Quantum channel discrimination is a fundamental information-processing task in quantum information theory [1, 2, 3, 4, 5, 6, 7, 8]. There are at least two ways of thinking about it: one in terms of quantifying error between an ideal channel and an experimental approximation of it [1, 2] and another in terms of symmetric hypothesis testing [5, 6]. In both scenarios, the diamond distance between channels  arises as the fundamental metric quantifying the distinguishability of two quantum channels. These interpretations of diamond distance are the main reason that it is employed as the primary theoretical quantifier of channel distance in applications such as fault-tolerant quantum computation , quantum complexity theory , and quantum Shannon theory .
To expand upon the first way of thinking about channel discrimination from [1, 2], suppose that the ideal channel to be implemented is (a completely positive, trace-preserving map taking operators for a system to operators for a system). Suppose further that the experimental approximation is . To interface with these channels and obtain classical data, the most general way for doing so is to prepare a state of a reference system and the channel input system , feed system into the unknown channel, and then perform a quantum measurement on the channel output system and the reference system . To be a legitimate quantum measurement, the set of operators should satisfy and for all . The result of this procedure (preparation, channel evolution, and measurement) is a classical outcome that occurs with probability if the channel is applied, while the outcome occurs with probability if the channel is applied. The error or difference between these probabilities is naturally quantified by the absolute deviation
We can then quantify the maximum possible error between the channels and by optimizing (1) with respect to all preparations and measurements:
where it is implicit that the channels and above have input system and output system . Mathematically, this has the effect of removing the dependence on the preparation and measurement such that the error is a function solely of the two channels and . It is a fundamental and well known result in quantum information theory [1, 2] that the error in (2) is equal to the normalized diamond distance:
where the diamond distance is defined as
In (4), the optimization is with respect to all pure bipartite states with system isomorphic to the channel input system , and the trace norm of an operator is given by , where . This interpretation of normalized diamond distance as error between channels is the main reason that it is employed in applications like fault-tolerant quantum computation .
The other setting in which diamond distance arises is in the context of symmetric hypothesis testing of quantum channels [5, 6]. We can also refer to this as “incoherent quantum channel discrimination,” a name that shall become clear later. This can be thought of as a guessing game between a prover and a verifier [5, 12], and here we describe the game with fully quantum-mechanical notation. Let us call it the “channel guessing game.” The game begins with the verifier flipping a fair coin described by the state . Meanwhile the prover prepares a pure state and sends system to the verifier. The verifier then performs the conditional channel
on systems and , so that the resulting global state is . The verifier sends the channel output system to the prover, whose task it is to guess which channel was applied by the verifier. The prover can act on the systems in his possession, which are and . The prover performs a quantum-to-classical channel , where for and , and sends the system back to the verifier. Finally, the verifier performs the measurement
and declares “success” if the first outcome of the measurement occurs. If success occurs, we interpret this outcome as meaning that the prover is able to distinguish the channels. Running through the calculation, the probability that the prover wins (verifier declares “success”) is equal to
Figure 1 depicts the channel guessing game (in order to understand it fully, it is necessary to read the next section).
The prover can optimize over all input states and measurements , and a well known result in quantum information  is that the optimal success probability of incoherent channel discrimination is given by
thus endowing the normalized diamond distance with another operational meaning as the relative bias away from random guessing in a channel guessing game of the above form. That is, a random guessing strategy leads to a success probability of and can be employed when the channels are the same or indistinguishable. However, when the channels have some distinguishability so that , then the success probability changes as a linear function of the normalized diamond distance and reaches its peak value when the channels are orthogonal to each other (perfectly distinguishable). This guessing game is a basic channel discrimination task in quantum information theory and has found application in the setting of quantum illumination [7, 8, 13].
A useful fact about the diamond distance is that it can be computed by means of a semi-definite program :
where is the Choi operator of the channel , with and , for orthonormal bases and . Thus, calculating the diamond distance is efficient in the dimensions of the input and output .
Ii Coherent Quantum Channel Discrimination
The main aim of the present paper is to introduce and analyze a fully quantum or coherent version of the channel guessing game presented above. Let us call it coherent quantum channel discrimination, in contrast to the incoherent channel discrimination task presented above. The primary modification that I make to it is to replace all classical steps of the verifier with their coherent counterparts, much like what was done previously in  to produce coherent versions of basic protocols in quantum information such as superdense coding and teleportation (see also  in this context). The resulting protocol is related to the fully quantum reading protocol from . A recent series of works have considered coherent control of quantum channels [18, 19, 20, 21], but coherent quantum channel discrimination is different from the protocols considered in these prior works.
