I Introduction
The quantum channel discrimination problem is a fundamental problem in quantum information science [1, 2, 3, 4, 5]. In this study, we consider a general quantum channel discrimination problem, and derive the lower bounds of the error probability.
From Helstrom’s seminar work [6] and Holevo’s subsequent work [7], the error probability of the discrimination of two quantum states and is characterized by the trace distance , which can be computed efficiently. As a consequence, the error probability of the discrimination of two quantum channels and by algorithms invoking the quantum channel once can be characterized by the maximum of the trace distance, , which is called the diamond distance. The diamond distance can be computed efficiently as well [8]. On the other hand, if we consider algorithms that use the quantum channel times, the analysis is more complicated. If we restrict the discrimination algorithms to be nonadaptive, the error probability is characterized by the diamond distance . To compute the above diamond distance, we have to solve an optimization problem on exponentially highdimensional linear space in . Hence, it is computationally hard to compute the error probability of nonadaptive algorithms with queries when is large. Furthermore, for general discrimination algorithms, no simple representation of the error probability has been known even for the discrimination of two quantum channels.
The exact error probability for a fixed number of queries is known if two unitary channels are given with uniform probabilities [3]. In this case, a nonadaptive algorithm achieves the minimum error probability. For general quantum channel discriminations, however, there exists some discrimination problem in which some adaptive algorithm gives a strictly smaller error probability than any nonadaptive algorithms [1].
The wellknown Grover’s search problem can be regarded as an instance of a multiple quantum channel discrimination problem if there is exactly one marked element. Grover’s search algorithm solves this problem with queries, where is the number of elements. The asymptotic optimality of Grover’s algorithm was shown in [9]. Zalka showed that the error probability of Grover’s algorithm for a fixed number of queries is optimal when there is exactly one marked element [10].
Recently, lower bounds on the error probability of general quantum channel discrimination have been studied [4, 5, 11]. In this study, we consider a quantum channel group discrimination problem, which is a generalization of the quantum channel discrimination problem, and derive the lower bounds of the error probability. On the lines of previous studies [9, 12, 10], the main technique we use to derive the lower bounds is the application of triangle inequalities for some distance measures to series of quantum states. In this work, we use two distance measures, namely, the Bures angle and the trace distance. Our lower bound based on the Bures angle shows the optimality of Grover’s algorithm if the number of marked elements is fixed to some integer . This result generalizes Zalka’s result for . For the lower bound based on the trace distance, we introduce “weights” of the internal quantum states to tighten the triangle inequality. This yields better lower bounds for some problems considered in [5].
Moreover, we derive another type of lower bound on the error probability of two quantum channel discriminations. Our technique is a natural generalization using the Bures angle of the known technique for unitary channel discrimination [3]. For the discrimination of two amplitude damping channels with damping rates and , we obtain a simple analytic lower bound if , where . This lower bound is better than the known lower bounds [4, 13] for some choices of the damping rates and and the number of queries. We also present several numerical results in which our lower bounds for two quantum channel discrimination based on the trace distance outperform recently obtained lower bounds.
This paper is organized as follows. Distance measures for quantum states, including the Bures angle and the trace distance, are introduced in Section II. The quantum channel discrimination problem and the quantum channel group discrimination problem are introduced in Section III. The main results in this paper are presented in Section IV. The applications of our lower bounds are described in Section V. The proofs of the main results are presented in Section VI. In Section VII, we show that the lower bounds derived in this paper can be computed efficiently using semidefinite programming (SDP). We conclude the paper in Section VIII.
Ii The trace distance and the Bures angle
In this section, we introduce the notion of distance measures for quantum states, and several examples of distance measures, including the trace distance and the Bures angle. Let be a set of linear operators on a quantum system . Let be a set of density operators on a quantum system .
Definition 1 (Distance measure for quantum states).
A function
is called a distance function if it satisfies the following conditions.^{1}^{1}1In fact, we only need the triangle inequality and the monotonicity in this paper.

(Identity of indiscernibles) For any , if and only if .

(Symmetry) For any , .

(Triangle inequality) For any , .

(Monotonicity) For any and any quantum channel from a quantum system to a quantum system , .
