Lower bounds on the error probability of multiple quantum channel discrimination by the Bures angle and the trace distance

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 $ρ$ and $σ$ is characterized by the trace distance $\frac{1}{2} ∥ ρ - σ ∥_{1}$ , which can be computed efficiently. As a consequence, the error probability of the discrimination of two quantum channels $Ψ$ and $Φ$ by algorithms invoking the quantum channel once can be characterized by the maximum of the trace distance, ${max}_{ρ} \frac{1}{2} ∥ Φ \otimes i d (ρ) - Ψ \otimes i d (ρ) ∥_{1} =: \frac{1}{2} ∥ Φ - Ψ ∥_{⋄}$ , 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 $n$ times, the analysis is more complicated. If we restrict the discrimination algorithms to be non-adaptive, the error probability is characterized by the diamond distance $\frac{1}{2} ∥ Φ^{\otimes n} - Ψ^{\otimes n} ∥_{⋄}$ . To compute the above diamond distance, we have to solve an optimization problem on exponentially high-dimensional linear space in $n$ . Hence, it is computationally hard to compute the error probability of non-adaptive algorithms with $n$ queries when $n$ 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 non-adaptive 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 non-adaptive algorithms [1].

The well-known 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 $O (\sqrt{N})$ queries, where $N$ 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 $k$ . This result generalizes Zalka’s result for $k = 1$ . 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 $r_{0}$ and $r_{1}$ , we obtain a simple analytic lower bound $\frac{1}{2} (1 - sin (n Δ (r_{0}, r_{1})))$ if $n Δ (r_{0}, r_{1}) \leq \frac{π}{2}$ , where $Δ (r_{0}, r_{1}) := arccos (\sqrt{r_{0} r_{1}} + \sqrt{(1 - r_{0}) (1 - r_{1})})$ . This lower bound is better than the known lower bounds [4, 13] for some choices of the damping rates $r_{0}$ and $r_{1}$ and the number $n$ 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 $L (A)$ be a set of linear operators on a quantum system $A$ . Let $D (A)$ be a set of density operators on a quantum system $A$ .

Definition 1 (Distance measure for quantum states).

A function

D : {ρ_{A} \in D (A), σ_{A} \in D (A) ∣ A : % a system} \to [0, \infty)

is called a distance function if it satisfies the following conditions.¹¹1In fact, we only need the triangle inequality and the monotonicity in this paper.

(Identity of indiscernibles) For any $ρ_{A}, σ_{A} \in D (A)$ , $D (ρ_{A}, σ_{A}) = 0$ if and only if $ρ_{A} = σ_{A}$ .
(Symmetry) For any $ρ_{A}, σ_{A} \in D (A)$ , $D (ρ_{A}, σ_{A}) = D (σ_{A}, ρ_{A})$ .
(Triangle inequality) For any $ρ_{A}, σ_{A}, τ_{A} \in D (A)$ , $D (ρ_{A}, σ_{A}) \leq D (ρ_{A}, τ_{A}) + D (τ_{A}, σ_{A})$ .
(Monotonicity) For any $ρ_{A}, σ_{A} \in D (A)$ and any quantum channel $Ψ_{A \to B}$ from a quantum system $A$ to a quantum system $B$ , $D (Ψ_{A \to B} (ρ_{A}), Ψ_{A \to B} (σ_{A})) \leq D (ρ_{A}, σ_{A})$ .

The trace distance is a well-known distance measure for quantum states.

Definition 2 (Trace distance).

For a linear operator $L_{A} \in L (A)$ , the trace norm is defined as $∥ L_{A} ∥_{1} := T r \sqrt{L_{A} L_{A}^{†}}$ . For density operators $ρ_{A}, σ_{A} \in D (A)$ , the trace distance is defined as $D (ρ_{A}, σ_{A}) = \frac{1}{2} ∥ ρ_{A} - σ_{A} ∥_{1}$ .