I now briefly summarize coherent channel discrimination. The main idea is to replace the initial state of the verifier with , the conditional channel of the verifier with a controlled unitary, and the final measurement with a projection onto the Bell state
(here and throughout the rest of the paper, we refer to both state vectors and density operators as states, as is conventional in the quantum information literature). Later, we shall see that it is sensible to include an uncomputing step to uncompute the controlled channel at the end before performing the Bell projection.
The modifications of the guessing game presented here could potentially have applications in quantum computation, where gates are often promoted to controlled gates and used in superposition. In particular, some works have recently investigated the question of compiling quantum circuits on quantum computers [22, 23]. The coherent games presented here could be used as benchmarks to assess how well an approximate implementation of a circuit could be used instead of the ideal one, even when it is employed in superposition (i.e., in controlled form). We do not investigate this particular application here but instead leave it for future work.
Before presenting details of the coherent version of the channel guessing game, let us recall some fundamental facts about quantum channels (see, e.g., ). First, every quantum channel has a Kraus representation as , where is a set of Kraus operators satisfying . Another fundamental fact is that every quantum channel has an isometric extension. That is, to every quantum channel , there exists an isometry (satisfying ) such that for all input states . Equivalently, there exists an environment system and a unitary such that
Thus, we can set . Any two isometric extensions of the original channel are related by an isometry acting on the environment system .
The coherent version of the channel guessing game proceeds as follows. The verifier prepares the state and the prover prepares . The prover sends the system to the verifier. The verifier then adjoins the state and performs the controlled unitary
where is a unitary that extends the channel as in (10). Let be the corresponding isometric extension. The resulting state is then
The verifier transmits system to the prover, who then adjoins an environment system in the state
, a qubit systemin the state , and performs a unitary . The resulting state is then
The prover sends systems and back to the verifier, who uncomputes the controlled unitary in (11) by performing
The state at this point is then
where we omit system labels for brevity. The verifier finally performs the measurement
on systems , where , and declares “success” (or “prover wins!”) if the first outcome occurs. The probability of success is equal to
where the second expression follows from the fact that . Figure 2 depicts coherent quantum channel discrimination.
We can already observe that the success probability in (17) is independent of the particular isometric extension of the original channel for both and . It is thus solely a function of the channels and , as well as the particular strategy of the prover (as indicated by the notation in (17)). This follows because the unitary that the prover performs does not act on the environment system . Thus, letting be some other isometric extension of , it follows that by employing the previously stated fact that there exists an isometry (satisfying ) such that .
Just as in the guessing game presented in Section I, the prover can optimize the success probability in (17) with respect to all possible strategies . Let us denote the resulting success probability as follows:
The main goal of this paper is to understand this quantity in more detail and relate it to the success probability in other forms of channel discrimination.
As a very simple example to demonstrate the task of coherent channel discrimination, suppose that the first channel is the identity channel and the second is the deterministic bit-flip channel, i.e., , where is the Pauli flip operator. These channels are orthogonal to each other, and a simple strategy for distinguishing them perfectly in incoherent channel discrimination is to input the state and perform a computational basis measurement . If the first channel is applied, the output state is , while if the second channel is applied, then the output state is , and these two states are perfectly distinguishable.
For coherent channel discrimination, the same input state is optimal. To see this, consider that the initial state of the verifier and prover’s systems is (there is no reference system needed in this case). The controlled unitary in (11), implemented by the verifier, is then a controlled-NOT gate , and there is no environment system because the channels are unitary channels. The resulting state after the controlled unitary is . The prover can then perform a controlled-NOT gate from system to system , and the resulting state is a GHZ state: . The verifier then performs the inverse of the controlled-NOT gate (itself a controlled-NOT), and the resulting state is , so that the Bell projection at the end succeeds with probability one; we thus arrive at the sensible conclusion that these channels are perfectly distinguishable in coherent channel discrimination.
This key example illustrates the necessity and sensibility of the uncomputing step in coherent channel discrimination. Without it, in this example, the final Bell projection would succeed only with probability , leading to the unreasonable conclusion that these channels would not be perfectly distinguishable in coherent channel discrimination. Uncomputing is commonly employed in reversible and quantum computation as a “clean-up” step [25, 26, 27], and it serves the same purpose here.
All proofs of the ensuing results appear in appendices.
Iv-a Alternate expression
For quantum channels and , the success probability in (18) is equal to
The operators act on the Hilbert space for and take them to the Hilbert space for . The dimension of need not be any larger than .
In the above, the -norm of an operator is defined as and the adjoint of a quantum channel is defined to be the unique linear map satisfying for all operators and .
It is interesting to contrast the expression in (19) with the following expression for the success probability of incoherent channel discrimination:
where and . This expression comes about from that in (8) by employing the definition of the -norm and the adjoint of a quantum channel. Even by examining these expressions, we can see how (19) is a coherent version of (20). The expression in (19) is like the square of a probability amplitude (the latter being the expression inside the -norm), and it involves operators for which the sum of their squares is equal to the identity instead of their sums being equal to the identity.