The trace distance is a wellknown distance measure for quantum states.
Definition 2 (Trace distance).
For a linear operator , the trace norm is defined as . For density operators , the trace distance is defined as .
The trace distance has a useful property called the joint convexity, which is
for any probability distribution
and density operators and .The fidelity of quantum states is defined as follows.
Definition 3 (Fidelity).
For density operators , the fidelity is defined as
The Bures angle is defined as follows.
Definition 4 (Bures angle).
For density operators , the Bures angle is defined as
The Bures angle satisfies the conditions on the distance measure for quantum states [14].
There are several distance measures induced by the fidelity. Indeed, the Bures distance and the sine distance , defined as
satisfy the conditions on the distance measure for quantum states. These distance measures can be naturally represented in terms of the Bures angle. Indeed, for , it holds and . The triangle inequalities for the Bures distance and the sine distance are obtained from the triangle inequality for the Bures angle (see Appendix A). This fact means that the triangle inequality for the Bures angle gives better bounds than the triangle inequalities for the Bures distance and the sine distance. Hence, in this study, we use the Bures angle, and do not use the Bures distance or the sine distance to derive inequalities.
Although the fidelitybased distances , , and are not jointly convex, the fidelity has a useful property called the joint concavity, which is for any probability distribution and density operators and .
Iii Quantum channel discrimination problems
Let and be finite sets. Let , and be finitedimensional quantum systems. The quantum channel discrimination problem is defined as follows.
Definition 5 (Quantum channel discrimination problem).
Let be a finite family of quantum channels and be a probability distribution on the quantum channels. Assume that one of the quantum channels is given as an oracle with probability . The given oracle can be used multiple times. “The quantum channel discrimination problem” is the problem of determining the index of the given oracle , and is denoted by .
We also consider the quantum channel group discrimination problem, which is a natural generalization of the quantum channel discrimination problem.
Definition 6 (Quantum channel group discrimination problem).
Let be a finite family of quantum channels and be a probability distribution on the quantum channels. Let be a family of subsets of . Assume that one of the quantum channels is given in the same way as . “The quantum channel group discrimination problem” is the problem of finding one of the indexes of a subset that satisfies . This problem is denoted by .
Note that is not necessarily a partition of . When there are multiple satisfying , all of them are regarded as correct answers. When there is no satisfying , any is incorrect answer. The quantum channel discrimination problem can be regarded as the quantum channel group discrimination problem where and for all .
We consider algorithms that solve the quantum channel group discrimination problem. The general algorithm for the quantum channel group discrimination problem is defined as follows.
Definition 7 (Discrimination algorithm).
Let be a working system. Let be a family of quantum channels, and be a positive operatorvalued measure (POVM). Then a pair represents the following algorithm for .
An algorithm succeeds if and only if its output satisfies , where is the index of the given oracle . The minimum error probability of a QCGDP is then defined as follows.
Definition 8 (The minimum error probability of a QCGDP).
Let be an algorithm for with queries. For each , a density operator is defined as
The minimum error probability of with queries is defined as
The minimum error probability of is defined in the same way.
An algorithm that achieves the minimum error probability exists for any QCGDP because the set of algorithms is compact and the error probability is continuous. The main goal of this study is to derive a lower bound of the minimum error probability .
The error probability without calling the oracle can be evaluated as follows:
Once the quantum channels in a discrimination algorithm are fixed, the problem is reduced to the quantum state discrimination problem. The optimal measurement and the error probability of discrimination of two quantum states were known by Helstrom [6] and Holevo [7].
Proposition 1 (Holevo–Helstrom theorem [6, 7, 15]).
Let and be density operators, and let and be nonnegative real numbers that satisfy . For any POVM , the following holds:
Moreover, a POVM exists that satisfies the equality.
From the Holevo–Helstrom theorem, the error probability of discrimination of two quantum channels with a single query is given by the following proposition.
Proposition 2 ([15]).
Let be the minimum error probability for with one query. The following then holds:
where denotes the identity operator on .
If , the maximum of the trace norm is called the diamond norm.
Definition 9 (Diamond norm).
Let be a linear operator from to . The diamond norm of is then defined as
for a quantum system whose dimension is equal to .