The trace distance has a useful property called the joint convexity, which is $D (\sum_{i} p_{i} ρ_{A}^{(i)}, \sum_{i} p_{i} σ_{A}^{(i)}) \leq \sum_{i} p_{i} D (ρ_{A}^{(i)}, σ_{A}^{(i)})$

for any probability distribution

{p_{i}}_{i}

and density operators

{ρ_{A}^{(i)}}_{i}

and

{σ_{A}^{(i)}}_{i}

The fidelity of quantum states is defined as follows.

Definition 3 (Fidelity).

For density operators $ρ_{A}, σ_{A} \in D (A)$ , the fidelity is defined as

F (ρ_{A}, σ_{A}) := ∥ \sqrt{ρ_{A}} \sqrt{σ_{A}} ∥_{1} .

The Bures angle is defined as follows.

Definition 4 (Bures angle).

For density operators $ρ_{A}, σ_{A} \in D (A)$ , the Bures angle is defined as

A (ρ_{A}, σ_{A}) := arccos F (ρ_{A}, σ_{A}) .

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 $B (ρ_{A}, σ_{A})$ and the sine distance $C (ρ_{A}, σ_{A})$ , defined as

	$B (ρ_{A}, σ_{A})$	$:= \sqrt{2 - 2 F (ρ_{A}, σ_{A})}$
	$C (ρ_{A}, σ_{A})$	$:= \sqrt{1 - F (ρ_{A}, σ_{A})^{2}},$

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 $θ = A (ρ_{A}, σ_{A})$ , it holds $B (ρ_{A}, σ_{A}) = 2 sin (θ / 2)$ and $C (ρ_{A}, σ_{A}) = sin (θ)$ . 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 fidelity-based distances $A$ , $B$ , and $C$ are not jointly convex, the fidelity has a useful property called the joint concavity, which is $F (\sum_{i} p_{i} ρ_{A}^{(i)}, \sum_{i} p_{i} σ_{A}^{(i)}) \geq \sum_{i} p_{i} F (ρ_{A}^{(i)}, σ_{A}^{(i)})$ for any probability distribution ${p_{i}}_{i}$ and density operators ${ρ_{A}^{(i)}}_{i}$ and ${σ_{A}^{(i)}}_{i}$ .

Iii Quantum channel discrimination problems

Let $Ξ$ and $H$ be finite sets. Let $A, B, E, R$ , and $W$ be finite-dimensional quantum systems. The quantum channel discrimination problem is defined as follows.

Definition 5 (Quantum channel discrimination problem).

Let ${O_{A \to B}^{ξ}}_{ξ \in Ξ}$ be a finite family of quantum channels and ${p_{ξ}}_{ξ \in Ξ}$ be a probability distribution on the quantum channels. Assume that one of the quantum channels $O_{A \to B}^{ξ}$ is given as an oracle with probability $p_{ξ}$ . The given oracle can be used multiple times. “The quantum channel discrimination problem” is the problem of determining the index of the given oracle $ξ \in Ξ$ , and is denoted by $QCDP({pξ},\allowbreak{OξA→B})$ .

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 ${O_{A \to B}^{ξ}}_{ξ \in Ξ}$ be a finite family of quantum channels and ${p_{ξ}}_{ξ \in Ξ}$ be a probability distribution on the quantum channels. Let ${C_{η} \subseteq Ξ}_{η \in H}$ be a family of subsets of $Ξ$ . Assume that one of the quantum channels $O_{A \to B}^{ξ}$ is given in the same way as $QCDP({pξ},\allowbreak{OξA→B})$ . “The quantum channel group discrimination problem” is the problem of finding one of the indexes of a subset $η \in H$ that satisfies $ξ \in C_{η}$ . This problem is denoted by $Q C G D P ({p_{ξ}}, {O_{A \to B}^{ξ}}, {C_{η}})$ .

Note that ${C_{η}}_{η \in H}$ is not necessarily a partition of $Ξ$ . When there are multiple $η \in H$ satisfying $ξ \in C_{η}$ , all of them are regarded as correct answers. When there is no $η \in H$ satisfying $ξ \in C_{η}$ , any $η$ is incorrect answer. The quantum channel discrimination problem can be regarded as the quantum channel group discrimination problem where $H = Ξ$ and $C_{η} = {η}$ for all $η \in H$ .

