Abstract
What is the ultimate performance for discriminating two arbitrary quantum channels acting on a finitedimensional Hilbert space? Here we address this basic question by deriving a general and fundamental lower bound. More precisely, we investigate the symmetric discrimination of two arbitrary qudit channels by means of the most general protocols based on adaptive (feedbackassisted) quantum operations. In this general scenario, we first show how portbased teleportation can be used to simplify these adaptive protocols into a much simpler nonadaptive form, designing a new type of teleportation stretching. Then, we prove that the minimum error probability affecting the channel discrimination cannot beat a bound determined by the Choi matrices of the channels, establishing a general, yet computable formula for quantum hypothesis testing. As a consequence of this bound, we derive ultimate limits and nogo theorems for adaptive quantum illumination and singlephoton quantum optical resolution. Finally, we show how the methodology can also be applied to other tasks, such as quantum metrology, quantum communication and secret key generation.
Introduction
Quantum hypothesis testing^{1} is a central area in quantum information theory,^{2,3} with many studies for both discrete variable (DV)^{4} and continuous variable (CV) systems.^{5} A number of tools^{6,7,8,9,10} have been developed for its basic formulation, known as quantum state discrimination. In particular, since the seminal work of Helstrom in the 70 s,^{1} we know how to bound the error probability affecting the symmetric discrimination of two arbitrary quantum states. Remarkably, after about 40 years, a similar bound is still missing for the discrimination of two arbitrary quantum channels. There is a precise motivation for that: The main problem in quantum channel discrimination (QCD)^{11,12,13,14,15} is that the strategies involve an optimization over the input states and the output measurements, and this process may be adaptive in the most general case, so that feedback from the output can be used to update the input.
Not only the ultimate performance of adaptive QCD is still unknown due to the difficulty of handling feedbackassistance, but it is also known that adaptiveness needs to be considered in QCD. In fact, apart from the cases where two channels are classical,^{16} jointly programmable or teleportation covariant,^{17,18} feedback may greatly improve the discrimination. For instance, ref. ^{19} presented two channels which can be perfectly distinguished by using feedback in just two adaptive uses, while they cannot be perfectly discriminated by any number of uses of a block (nonadaptive) protocol, where the channels are probed in an identical and independent fashion. This suggests that the best discrimination performance is not directly related to the diamond distance,^{20} when computed over multiple copies of the quantum channels.
In this work we finally fill this fundamental gap by deriving a universal computable lower bound for the error probability affecting the discrimination of two arbitrary quantum channels. To derive this bound we adopt a technique which reduces an adaptive protocol over an arbitrary finitedimensional quantum channel into a block protocol over multiple copies of the channel’s Choi matrix. This is obtained by using portbased teleportation (PBT)^{21,22,23,24} for channel simulation and suitably generalizing the technique of teleportation stretching.^{25,26,27} This reduction is shown for adaptive protocols with any task (not just QCD). When applied to QCD, it allows us to bound the ultimate error probability by using the Choi matrices of the channels.
As a direct application, we bound the ultimate adaptive performance of quantum illumination^{28–35} and the ultimate adaptive resolution of any singlephoton diffractionlimited optical system, setting corresponding nogo theorems for these applications. We then apply our result to adaptive quantum metrology showing an ultimate bound which has an asymptotic Heisenberg scaling. As an example, we also study the adaptive discrimination of amplitude damping channels, which are the most difficult channels to be simulated. Finally, other implications are for the twoway assisted capacities of quantum and private communications.
Results
Adaptive protocols
Let us formulate the most general adaptive protocol over an arbitrary quantum channel \({\cal{E}}\) defined between Hilbert spaces of dimension d (more generally, this can be taken as the dimension of the input space). We first provide a general description and then we specify the protocol to the task of QCD. A general adaptive protocol involves an unconstrained number of quantum systems which may be subject to completely arbitrary quantum operations (QOs). More precisely, we may organize the quantum systems into an input register a and an output register b, which are prepared in an initial state ρ_{0} by applying a QO Λ_{0} to some fundamental state of a and b. Then, a system a_{1} is picked from the register a and sent through the channel \({\cal{E}}\). The corresponding output b_{1} is merged with the output register b_{1}b → b. This is followed by another QO Λ_{1} applied to a and b. Then, we send a second system a_{2} ∈ a through \({\cal{E}}\) with the output b_{2} being merged again b_{2}b → b and so on. After n uses, the registers will be in a state ρ_{n} which depends on \({\cal{E}}\) and the sequence of QOs {Λ_{0}, Λ_{1}, …, Λ_{n}} defining the adaptive protocol \({\cal{P}}_n\) with output state ρ_{n} (see Fig. 1).
