixed quantum state $\omega$, independent of the quantum states sent by Alice. This setting is broadly assumed in other communication models based on superposition of channels (52; 7; 8; 13).

It is also worth noting that Charlie’s assistance is limited to the transmission of classical information. This constraint is similar to the constraint adopted in the study of quantum communication with the assistance of classical information from the environment (75; 76), with the only difference that in our case we do not assume assistance from the whole environment of the channel between Alice and Bob, but only from a two-dimensional system that controls the direction of the information flow within a given device.

On a more formal level, one can regard our communication model as part of a resource theory (77). More specifically, our model is an instance of a resource theory of quantum communication (52), in which communication devices are assembled by the communication provider to build an effective communication channel connecting the sender to the receiver. In this framework, the operations performed by the communication provider are described by quantum supermaps (35; 56; 78; 79), that is, higher order operations that take in input the quantum channels associated to the initial devices, and produce in output a new quantum channel connecting the sender to the receiver.

The supermaps implementable by the communication provider are called free, in the sense that they are regarded as achievable in the resource theory under consideration. Often, free operations are motivated by practical constraints arising in real-world applications. Note, however, that in general this needs not be the case: the resource-theoretic framework is useful not only as a way to analyze practical constraints, but also as a theoretical lens for understanding the power of certain sets of operations, conventionally referred to as “free” even though they are not necessarily easy to implement.

Different resource theories of communication correspond to different choices of free supermaps. For example, standard quantum Shannon theory can be formulated as a resource theory of communication where the free supermaps combine the original devices in a well-defined configuration, typically in parallel and without the addition of any other channel on the side (52). Various extension of standard quantum Shannon theory can be obtained by enlarging the set of free operations, e.g. by including the ability to connect quantum channels in a coherent superposition of orders (11; 12; 13; 14; 15; 16; 20) or to coherently control which quantum channel is used to transmit information (7; 8). Communication advantages over standard quantum Shannon theory can then be understood as a result of the enlargement of the set of free supermaps, corresponding to more general ways to assemble the initial devices into an effective channel connecting the sender to the receiver.

The communication model considered in this paper corresponds to a new way to enlarge the set of free supermaps, including the possibility of using bidirectional devices in a coherent superposition of two alternative directions. As we will see later in the paper, this enlargement provides some interesting advantages over the standard model of quantum Shannon theory. On the other hand, the model still satisfies a constraint put forward in Ref. (52): free operations should not allow Alice and Bob to communicate independently of the initial devices used by Charlie to set up the communication link between them. The proof that our communication model satisfies this constraint is provided in Appendix A.

A qubit depolarising channel with depolarisation probability $q$ is a linear map $\mathcal{N}_{q}:M_{2}(\mathbb{C})\to M_{2}(\mathbb{C})$ of the form

where $X,Y,Z$ are the three Pauli matrices. Like all Pauli channels, the depolarising channel is bistochastic and invariant under transposition.

Now, suppose that Charlie uses a depolarising channel to set up a communication link between Alice and Bob, as in the model of Fig. 2. Charlie’s optimal strategy, provided by Proposition 2, results into the effective channel (10), which in this case reads

with

We now compare the classical capacities of the channels $\mathcal{N}_{q}$ and $\mathcal{F}_{|+\rangle\langle+|}(\mathcal{N}_{q})$. The classical capacity of channel $\mathcal{N}_{q}$ is well-known (86), and is given by

where $\log$ denotes the logarithm in base 2.

The classical capacity of the channel $\mathcal{F}_{|+\rangle\langle+|}(\mathcal{N}_{q})$ is derived in Appendix C, where we also address the case of general target systems of dimension $d\geq 2$. There, we prove that the classical capacity of the channel $\mathcal{F}_{|+\rangle\langle+|}(\mathcal{N}_{q})$ is a convex combination of the capacities of the channels $\mathcal{C}_{+}$ and $\mathcal{C}_{-}$, and takes the value

This equation shows that the classical capacity $C\big{(}\mathcal{F}_{|+\rangle\langle+|}(\mathcal{N}_{q})\big{)}$ is strictly larger than the classical capacity of the original depolarising channel $\mathcal{N}_{q}$ for every positive value of the depolarisation probability. A comparison between the capacities of the two channels is illustrated in Fig. 3. In the most extreme case, $q=1$, input-output indefiniteness in the channel $\mathcal{F}_{|+\rangle\langle+|}(\mathcal{N}_{1})$ permits communication with a capacity of $0.3113$ bits per channel use, even if the original channel $\mathcal{N}_{1}$ has zero classical capacity.