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 $R$ be a working system. Let ${Φ_{B R \to A R}^{k}}_{k = 1}^{n}$ be a family of quantum channels, and ${M_{B R}^{η}}_{η \in H}$ be a positive operator-valued measure (POVM). Then a pair $({Φ_{B R \to A R}^{k}}, {M_{B R}^{η}})$ represents the following algorithm for $Q C G D P ({p_{ξ}}, {O_{A \to B}^{ξ}}, {C_{η}})$ .

1:Set the initial state

| 0 ⟩ ⟨ 0 |_{B R}

in a quantum computer.

2:for

k = 1, 2, \dots, n

3: Apply the quantum channel

Φ_{B R \to A R}^{k}

4: Apply the given oracle

O_{A \to B}^{ξ}

5:end for

6:Apply a measurement

{M_{B R}^{η}}_{η \in H}

to the state and output the observed value

η \in H

An algorithm succeeds if and only if its output $η \in H$ satisfies $ξ \in C_{η}$ , where $ξ \in Ξ$ is the index of the given oracle $O_{A \to B}^{ξ}$ . The minimum error probability of a QCGDP is then defined as follows.

Definition 8 (The minimum error probability of a QCGDP).

Let $({Φ_{B R \to A R}^{k}}, {M_{B R}^{η}})$ be an algorithm for $Q C G D P ({p_{ξ}}, {O_{A \to B}^{ξ}}, {C_{η}})$ with $n$ queries. For each $ξ \in Ξ$ , a density operator $ρ_{B R}^{ξ}$ is defined as

	$ρ_{B R}^{ξ}$	$\coloneqqOξA→B∘ΦnBR→AR∘OξA→B∘Φn−1BR→AR$
		$\circ \dots \circ O_{A \to B}^{ξ} \circ Φ_{B R \to A R}^{1} (\| 0 ⟩ ⟨ 0 \|_{B R}) .$

The minimum error probability of $Q C G D P ({p_{ξ}}, {O_{A \to B}^{ξ}}, {C_{η}})$ with $n$ queries is defined as

The minimum error probability of $QCDP({pξ},\allowbreak{OξA→B})$ 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 $p_{e r r} (n)$ .

The error probability $p_{e r r} (0)$ without calling the oracle can be evaluated as follows:

	$p_{e r r} (0)$	$= min {M_{B R}^{η}} \sum ξ \in Ξ p_{ξ} \sum η : ξ \notin C_{η} T r (M_{B R}^{η} \| 0 ⟩ ⟨ 0 \|_{B R})$
		$= min {M_{B R}^{η}} \sum η \in H T r (M_{B R}^{η} \| 0 ⟩ ⟨ 0 \|_{B R}) \sum ξ \notin C_{η} p_{ξ}$
		$= min η \in H ⎛ ⎝ \sum ξ \notin C_{η} p_{ξ} ⎞ ⎠ .$

Once the quantum channels ${Φ_{B R \to A R}^{k}}_{k = 1}^{n}$ 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 $ρ_{A}^{0}$ and $ρ_{A}^{1}$ be density operators, and let $p_{0}$ and $p_{1}$ be non-negative real numbers that satisfy $p_{0} + p_{1} = 1$ . For any POVM ${M_{A}^{0}, M_{A}^{1}}$ , the following holds:

\sum ξ \in {0, 1} p_{ξ} T r (M_{A}^{ξ} ρ_{A}^{ξ}) \leq \frac{1}{2} (1 + {∥ ∥ p_{0} ρ_{A}^{0} - p_{1} ρ_{A}^{1} ∥ ∥}_{1}) .

Moreover, a POVM ${M_{A}^{0}, M_{A}^{1}}$ 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 $p_{e r r} (1)$ be the minimum error probability for $QCDP({p0,p1},\allowbreak{O0A→B,O1A→B})$ with one query. The following then holds:

	$p_{e r r} (1) = \frac{1}{2} (1$
		$- p_{1} (O_{A \to B}^{1} \otimes {i d}_{R}) (ρ_{A R}) {∥ ∥}_{1}),$

where ${i d}_{R}$ denotes the identity operator on $L (R)$ .