In a protocol of quantum communication, the registers belong to remote users and, in absence of entanglementassistance, the QOs are local operations (LOs) assisted by twoway classical communication (CC), also known as adaptive LOCCs. The output is generated in such a way to approximate some target state.^{25} In a protocol of quantum channel estimation, the channel is labelled by a continuous parameter \({\cal{E}} = {\cal{E}}_\theta\) and the QOs include the use of entanglement across the registers. The output state will encode the unknown parameter ρ_{n} = ρ_{n}(θ), which is detected and the outcome processed into an optimal estimator.^{17} Here, in a protocol of binary and symmetric QCD, the channel is labelled by a binary digit, i.e., \({\cal{E}} = {\cal{E}}_u\) where u ∈ {0, 1} has equal priors. The QOs are generally entangled and they generate an output state encoding the information bit, i.e., ρ_{n} = ρ_{n}(u).
The output state ρ_{n}(u) of an adaptive discrimination protocol \({\cal{P}}_n\) is finally detected by an optimal positiveoperator valued measure (POVM). For binary discrimination, this is the Helstrom POVM, which leads to the conditional error probability
where D(ρ, σ) := ρ − σ/2 is the trace distance.^{4} The optimization over all discrimination protocols \({\cal{P}}_n\) defines the minimum error probability affecting the nuse adaptive discrimination of \({\cal{E}}_0\) and \({\cal{E}}_1\), i.e., we may write
This is generally less than the ncopy diamond distance between the two channels \({\cal{E}}_0^{ \otimes n}\) and \({\cal{E}}_1^{ \otimes n}\)
where^{2}
with \({\cal{I}}\) being an identity map acting on a reference system r. The upper bound in Eq. (3) is achieved by a nonadaptive protocol, where an (optimal) input state ρ_{ar} is prepared and its aparts transmitted through \({\cal{E}}_u^{ \otimes n}\). Note that Eq. (3) is very difficult to compute, which is why we usually compute larger but simpler singleletter upper bounds such as
where F is the fidelity between the Choi matrices, \(\rho _{{\cal{E}}_0}\) and \(\rho _{{\cal{E}}_1}\), of the two channels.
Our question is: Can we complete Eq. (3) with a corresponding lower bound? Up to today this has been only proven for jointly programmable channels, i.e., channels \({\cal{E}}_0\) and \({\cal{E}}_1\) admitting a simulation \({\cal{E}}_u(\rho ) = {\cal{S}}(\rho \otimes \pi _u)\) with a tracepreserving QO \({\cal{S}}\) and different program states π_{0} and π_{1}. In this case, we have \(p_n \ge [1  D(\pi _0^{ \otimes n},\pi _1^{ \otimes n})]/2\).^{17} In particular, this is true if the channels are jointly teleportation covariant, so that \({\cal{S}}\) becomes teleportation and the program state is a Choi matrix \(\rho _{{\cal{E}}_u}\). For these channels, ref. ^{17} found that Eq. (3) holds with an equality and we may write \({\cal{E}}_0^{ \otimes n}  {\cal{E}}_1^{ \otimes n}_\diamondsuit = \rho _{{\cal{E}}_0}^{ \otimes n}  \rho _{{\cal{E}}_1}^{ \otimes n}\). More precisely, the question to ask is therefore the following: Can we establish a universal lower bound for \(p_n({\cal{E}}_0 \ne {\cal{E}}_1)\) which is valid for arbitrary channels? As we show here, this is possible by resorting to a more general (multiprogram) simulation of the channels, i.e., of the type \({\cal{S}}(\rho \otimes \pi _u^{ \otimes M})\).
PBT and simulation of the identity
Let us describe the protocol of PBT with qudits of arbitrary dimension d ≥ 2. More technical details can be found in the original proposals.^{22,23} The parties exploit two ensembles of M ≥ 2 qudits, i.e., Alice has A := {A_{1}, …, A_{M}} and Bob has B := {B_{1}, …, B_{M}} representing the output “ports”. The generic ith pair (A_{i}, B_{i}) is prepared in a maximally entangled state, so that we have the global state
To teleport the state of a qudit C, Alice performs a joint measurement on C and her ensemble A. This is a POVM \(\{ {\Pi}_{C{\mathbf{A}}}^i\} _{i = 1}^M\) with M possible outcomes (see refs ^{22,23} for the details). In the standard protocol considered here, this POVM is a square root measurement (known to be optimal in the qubit case). Once Alice communicates the outcome i to Bob, he discards all the ports but the ith one, which contains the teleported state (see Fig. 2a).