Iv-B Bounds on success probability
The following bounds hold for the success probability in (18):
The upper bound is saturated if and only if the channels are orthogonal (i.e., there exists a pure state such that ). The lower bound is saturated if the channels are identical (i.e., indistinguishable).
The upper bound is obvious since is a probability, and the necessary and sufficient condition for saturation follows by employing the bounds
discussed later. The lower bound follows by setting for in (19), which corresponds to “not even trying to distinguish,” and the sufficient saturation condition follows by direct evaluation.
Iv-C Non-increase under a superchannel
A key property of the success probability in (18) is that it does not increase under the action of a quantum superchannel. This is a basic property expected of any channel distinguishability measure, and it was recently shown that the diamond distance (and thus the success probability in (8)) satisfies this property .
To expand upon this statement, recall from  that a quantum superchannel is a physical mapping of a quantum channel to a quantum channel, and it should be this way even when acting on one share of an arbitrary bipartite channel. In more detail, a superchannel is a linear map that completely preserves the properties of complete positivity and trace preservation. Then for an arbitrary input bipartite channel , the output is a bipartite channel from systems to systems . The fundamental theorem of superchannels is that any superchannel has a physical realization in terms of a pre-processing channel and a post-processing channel :
With the fundamental theorem of superchannels in hand, we can arrive at an operational proof that the success probability in (18) does not increase under the action of a superchannel. To see this, consider that a particular strategy of the prover for coherently distinguishing the channels and is to prepare a state and act with an isometric extension of the pre-processing on system . Then the verifier performs the controlled unitary in (11), and the prover performs an isometric extension of the post-processing , the unitary , and the adjoint of (the last being implemented by a unitary and a projection). The verifier finally performs the inverse of (11) and the projective measurement in (16). Since the success probability does not increase under the action of the adjoint of and since this is a particular strategy for coherent discrimination of and , while being a general strategy for coherent discrimination of and , we conclude that the success probability does not increase under the action of a superchannel:
Let and be quantum channels, and let be a quantum superchannel. Then the success probability of coherent channel discrimination in (18) does not increase under the action of :
Iv-D Computable by semi-definite programming
The success probability in (18) can be computed by means of the following semi-definite program:
where and are density operators and
with a set of Kraus operators for the channel for . This follows from the observation that coherent channel discrimination is a quantum interactive proof, and the acceptance probability of any quantum interactive proof can be calculated by means of a semi-definite program [30, 10]. In the above semi-definite program, the density operator can be understood as the reduction of the initial state of the prover on system , and the density operator is the reduced state from (12) on systems . The projection corresponds to the concatenation of the inverse unitary in (14) followed by the projection in (16) onto the accepting subspace. The equality constraint in (25) corresponds to the fact that the state of the verifier on systems and should be the same before and after the prover acts with the unitary .
The dual semi-definite program is given by
where , the operator is Hermitian, and . This follows by the standard Lagrange multiplier method.
V Incoherent Channel Discrimination with Uncomputing
Another variation of channel discrimination is to follow the same protocol for coherent channel discrimination but have the initial state be the maximally mixed state and the final measurement be as in (6), with the first outcome indicating success. So this is the main difference with coherent channel discrimination, and the main difference with incoherent channel discrimination is that we include a step for uncomputing. Let denote the success probability for this case. We then have the following bounds, implying (22):
where the channel arguments are left implicit for brevity.
This paper has introduced a coherent version of quantum channel discrimination and investigated various aspects of the success probability. I have proven an alternate expression for it in Proposition 1, some bounds in Proposition 2 and Eq. (29), that it does not increase under the action of a quantum superchannel, and that it can be calculated by means of a semi-definite program. An intriguing open question is to determine if is a metric on quantum channels. Consider that is, as is clear from (8).
I thank Stefan Bäuml, Siddhartha Das, Felix Leditzky, and Xin Wang for discussions related to the topic of this paper. I also acknowledge support from the National Science Foundation under grant no. 1907615.
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Appendix A Proof of Proposition 1
Let us begin with the expression in (17) for the unoptimized success probability:
where we used the fact that can be expressed in terms of the channel adjoint as . Let us write
where the operators satisfy
in order for to be unitary. Then we find that
which leads to
Now optimizing over all input states and unitaries , while setting
we find that
Also, note that to any set satisfying , we can complete it to a unitary .
Since the unitary implements a quantum channel from systems to , and since the dimension of the environment of any quantum channel need not be larger than the product of the input and output dimensions, it suffices to take . Since and , it suffices to take as claimed.
Appendix B Proof of Proposition 2
The first inequality follows by picking as indicated. The first equality follows because is a unital map.
If the channels and are the same (so that ), then consider for satisfying that