If $dim (R) \geq dim (A)$ , the maximum of the trace norm is called the diamond norm.

Definition 9 (Diamond norm).

Let $Ψ_{A \to B}$ be a linear operator from $L (A)$ to $L (B)$ . The diamond norm of $Ψ_{A \to B}$ is then defined as

∥ Ψ_{A \to B} ∥_{⋄} := max ρ_{A R} \in D (A \otimes R) ∥ (Ψ_{A \to B} \otimes {i d}_{R}) (ρ_{A R}) ∥_{1},

for a quantum system $R$ whose dimension is equal to $dim (A)$ .

An algorithm is said to be non-adaptive if it calls the oracle in parallel. In general, the minimum error probability of non-adaptive algorithms with $n$ queries for discriminating two quantum channels is expressed as

\frac{1}{2} (1 - {∥ ∥ ∥ p_{0} {(O_{A \to B}^{0})}^{\otimes n} - p_{1} {(O_{A \to B}^{1})}^{\otimes n} ∥ ∥ ∥}_{⋄}^{\otimes n}),

if $dim (R) \geq dim (A)^{n}$ . Although the diamond norm $∥ Ψ_{A \to B} ∥_{⋄}$ can be evaluated in polynomial time with respect to the dimensions of $A$ and $B$ via SDP [8], the dimension of $A^{\otimes n}$ is exponentially large, so that SDP does not give an efficient algorithm when $n$ 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 $p_{e r r} (n)$ . The Stinespring representation $O_{A \to B E}$ of a quantum channel $O_{A \to B}$ is a linear isometry from $A$ to $B \otimes E$ satisfying

O_{A \to B} (ρ_{A}) = {T r}_{E} (O_{A \to B E} ρ_{A} O_{A \to B E}^{†}),

for some quantum system $E$ . Let $S (A)$

be a set of state vectors on a quantum system

A

. We first show a lower bound of the error probability for QCGDP by using the Bures angle.

Theorem 1.

Let $p_{e r r} (n)$ be the minimum error probability for $Q C G D P ({p_{ξ}}, {O_{A \to B}^{ξ}}, {C_{η}})$ with $n$ queries. For a quantum channel $Ψ_{A R \to B R}$ and $m \in {0, 1, \dots, n}$ , $θ_{A} (Ψ_{A R \to B R})$ and $θ_{m}$ are defined as

	$θA(ΨAR→BR)\coloneqqarccosmin\|ϕ⟩AR∈S(AR)∑ξ∈Ξpξ$

	$θm\coloneqqarccos√perr(m).$

The following then holds:

p_{e r r} (n) \geq {cos}^{2} ((n - m) θ_{A} (Ψ_{A R \to B R}) + θ_{m}),

for an arbitrary $Ψ_{A R \to B R}$ and $m$ satisfying $(n - m) θ_{A} (Ψ_{A R \to B R}) + θ_{m} \in [0, π / 2]$ .

Furthermore, if $dim (R) \geq dim (A)$ , the following then holds:

	$θ_{A} (Ψ_{A \to B} \otimes {i d}_{R})$	$= arccos min σ_{A} \in D (A) \sum ξ \in Ξ p_{ξ}$
		${∥ ∥ {T r}_{B} (O_{A \to B E}^{ξ} σ_{A} (V_{A \to B E})^{†}) ∥ ∥}_{1},$

where $O_{A \to B E}^{ξ}$ denotes the Stinespring representation of $O_{A \to B}^{ξ}$ for $ξ \in Ξ$ , and $V_{A \to B E}$ denotes the Stinespring representation of $Ψ_{A \to B}$ .

We next show a lower bound of the error probability for QCGDP by using the trace distance.

Theorem 2.