The measurement outcomes are equiprobable and independent of the input, and the output state is invariant under permutation of the ports (this can be understood by the fact that the scheme is invariant under permutation of the Bell states and, therefore, of the ports). Averaging over the outcomes, we define the teleported state \(\rho _B^M = \Gamma _M(\rho _C)\), where Γ_{M} is the corresponding PBT channel. Explicitly, this channel takes the form
where \({\mathrm{Tr}}_{\bar B_i}\) denotes the trace over all ports B but B_{i}.
As shown in ref. ^{22}, the standard protocol gives a depolarizing channel^{4} whose probability ξ_{M} decreases to zero for increasing number of ports M. Therefore, in the limit of many ports \(M \gg 1\), the Mport PBT channel Γ_{M} tends to an identity channel \({\cal{I}}\), so that Bob’s output becomes a perfect replica of Alice’s input. Here we prove a stronger result in terms of channel uniform convergence.^{26,27} In fact, for any M, we show that the simulation error, expressed in terms of the diamond distance between Γ_{M} and \({\cal{I}}\), is onetoone with the entanglement fidelity of the PBT channel Γ_{M}. In turn, this result allows us to write a simple upper bound for this error. Moreover, we can fully characterize the simulation error with an exact analytical expression for qubits (see Methods for the proof, with further details given in Supplementary Section 1).
Lemma 1
In arbitrary (finite) dimension d, the diamond distance between the Mport PBT channel Γ_{M} and the identity channel \({\cal{I}}\) satisfies
where \(f_e(\Gamma _M): = \langle \Phi [{\cal{I}} \otimes \Gamma _M(\Phi \rangle \langle \Phi )]\Phi \rangle\) is the entanglement fidelity of Γ_{M}. This gives the upper bound
More precisely, we can write the exact result
where ξ_{M} is the depolarizing probability of the PBT channel Γ_{M}. For qubits (d = 2), the “PBT number” ξ_{M} has the closed analytical expression
where s_{min} = 1/2 for even M and 0 for odd M.
General channel simulation via PBT
Let us discuss how PBT can be used for channel simulation. This was first shown in ref. ^{21} where PBT was introduced as a possible design for a programmable quantum gate array.^{36} As depicted in Fig. 2b, suppose that Bob applies an arbitrary channel \({\cal{E}}\) to the teleported output, so that Alice’s input ρ_{C} is subject to the approximate channel
Note that the port selection commutes with \({\cal{E}}\), because the POVM acts on a different Hilbert space.^{21} Therefore, Bob can equivalently apply \({\cal{E}}\) to each port before Alice’s CC, i.e., apply \({\cal{E}}^{ \otimes M}\) to his B qudits before selecting the output port, as shown in Fig. 2c. This leads to the following simulation for the approximate channel
where \({\cal{T}}^M\) is a tracepreserving LOCC and \(\rho _{\cal{E}}\) is the channel’s Choi matrix (see Fig. 2d). By construction, the simulation LOCC \({\cal{T}}^M\) is universal, i.e., it does not depend on the channel \({\cal{E}}\). This means that, at fixed M, the channel \({\cal{E}}^M\) is fully determined by the program state \(\rho _{\cal{E}}\). One can bound the accuracy of the simulation. From Eq. (12) and the monotonicity of the diamond norm, we get
where δ_{M} is the simulation error in Eq. (9), with the dimension d being the one of the input Hilbert space. It is worth to remark that, while the simulation in Eq. (13) relies on a number of copies of the channel’s Choi matrix, it can be applied to an arbitrary quantum channel \({\cal{E}}\) without the condition of teleportation covariance.^{25}
PBT stretching of an adaptive protocol
Channel simulation is a preliminary tool for the following technique of teleportation stretching, where an arbitrary adaptive protocol is reduced into a simpler block version. There are two main steps. First of all, we need to replace each channel \({\cal{E}}\) with its Mport approximation \({\cal{E}}^M\) while controlling the propagation of the simulation error δ_{M} from the channel to the output state. This step is crucial also in simulations via standard teleportation^{18,26} (see also refs ^{37,38,39,40,41}). Second, we need to “stretch” the protocol^{25} by replacing the various instances of the approximate channel \({\cal{E}}^M\) with a collection of Choi matrices \(\rho _{\cal{E}}^{ \otimes M}\) and then suitably reorganizing all the remaining QOs. Here we describe the technique for a generic task, before specifying it to QCD.