An algorithm is said to be nonadaptive if it calls the oracle in parallel. In general, the minimum error probability of nonadaptive algorithms with queries for discriminating two quantum channels is expressed as
if . Although the diamond norm can be evaluated in polynomial time with respect to the dimensions of and via SDP [8], the dimension of is exponentially large, so that SDP does not give an efficient algorithm when is large. For general adaptive algorithms, no simple formula expressing the minimum error probability is known even for the discrimination of two quantum channels.
Iv Main results
In this paper, we present four theorems on the lower bounds on the minimum error probability . The Stinespring representation of a quantum channel is a linear isometry from to satisfying
for some quantum system . Let
be a set of state vectors on a quantum system
. We first show a lower bound of the error probability for QCGDP by using the Bures angle.Theorem 1.
Let be the minimum error probability for with queries. For a quantum channel and , and are defined as
The following then holds:
for an arbitrary and satisfying .
Furthermore, if , the following then holds:
where denotes the Stinespring representation of for , and denotes the Stinespring representation of .
We next show a lower bound of the error probability for QCGDP by using the trace distance.
Theorem 2.
Let be the minimum error probability for with queries. Let and be integers that satisfy . Let and be nonnegative real numbers that satisfy . For a quantum channel , and are defined as
The following then holds:
for an arbitrary , , , , and .
For the discrimination of two channels, we show another type of lower bound. The following is a lower bound based on the Bures angle.
Theorem 3.
Let be the minimum error probability for with queries. Let be
The following then holds:
under the condition . Furthermore, if both the quantum channels are unitary channels, then a nonadaptive algorithm exists that satisfies the equality.
Furthermore, if , the following then holds:
where and denote the Stinespring representations of and , respectively.
Finally, we show a lower bound based on the trace distance for the discrimination of two quantum channels.
Theorem 4.
Let be the minimum error probability for with queries. Let be an integer. Let and be nonnegative real numbers that satisfy . Let and be
The following then holds:
for arbitrary , and, .
Furthermore, if , the following then holds:
V Applications
v.1 Application of Theorem 1: Grover’s search problem
We prove the optimality of Grover’s algorithm by using Theorem 1. Let be an dimensional quantum system, and be its computational basis. For some fixed , . Let be a family of unitary operators defined as . Let be a family of quantum channels defined as
Let be a family of subsets of defined as . The problem is then called Grover’s search problem. The following holds:
Hence, we obtain
Moreover, we obtain
Therefore, we obtain the lower bound
from Theorem 1 under the condition . This lower bound is achieved by Grover’s algorithm [16] if and by a modification of Grover’s algorithm [17] otherwise. The lower bound for was obtained by Zalka [10].^{2}^{2}2Zalka wrote: “It seems very plausible that the proof can be extended to oracles with any known number of marked elements.” However, no formal proof is known to the best of our knowledge.
v.2 Application of Theorem 2: Channel position finding for amplitude damping channels
We consider the discrimination of amplitude damping channels to demonstrate the lower bound given by Theorem 2.
Definition 10.
The amplitude damping channel with damping rate is defined by the Stinespring representation
In the following, we consider the channel position finding problem introduced in [5]. Let be an integer. Let , and be twodimensional quantum systems. Let be the composite system of , i.e., . Let , and . For each , a quantum channel is defined as
We consider the channel discrimination problem using a working system . Let be an integer that satisfies . Let and be nonnegative real numbers that satisfy . We then obtain
The second equality follows from the multiplicativity of the diamond norm [15].
Hence, in Theorem 2 is bounded as
Note that satisfies the equality. Similarly, we obtain
From , we obtain
The numerical result for multiple amplitude damping channel discrimination is shown in Figs. 1 and 2. The distribution of the channels is uniform, i.e., . The number of oracles is . In Fig. 1, the damping rates of the channels are and . The parameters and are optimized to maximize the lower bound. In Fig. 1, the number of queries is . In Fig. 2, the damping rates of the channels are and . Compared with a lower bound obtained by Zhuang and Pirandola [5], the lower bound given by Theorem 2 is better for all in Figs. 1 and for all in Fig. 2.