Let $p_{e r r} (n)$ be the minimum error probability for $Q C G D P ({p_{ξ}}, {O_{A \to B}^{ξ}}, {C_{η}})$ with $n$ queries. Let $m$ and $k$ be integers that satisfy $0 \leq m \leq k \leq n$ . Let $α_{0}$ and $α_{1}$ be non-negative real numbers that satisfy $α_{0}^{n - k} = α_{1}^{k - m}$ . For a quantum channel $Ψ_{A R \to B R}$ , $θ_{⋄}^{0} (Ψ_{A R \to B R})$ and $θ_{⋄}^{1} (Ψ_{A R \to B R})$ are defined as

	$θ0⋄(ΨAR→BR)\coloneqqmax\|ϕ⟩AR∈S(AR)∑ξ∈Ξpξ$
	$\frac{1}{2} {∥ ∥ (O_{A \to B}^{ξ} - α_{0} Ψ_{A R \to B R}) (\| ϕ ⟩ ⟨ ϕ \|_{A R}) ∥ ∥}_{1},$
	$θ1⋄(ΨAR→BR)\coloneqqmax\|ϕ⟩AR∈S(AR)∑ξ∈Ξpξ$
	$\frac{1}{2} {∥ ∥ (α_{1} O_{A \to B}^{ξ} - Ψ_{A R \to B R}) (\| ϕ ⟩ ⟨ ϕ \|_{A R}) ∥ ∥}_{1} .$

The following then holds:

	$p_{e r r} (n) \geq p_{e r r} (m)$	$- (n - k - 1 \sum i = 0 α_{0}^{i}) θ_{⋄}^{0} (Ψ_{A R \to B R})$
		$- (k - m - 1 \sum i = 0 α_{1}^{i}) θ_{⋄}^{1} (Ψ_{A R \to B R}),$

for an arbitrary $Ψ_{A R \to B R}$ , $m$ , $α_{0}$ , $α_{1}$ , and $k$ .

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 $p_{e r r} (n)$ be the minimum error probability for $QCDP({p0,p1},\allowbreak{O0A→B,O1A→B})$ with $n$ queries. Let $τ_{A}$ be

	$τ_{A}$	$\coloneqqarccos$
		$min \| ϕ ⟩_{A R} \in S (A R) F (O_{A \to B}^{0} (\| ϕ ⟩ ⟨ ϕ \|_{A R}), O_{A \to B}^{1} (\| ϕ ⟩ ⟨ ϕ \|_{A R})) .$

The following then holds:

p_{e r r} (n) \geq \frac{1}{2} (1 - \sqrt{1 - 4 p_{0} p_{1} {cos}^{2} (n τ_{A})}),

under the condition $n τ_{A} \in [0, π / 2]$ . Furthermore, if both the quantum channels are unitary channels, then a non-adaptive algorithm exists that satisfies the equality.

Furthermore, if $dim (R) \geq dim (A)$ , the following then holds:

where $O_{A \to B E}^{0}$ and $O_{A \to B E}^{1}$ denote the Stinespring representations of $O_{A \to B}^{0}$ and $O_{A \to B}^{1}$ , respectively.

Finally, we show a lower bound based on the trace distance for the discrimination of two quantum channels.

Theorem 4.

Let $p_{e r r} (n)$ be the minimum error probability for $QCDP({p0,p1},\allowbreak{O0A→B,O1A→B})$ with $n$ queries. Let $k \in {0, 1, \dots, n}$ be an integer. Let $α_{0}$ and $α_{1}$ be non-negative real numbers that satisfy $p_{0} α_{0}^{k} = p_{1} α_{1}^{n - k}$ . Let $τ_{⋄}^{0}$ and $τ_{⋄}^{1}$ be

	$τ_{⋄}^{0}$	$\coloneqqmax\KetϕAR∥∥(O0A→B−α0O1A→B)(\Ketϕ\BraϕAR)∥∥1,$
	$τ_{⋄}^{1}$	$\coloneqqmax\KetϕAR∥∥(α1O0A→B−O1A→B)(\Ketϕ\BraϕAR)∥∥1.$

The following then holds:

p_{e r r} (n) \geq \frac{1}{2} [1 - p_{0} (k - 1 \sum i = 0 α_{0}^{i}) τ_{⋄}^{0} - p_{1} (n - k - 1 \sum i = 0 α_{1}^{i}) τ_{⋄}^{1}],

for arbitrary $α_{0}$ , $α_{1}$ and, $k$ .