Given an adaptive protocol \({\cal{P}}_n\) over a channel \({\cal{E}}\) with output ρ_{n}, consider the same protocol over the simulated channel \({\cal{E}}^M\), so that we get the different output \(\rho _n^M\). Using a “peeling” argument (see Methods), we bound the output error in terms of the channel simulation error
Once understood that the output state can be closely approximated, let us simplify the adaptive protocol over \({\cal{E}}^M\). Using the simulation in Eq. (13), we may replace each channel \({\cal{E}}^M\) with the resource state \(\rho _{\cal{E}}^{ \otimes M}\), iterate the process for all n uses, and collapse all the simulation LOCCs and QOs as shown in Fig. 3. As a result, we may write the multicopy Choi decomposition
for a tracepreserving QO \(\bar \Lambda\). Now, we can combine the two ingredients of Eqs. (15) and (16), into the following.
Lemma 2 (PBT stretching)
Consider an adaptive quantum protocol (with arbitrary task) over an arbitrary ddimensional quantum channel \({\cal{E}}\) (which may be unknown and parametrized). After n uses, the output ρ_{n} of the protocol can be decomposed as follows
where \(\bar \Lambda\) is a tracepreserving QO, \(\rho _{\cal{E}}\) is the Choi matrix of \({\cal{E}}\), and δ_{M} is the Mport simulation error in Eq. (9).
When we apply the lemma to protocols of quantum or private communication, where the QOs Λ_{i} are LOCCs, then we may write Eq. (17) with \(\bar \Lambda\) being a LOCC. In protocols of channel estimation or discrimination, where \({\cal{E}}\) is parametrized, we may write Eq. (17) with \(\rho _{\cal{E}}\) storing the parameter of the channel. In particular, for QCD we have \(\{ {\cal{E}}_u\} _{u = 0,1}\) and the output ρ_{n}(u) of the adaptive protocol \({\cal{P}}_n\) can be decomposed as follows
Ultimate bound for channel discrimination
We are now ready to show the lower bound for minimum error probability \(p_n({\cal{E}}_0 \ne {\cal{E}}_1)\) in Eq. (3). Consider an arbitrary protocol \({\cal{P}}_n\), for which we may write Eq. (1). Combining Lemma 2 with the triangle inequality leads to
where we also use the monotonicity of the trace distance under channels. Because \(\bar \Lambda\) is lost, the bound does no longer depend on the details of the protocol \({\cal{P}}_n\), which means that it applies to all adaptive protocols. Thus, using Eq. (19) in Eqs. (1) and (2), we get the following.
Theorem 3
Consider the adaptive discrimination of two channels \(\{ {\cal{E}}_u\} _{u = 0,1}\) in dimension d. After n probings, the minimum error probability satisfies the bound
where M may be chosen to maximize the right hand side.
Not only this is the first universal bound for adaptive QCD, but also its analytical form is rather surprising. In fact, its tighest value is given by an optimal (finite) number of ports M for the underlying protocol of PBT.
Let us bound the trace distance in Eq. (20) as
where F is the fidelity between the Choi matrices of the channels. This comes from the Fuchsvan de Graaf relations^{42} and the multiplicativity of the fidelity over tensor products. Other bounds that can be written are
from the subadditivity of the trace distance, and
from the Pinsker inequality,^{43,44} where \(S(\rho \sigma ) = \mathrm{Tr}[\rho (\log _{2}\rho  \log _{2}\sigma )]\) is the relative entropy.^{4}
If we exploit Eqs. (9) and (21) in Eq. (20), we may write the following simplified bound
In the previous formula there are terms with opposite monotonicity in M, so that the maximum value of the bound B is achieved at some intermediate value of M. Setting M = xd(d − 1)n for some x > 2, we get
One good choice is therefore M = 4d(d − 1)n, so that
In particular, consider two infinitesimallyclose channels, so that \(F \simeq 1  \epsilon\) where \(\epsilon \simeq 0\) is the infidelity. By expanding in \(\epsilon\) for any finite n, we may write
For instance, in the case of qubits this becomes \([\exp (  8n\sqrt \epsilon )]/4\), to be compared with the upper bound \([\exp (  2n\epsilon )]/2\) computed from Eq. (5). Discriminating between two close quantum channels is a problem in many physical scenarios. For instance, this is typical in quantum optical resolution^{45,46,47} (discussed below), quantum illumination^{28,29,30,31,32,33,34,35,48,49} (discussed below), ideal quantum reading,^{50,51,52,53,54} quantum metrology^{55,56,57,58,59} (discussed below), and also tests of quantum field theories in noninertial frames,^{60} e.g., for detecting effects such as the Unruh or the Hawking radiation.