The $0.3113$ capacity achieved with input-output indefiniteness can be compared with the maximum capacity achievable by other communication protocols using coherent superpositions of configurations. For example, when two completely depolarising channels are used in a superposition of two alternative orders, the maximum classical capacity is 0.049 bits (43; 19). If instead two completely depolarising channels are placed on two alternative paths, and a quantum particle is sent through these paths in a quantum superposition, then the maximum classical capacity is $0.16$ bits (87; 10). Remarkably, comparison between these setups shows that a single channel in a superposition of two input-output directions can offer larger benefits than two channels in a superposition of two orders, or in a superposition of two paths.

We now consider the transmission of quantum data through depolarising channels in a superposition of input-output directions. One challenge here is that no close-form expression for the quantum capacity of the depolarising channel has been found up to date. In order to establish an advantage of input-output indefiniteness, we will use an upper bound on the quantum capacity, derived by Smith and Smolin in Ref. (88). There, they showed that the quantum capacity $Q(\mathcal{N}_{q})$ is upper bounded as

where $p=3q/4$, $\gamma(p)=4\sqrt{1-p}(1-\sqrt{1-p})$, $H(p)=-p\log p-(1-p)\log(1-p)$ and $\operatorname{co}[f_{1}(p),\cdots,f_{n}(p)]$ denotes the maximal convex function that is less than or equal to all $f_{i}(p)$, $i=1,\cdots,n$.

We now compare this upper bound with a lower bound on the quantum capacity of the effective channel $\mathcal{F}_{|+\rangle\langle+|}(\mathcal{N}_{q})$. By Eq. (16), the quantum capacity $Q(\mathcal{F}_{|+\rangle\langle+|}(\mathcal{N}_{q}))$ is lower bounded by the coherent information $I_{\rm c}(\mathcal{F}_{|+\rangle\langle+|}(\mathcal{N}_{q}))$, which in turn can be lower bounded by setting $\rho=I/2$ in Eq. (14). The resulting lower bound is

The derivation of this equation is provided in Appendix D, where we also extend the bound to general quantum systems of dimension $d\geq 2$.

Comparing Eqs. (25) and (26), we obtain that the strict inequality $Q\big{(}\mathcal{F}_{|+\rangle\langle+|}(\mathcal{N}_{q})\big{)}>Q(\mathcal{N}_{q})$ is valid for every value of $q$ in the interval $(0,0.495)$. Our lower bound on the quantum capacity is plotted in Fig. 4, along with the Smith-Smolin upper bound on the quantum capacity of the depolarising channel.

When the depolarising probability reaches the value $q=1/3$, the Smith-Smolin bound implies that the quantum capacity of the depolarising channel $\mathcal{N}_{q}$ is zero. In contrast, our lower bound shows that the quantum capacity of the flipped depolarising channel $\mathcal{F}_{|+\rangle\langle+|}(\mathcal{N}_{q})$ is at least 0.2063 qubits at $q=1/3$, and remains larger than zero until $q$ is approximately $0.5$.

Note that a non-zero quantum capacity also implies a non-zero private capacity $P\big{(}\mathcal{F}_{|+\rangle\langle+|}(\mathcal{N}_{q})\big{)}$, due to the bound (19). Hence, the flipped depolarising channel $\mathcal{F}_{|+\rangle\langle+|}(\mathcal{N}_{q})$ can be used to securely transmit private messages from Alice to Bob. Notably, the definition of private capacity guarantees that the transmission is secure even if the eavesdropper has access to the whole environment of the channel $\mathcal{F}_{|+\rangle\langle+|}(\mathcal{N}_{q})$, which includes in particular the case in which the eavesdropper has access to Charlie’s measurement outcome.