Furthermore, if $dim (R) \geq dim (A)$ , the following then holds:

τ_{⋄}^{0} = {∥ ∥ O_{A \to B}^{0} - α_{0} O_{A \to B}^{1} ∥ ∥}_{⋄}, τ_{⋄}^{1} = {∥ ∥ α_{1} O_{A \to B}^{0} - O_{A \to B}^{1} ∥ ∥}_{⋄} .

The proofs of Theorems 1, 2, 3, and 4 are presented in Section VI.

In Section VII, we show that all optimization problems on quantum states $| ψ ⟩_{A R}$ and $σ_{A}$ in Theorems 1, 2, 3, and 4 can be written as SDP when $dim (R) \geq dim (A)$ , so that the lower bounds can be evaluated efficiently.

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 $A$ be an $| H |$ -dimensional quantum system, and ${| η ⟩_{A}}_{η \in H}$ be its computational basis. For some fixed $k \in {1, 2, \dots, | H | / 2}$ , $Ξ\coloneqq{ξ ⊆ H∣|ξ|=k}$ . Let ${O_{A}^{ξ}}_{ξ \in Ξ}$ be a family of unitary operators defined as $OξA\coloneqqI−2∑u∈ξ\Ketu\BrauA$ . Let ${O_{A}^{ξ}}_{ξ \in Ξ}$ be a family of quantum channels defined as

OξA(ρA)\coloneqqOξAρAOξ†A.

Let ${C_{η}}_{η \in H}$ be a family of subsets of $Ξ$ defined as $Cη\coloneqq{ξ∈Ξ∣η∈ξ}$ . The problem $Q C G D P ({1 / | Ξ |}, {O_{A}^{ξ}}, {C_{η}})$ is then called Grover’s search problem. The following holds:

	$\frac{1}{\| Ξ \|} \sum ξ \in Ξ O_{A}^{ξ}$	$= {(\frac{\| H \|}{k})}^{- 1} [(\frac{\| H \| - 1}{k}) - (\frac{\| H \| - 1}{k - 1})] I_{A}$
		$= (1 - \frac{2 k}{\| H \|}) I_{A} .$

Hence, we obtain

	$θ_{A} ({i d}_{A})$	$= arccos min \| ϕ ⟩_{A R} \sum ξ \in Ξ \frac{1}{\| Ξ \|} F (O_{A}^{ξ} (\| ϕ ⟩ ⟨ ϕ \|_{A R}), \| ϕ ⟩ ⟨ ϕ \|_{A R})$
		$=arccosmin\|ϕ⟩AR∑ξ∈Ξ1\|Ξ\|∣∣\BraketϕOξAϕAR∣∣$
		$≤arccosmin\|ϕ⟩AR∣∣ ∣∣∑ξ∈Ξ1\|Ξ\|\BraketϕOξAϕAR∣∣ ∣∣$
		$= arccos (1 - \frac{2 k}{\| H \|})$
		$= 2 arcsin \sqrt{\frac{k}{\| H \|}} .$

Moreover, we obtain

	$θ_{0}$	$= arccos \sqrt{1 - max η \in H ⎛ ⎝ \sum ξ \in C_{η} \frac{}{1} \| Ξ \| ⎞ ⎠}$
		$= arcsin \sqrt{max η \in H ⎛ ⎝ \sum ξ \in C_{η} \frac{1}{\| Ξ \|} ⎞ ⎠}$
		$= arcsin \sqrt{(\frac{\| H \| - 1}{k - 1}) {(\frac{\| H \|}{k})}^{- 1}}$
		$= arcsin \sqrt{\frac{k}{\| H \|}} .$

Therefore, we obtain the lower bound

p_{e r r} (n) \geq {cos}^{2} ((2 n + 1) arcsin \sqrt{\frac{k}{} | H |}),

from Theorem 1 under the condition $(2n+1)\allowbreakarcsin(√k/|H|)∈[0,π/2]$ . This lower bound is achieved by Grover’s algorithm [16] if $(2 n + 1) arcsin (\sqrt{k / | H |}) \in [0, π / 2]$ and by a modification of Grover’s algorithm [17] otherwise. The lower bound for $k = 1$ was obtained by Zalka [10].²²2Zalka 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.