Limits of singlephoton quantum optical resolution
Consider a microscopetype problem where we aim at locating a point in two possible positions, either s/2 or −s/2, where the separation s is very small. Assume we are limited to use probe states with at most one photon and an output finiteaperture optical system (this makes the optical process to be a qubittoqutrit channel, so that the input dimension is d = 2). Apart from this, we are allowed to use an arbitrary large quantum computer and arbitrary QOs to manipulate its registers. We may apply Eq. (27) with \(\epsilon \simeq \eta s^{2}/16\), where η is a diffractionrelated loss parameter. In this way, we find that the error probability affecting the discrimination of the two positions is approximately bounded by \(B \gtrsim {\frac{1}{4}}\exp (  2ns\sqrt \eta )\). This bound establishes a nogo for perfect quantum optical resolution. See Supplementary Section 2 for more mathematical details on this specific application.
Limits of adaptive quantum illumination
Consider the protocol of quantum illumination in the DV setting.^{28} Here the problem is to discriminate the presence or not of a target with low reflectivity η ≃ 0 in a thermal background which has \(b \ll 1\) mean thermal photons per optical mode. One assumes that d modes are used in each probing of the target and each of them contains at most one photon. This means that the Hilbert space is (d + 1)dimensional with basis \(\{ \left 0 \right\rangle ,\left 1 \right\rangle , \ldots ,\left d \right\rangle \}\), where \(\left i \right\rangle : = \left {0 \cdots 010 \cdots 0} \right\rangle\) has one photon in the ith mode. If the target is absent (u = 0), the receiver detects thermal noise; if the target is present (u = 1), the receiver measures a mixture of signal and thermal noise.
In the most general (adaptive) version of the protocol, the receiver belongs to a large quantum computer where the (d + 1)dimensional signal qudits are picked from an input register, sent to target, and their reflection stored in an output register, with adaptive QOs performed between each probing. After n probings, the state of the registers ρ_{n}(u) is optimally detected. Assuming the typical regime of quantum illumination,^{28} we find that the error probability affecting target detection is approximately bounded by \(B \gtrsim {\frac{1}{4}}\exp (  4nd\sqrt \eta )\). This bound establishes a nogo for exponential improvement in quantum illumination. Entanglement and adaptiveness can at most improve the error exponent with respect to separable probes, for which the error probability is \(\lesssim {\frac{1}{2}}\exp [  n\eta /(8d)]\). See also Supplementary Section 3.
Limits of adaptive quantum metrology
Consider the adaptive estimation of a continuous parameter θ encoded in a quantum channel \({\cal{E}}_\theta\). After n probings, we have a θdependent output state ρ_{n}(θ) generated by an adaptive quantum estimation protocol \({\cal{P}}_n\). This output state is then measured by a POVM \({\cal{M}}\) providing an optimal unbiased estimator \(\tilde \theta\) of parameter θ. The minimum error variance Var\((\tilde \theta ): = \langle (\tilde \theta  \theta )^2\rangle\) must satisfy the quantum CramerRao bound Var\((\tilde \theta ) \ge 1/\)\({\mathrm{QFI}}_\theta ({\cal{P}}_n)\), where \({\mathrm{QFI}}_\theta ({\cal{P}}_n)\) is the quantum Fisher information^{55} associated with \({\cal{P}}_n\). The ultimate precision of adaptive quantum metrology is given by the optimization over all protocols
This quantity can be simplified by PBT stretching. In fact, for any input state ρ_{C}, we may write the simulation \({\cal{E}}_\theta ^M(\rho _C) = {\cal{T}}^M(\rho _C \otimes \rho _{{\cal{E}}_\theta }^{ \otimes M})\), which is an immediate extension of Eq. (13). In this way, the output state can be decomposed following Lemma 2, i.e., we may write \(\rho _n(\theta )  \bar \Lambda (\rho _{{\cal{E}}_\theta }^{ \otimes nM}) \le n\delta _M\). Exploiting the latter inequality for large n, we find that the ultimate bound of adaptive quantum metrology takes the form
where \({\mathrm{QFI}}(\rho _{{\cal{E}}_\theta })\) is computed on the channel’s Choi matrix. In particular, we see that PBT allows us to write a simple bound in terms of the Choi matrix and implies a general nogo theorem for superHeisenberg scaling in quantum metrology. See Supplementary Section 4 for a detailed proof of Eq. (29).