In summary, input-output indefiniteness offers the possibility to achieve a reliable transmission of quantum states starting from depolarising channels that have zero quantum capacity. Remarkably, this feature cannot be achieved by using two depolarising channels in a superposition of alternative orders: as shown in Ref. (19), the quantum capacity obtained by placing two depolarising channels in a superposition of orders is non-zero only if the quantum capacity of the original channels is non-zero. In contrast, the superposition of two alternative input-output directions permits quantum communication even with a single depolarising channel with zero quantum capacity.

The increase of quantum capacity from zero to nonzero enables the use of quantum key distribution (QKD) protocols, such as BB84 (1) and E91 (2), or the use of quantum secure direct communication (QSDC) protocols (67; 68; 69) to transmit private messages from the sender to the receiver (see also (89) for a recent experimental demonstration and (90; 91; 92) for recent theoretical developments on this topic). In principle, these protocols can be implemented by encoding the appropriate quantum states in the multipartite states that achieve the quantum capacity $Q\big{(}\mathcal{F}_{|+\rangle\langle+|}(\mathcal{N}_{q})\big{)}$. In practice, however, this approach is still challenging because the capacity-achieving encoding is not known in general, and may involve large multipartite states that are hard to implement. Fortunately, a simpler approach is also possible: indeed, the superposition of input-output directions also enables a noiseless heralded transmission of quantum data. As one can see from Eq. (21), the evolution of the target system is unitary whenever the control system is found in the state $|-\rangle$. This means that, when Bob receives the outcome -, he can retrieve the state of Alice’s qubit without any error. Such heralded transmission allows Alice and Bob to implement QKD and QSDC protocols using the quantum states of the original protocols, which are much simpler to realise compared to their encoded version.

We now show a scenario where input-output indefiniteness enables a perfect, deterministic transmission of quantum information. Consider a qubit dephasing channel of the form

In normal conditions, this channel $\mathcal{M}_{p}$ can be used to transmit a classical bit, by encoding it into an eigenstate of the Pauli-$Y$ gate. However, transmitting a quantum bit per channel use of $\mathcal{M}_{p}$ is impossible except in the extreme cases $p=0$ or $p=1$, in which the channel $\mathcal{M}_{p}$ is noiseless. Indeed, the quantum capacity of the dephasing channel $\mathcal{M}_{q}$ is (3)

and is strictly smaller than 1 for every $q\in(0,1)$. In stark contrast, we now show that using the channel $\mathcal{M}_{q}$ in a coherent superposition of the forward and backward direction yields a quantum capacity of 1 qubit per channel use, for every possible value of $q$.

Suppose that the depolarising channel $\mathcal{M}_{q}$ is used in the scenario of Fig. 2, with the control qubit initialised in the maximally coherent state $\omega$. The resulting channel $\mathcal{F}_{|+\rangle\langle+|}(\mathcal{M}_{p})$ can be easily computed from Eq. (9), which yields

Using this channel, Alice can transmit a qubit to Bob deterministically and without error. To retrieve Alice’s input state $\rho$, Bob needs only to measure the control system in the Fourier basis $\{|+\rangle,|-\rangle\}$: if the measurement outcome is “$+$”, then Bob obtained $\rho$ directly; if the outcome is “$-$”, then Bob can apply Pauli-$Y$ gate to the state to recover $\rho$.

Summarising, the flipped channel $\mathcal{F}_{|+\rangle\langle+|}(\mathcal{M}_{p})$ offers an advantage over the original dephasing channel $\mathcal{M}_{p}$ for every value of the dephasing probability except $p=0$ and $p=1$. The most extreme advantage arises for $p=1/2$, when the original dephasing channel has zero quantum capacity.

The possibility of perfect quantum communication through zero-capacity channels was also observed for quantum communication with indefinite causal order (13), where it was later shown to enable a task called random-receiver quantum communication (20). The same task can be achieved also in the present scenario, using a single dephasing channel in a coherent superposition of two alternative input-output directions instead of two channels in a coherent superposition of two alternative orders.