Tightening the main formula
Let us note that the formula in Theorem 3 is expressed in terms of the universal error δ_{M} coming from the PBT simulation of the identity channel (Lemma 1). There are situations where the diamond distance \(\Delta _M: = {\cal{E}}  {\cal{E}}^M_\diamondsuit\) between a quantum channel \({\cal{E}}\) and its Mport simulation \({\cal{E}}^M\) is exactly computable. In these cases, we can certainly formulate a tighter version of Eq. (20) where δ_{M} is suitably replaced. In fact, from the peeling argument, we have \(\rho _n  \rho _n^M \le n\Delta _M\), so that a tighter version of Eq. (17) is simply \(\rho _n  \bar \Lambda (\rho _{\cal{E}}^{ \otimes nM}) \le n\Delta _M\). Then, for the two possible outputs ρ_{n}(0) and ρ_{n}(1) of an adaptive discrimination protocol over \({\cal{E}}_0\) and \({\cal{E}}_1\), we can replace Eq. (19) with
where \(\bar \Delta _M: = ({\cal{E}}_0  {\cal{E}}_0^M_\diamondsuit + {\cal{E}}_1  {\cal{E}}_1^M_\diamondsuit )/2\). It is now easy to check that Eq. (20) becomes the following
In the following section, we show that \(\bar \Delta _M\), and therefore the bound in Eq. (31), can be computed for the discrimination of amplitude damping channels.
Discrimination of amplitude damping channels
As an additional example of application of the bound, consider the discrimination between amplitude damping channels. These channels are not teleportation covariant, so that the results from ref. ^{17} do not apply and no bound is known on the error probability for their adaptive discrimination. Recall that an amplitude damping channel \({\cal{E}}_p\) transforms an input state ρ as follows
with Kraus operators
where \(\{ \left 0 \right\rangle ,\left 1 \right\rangle \}\) is the computational basis and p is the damping probability or rate.
Given two amplitude damping channels, \({\cal{E}}_{p_0}\) and \({\cal{E}}_{p_1}\), first assume a discrimination protocol where these channels are probed by n maximally entangled states and the outputs are optimally measured. The optimal error probability for this (nonadaptive) block protocol is given by \(p_n^{{\mathrm{block}}} = [1  D(\rho _{{\cal{E}}_{p_0}}^{ \otimes n},\rho _{{\cal{E}}_{p_1}}^{ \otimes n})]/2\) and satisfies
where \(F(p_0,p_1): = F(\rho _{{\cal{E}}_{p_0}},\rho _{{\cal{E}}_{p_1}})\) is the fidelity between the Choi matrices. In particular, we explicitly compute
It is clear that \(p_n^{{\mathrm{block}}}\) in Eq. (34) is an upper bound to ultimate (adaptive) error probability \(p_n({\cal{E}}_{p_0} \ne {\cal{E}}_{p_1})\) for the discrimination of the two channels.
To lowerbound the ultimate probability we employ Eq. (31). In fact, for the Mport simulation \({\cal{E}}_p^M\) of \({\cal{E}}_p\), we compute
where ξ_{M} are the PBT numbers defined in Eq. (11). For any two amplitude damping channels, \({\cal{E}}_{p_0}\) and \({\cal{E}}_{p_1}\), we can then compute \(\bar \Delta _M(p_0,p_1)\) and use Eq. (31) to bound \(p_n({\cal{E}}_{p_0} \ne {\cal{E}}_{p_1})\). More precisely, we can also exploit Eq. (21) and write the computable lower bound
In Fig. 4 we show an example of discrimination between two amplitude damping channels. In particular, we show how large is the gap between the upper bound \(p_n^{{\mathrm{block}}}\) of Eq. (34) and the lower bound in Eq. (37) suitably optimized over the number of ports M. It is an open question to find exactly \(p_n({\cal{E}}_{p_0} \ne {\cal{E}}_{p_1})\). At this stage, we do not know if this result may achieved by tightening the upper bound or the lower bound.
Discussion
In this work we have established a general and fundamental lower bound for the error probability affecting the adaptive discrimination of two arbitrary quantum channels acting on a finitedimensional Hilbert space. This bound is conveniently expressed in terms of the Choi matrices of the channels involved, so that it is very easy to compute. It also applies to many scenarios, including adaptive protocols for quantumenhance optical resolution and quantum illumination. In order to derive our result, we have employed portbased teleportation as a tool for channel simulation, and developed a methodology which simplifies adaptive protocols performed over an arbitrary finitedimensional channel. This technique can be applied to many other scenarios. For instance, in quantum metrology we are able to prove that adaptive protocols of quantum channel estimation are limited by a bound simply expressed in terms of the Choi matrix of the channel and following the Heisenberg scaling in the number of probings. Not only this shows that our bound is asymptotically tight but also draws an unexpected connection between portbased teleportation and quantum metrology. Further potential applications are in quantum and private communications, which are briefly discussed in our Supplementary Section 5.
Methods
Simulation error in diamond norm (proof of Lemma 1)
It is easy to check that the channel Γ_{M} associated with the qudit PBT protocol of ref. ^{21} is covariant under unitary transformations, i.e.,
for any input state ρ and unitary operator U. As discussed in ref. ^{61}, for a channel with such a symmetry, the diamond distance with the identity map is saturated by a maximally entangled state, i.e.,
where \(\Phi \rangle = d^{  1/2}\mathop {\sum}\limits_{k = 1}^d  k\rangle k\rangle\). Here we first show that
In fact, note that the map \(\Lambda _M = {\cal{I}} \otimes \Gamma _M\) is covariant under twirling unitaries of the form U ⊗ U^{*}, i.e.,
for any input state ρ and unitary operator U. This implies that the quantum state Λ_{M}(Φ〉〈Φ) is invariant under twirling unitaries, i.e.,
This is therefore an isotropic state of the form
where \({\Bbb I}\) is the twoqudit identity operator.
We may rewrite this state as follows
where ρ^{⊥} is state with support in the orthogonal complement of Φ, and F is the singlet fraction
Thanks to the decomposition in Eq. (44) and using basic properties of the trace norm,^{4} we may then write
where the last step exploits the fact that the singlet fraction F is the channel’s entanglement fidelity f_{e}(Γ_{M}). This completes the proof of Eq. (40).
Therefore, combining Eqs. (39) and (40), we obtain
which is Eq. (8) of the main text. Then, we know that the entanglement fidelity of Γ_{M} is bounded as^{21}
Therefore, using Eq. (48) in Eq. (47), we derive the following upper bound
which is Eq. (9) of the main text.
Let us now prove Eq. (10). It is known^{22} that implementing the standard PBT protocol over the resource state of Eq. (6) leads to a PBT channel Γ_{M}, which is a qudit depolarizing channel. Its isotropic Choi matrix \(\rho _{\Gamma _M}\), given in Eq. (43), can be written in the form
where ξ_{M} is the probability p of depolarizing, Φ〉^{0}〈Φ is the projector onto the initial maximally entangled state of two qudits (one system of which was sent through the channel), and Φ〉^{i}〈Φ are the projectors onto the other d^{2} − 1 maximally entangled states of two qudits (generalized Bell states). Since the Choi matrix of the identity channel is \(\rho _{\cal{I}} = \Phi \rangle ^0\langle \Phi \), it is easy to compute
From the previous equation, we derive
where we have used \({\mathrm{Tr}}_2\Phi \rangle ^i\langle \Phi  = d^{  1}\mathop {\sum}\limits_{j = 0}^{d  1}  j\rangle \langle j\) in the qudit computational basis {j〉} and we have summed over the d^{2} generalized Bell states. It is clear that Eq. (52) is a diagonal matrix with equal nonzero elements, i.e., it is a scalar. As a result, we can apply Proposition 1 of ref. ^{62} over the Hermitian operator \(\rho _{\cal{I}}  \rho _{\Gamma _M}\), and write
The final step of the proof is to compute the explicit expression of ξ_{M} for qubits, which is the formula given in Eq. (11). Because this derivation is technically involved, it is reported in Supplementary Section 1.
Propagation of the simulation error
For the sake of completeness, we provide the proof of the first inequality in Eq. (15) (this kind of proof already appeared in refs ^{25,26}). Consider the adaptive protocol described in the main text. For the nuse output state we may compactly write
where Λ’s are adaptive QOs and \({\cal{E}}\) is the channel applied to the transmitted signal system. Then, ρ_{0} is the preparation state of the registers, obtained by applying the first QO Λ_{0} to some fundamental state. Similarly, for the Mport simulation of the protocol, we may write
where \({\cal{E}}^M\) is in the place of \({\cal{E}}\).
Consider now two instances (n = 2) of the adaptive protocol. We may bound the trace distance between ρ_{2} and \(\rho _2^M\) using a “peeling” argument^{17,18,25,26,27}
In (1) we use the monotonicity of the trace distance under completely positive tracepreserving (CPTP) maps (i.e., quantum channels); in (2) we employ the triangle inequality; in (3) we use the monotonicity with respect to the the CPTP map \({\cal{E}} \circ \Lambda _1\) whereas in (4) we exploit the fact that the diamond norm is an upper bound for the trace norm computed on any input state. Generalizing the result of Eq. (56) to arbitrary n, we achieve the first inequality in Eq. (15). Note that the previous reasoning can also be applied to a classicallyparametrized unknown channel.
PBT simulation of amplitude damping channels
Here we show the result in Eq. (36) for \(\Delta _M(p) = {\cal{E}}_p  {\cal{E}}_p^M_\diamondsuit\), which is the error associated with the Mport simulation of an arbitrary amplitude damping channel \({\cal{E}}_p\). From ref. ^{22}, we know that the PBT channel Γ^{M} is a depolarizing channel. In the qubit computational basis \(\{ \left {i,j} \right\rangle \} _{i,j = 0,1}\), it has the following Choi matrix
where ξ_{M} are the PBT numbers of Eq. (11). Note that these take decreasing positive values, for instance
By applying the Kraus operators K_{0} and K_{1} of \({\cal{E}}_p\) locally to \(\rho _{\Gamma ^M}\) we obtain the Choi matrix of the Mport simulation \({\cal{E}}_p^M\), which is
where \(x: = \frac{1}{2}  \left( {1  p} \right)\frac{{\xi _M}}{4}\), \(y: = \sqrt {1  p} \left( {\frac{1}{2}  \frac{{\xi _M}}{2}} \right)\), \(z: = \left( {\frac{1}{2}  \frac{{\xi _M}}{4}} \right)\left( {1  p} \right)\), and \(w: = \left( {\frac{1}{2}  \frac{{\xi _M}}{4}} \right)p + \frac{{\xi _M}}{4}\). This has to be compared with the Choi matrix of \({\cal{E}}_p\), which is
Now, consider the Hermitian matrix \(J = \rho _{{\cal{E}}_p^M}  \rho _{{\cal{E}}_p}\). If the matrix \(\phi = {\mathrm{Tr}}_2\sqrt {J^\dagger J} = {\mathrm{Tr}}_2\sqrt {JJ^\dagger }\) is scalar (i.e., both of its eigenvalues are equal), then the trace distance between the Choi matrices J is equal to the diamond distance between the channels Δ_{M}(p) [^{62}, Proposition 1]. After simple algebra we indeed find
where \(a_ \pm = \sqrt {1  p} \sqrt {5 \pm 4\sqrt {1  p}  p}\). Because ϕ is scalar, the condition above is met and the expression of Δ_{M}(p) is twice the (degenerate) eigenvalue of ϕ, i.e.,
which simplifies to Eq. (36).
Data availability
The data sets generated in this study are available from the corresponding author upon reasonable request.
Code availability
The code used to generate data will be made available to the interested reader upon reasonable request.
Change history
02 August 2019
An amendment to this paper has been published and can be accessed via a link at the top of the paper.
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Acknowledgements
This work has been supported by the EPSRC via the ‘UK Quantum Communications Hub’ (EP/M013472/1) and by the European Commission via ‘Continuous Variable Quantum Communications’ (CiViQ, Project ID: 820466). The authors would like to thank Satoshi Ishizaka, Sam Braunstein, Seth Lloyd, Gaetana Spedalieri, and ZhiWei Wang for feedback.
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All authors contributed to prove the main theoretical results of the work. S.P. developed the basic idea and methodology, steered the research project, wrote the main manuscript and Supplementary Sections 4 and 5. R.L. wrote Supplementary Section 3. C.L wrote Supplementary Sections 2 and 4. J.P. contributed to refine Lemma 1 and to study the PBT simulation of amplitude damping channels, besides writing Supplementary Section 1.
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Pirandola, S., Laurenza, R., Lupo, C. et al. Fundamental limits to quantum channel discrimination. npj Quantum Inf 5, 50 (2019). https://doi.org/10.1038/s415340190162y
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