Figure 1: The lower bounds on the channel position finding for amplitude damping channels ( $ℓ = 3, n = 15, p_{ξ} = 1 / ℓ (ξ \in Ξ), r_{1} = r_{0} + 0.01$ ); $∙$ , the bound from Theorem 2; $\times$ , the bound from [5].

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 $E_{A}^{r} (ρ_{A})$ with damping rate $r \in [0, 1]$ is defined by the Stinespring representation

	$U_{A \to A E}^{r}$	$\coloneqq(\Ket0\Bra0A+√1−r\Ket1\Bra1A)⊗\Ket0E$
		$+(√r\Ket0\Bra1A)⊗\Ket1E.$

In the following, we consider the channel position finding problem introduced in [5]. Let $ℓ \geq 2$ be an integer. Let $A_{1}, \dots, A_{ℓ}$ , and $R$ be two-dimensional quantum systems. Let $A^{'}$ be the composite system of $A_{1}, \dots, A_{ℓ}$ , i.e., $A^{'} = ⨂_{i = 1}^{ℓ} A_{i}$ . Let $Ξ\coloneqq{1,…,ℓ}$ , and $r_{0}, r_{1} \in [0, 1]$ . For each $ξ \in Ξ$ , a quantum channel $O_{A^{'}}^{ξ}$ is defined as

OξA′\coloneqq(ξ−1⨂i=1Er0Ai)⊗Er1Aξ⊗⎛⎝ℓ⨂i=ξ+1Er0Ai⎞⎠.

We consider the channel discrimination problem $QCDP({1/ℓ},\allowbreak{OξA′})$ using a working system $R^{'} = R^{\otimes ℓ}$ . Let $k$ be an integer that satisfies $0 \leq k \leq n$ . Let $α_{0}$ and $α_{1}$ be non-negative real numbers that satisfy $α_{0}^{n - k} = α_{1}^{k}$ . We then obtain

	${∥ ∥ ∥ ∥ O_{A^{'}}^{ξ} - α_{0} ℓ ⨂ i = 1 E_{A_{i}}^{r_{0}} ∥ ∥ ∥ ∥}_{⋄}$
	$= {∥ ∥ ∥ ∥ (ξ - 1 ⨂ i = 1 E_{A_{i}}^{r_{0}}) \otimes (E_{A_{ξ}}^{r_{1}} - α_{0} E_{A_{ξ}}^{r_{0}}) \otimes ⎛ ⎝ ℓ ⨂ i = ξ + 1 E_{A_{i}}^{r_{0}} ⎞ ⎠ ∥ ∥ ∥ ∥}_{⋄}$
	$= (ξ - 1 \prod i = 1 ∥ ∥ E_{A_{i}}^{r_{0}} {∥ ∥}_{⋄}) ∥ ∥ E_{A_{ξ}}^{r_{1}} - α_{0} E_{A_{ξ}}^{r_{0}} {∥ ∥}_{⋄} ⎛ ⎝ ℓ \prod i = ξ + 1 ∥ ∥ E_{A_{i}}^{r_{0}} {∥ ∥}_{⋄} ⎞ ⎠$
	$= {∥ ∥ E_{A}^{r_{1}} - α_{0} E_{A}^{r_{0}} ∥ ∥}_{⋄} .$

The second equality follows from the multiplicativity of the diamond norm [15].

Figure 2: The lower bounds on the channel position finding for amplitude damping channels ( $ℓ = 3, p_{ξ} = 1 / ℓ (ξ \in Ξ), r_{0} = 0.10, r_{1} = 0.11$ ); $▲$ , the bound from Theorem 2 where $k = ⌊ n / 2 ⌋$ ; $∙$ , the bound from Theorem 2 where $k$ is optimal; $\times$ , the bound from [5].

Hence, $θ_{⋄}^{0} (⨂_{i = 1}^{ℓ} E_{A_{i}}^{r_{0}})$ in Theorem 2 is bounded as

	$θ_{⋄}^{0} (ℓ ⨂ i = 1 E_{A_{i}}^{r_{0}})$
	$= max ρ_{A^{'} R^{'}} \sum ξ \in Ξ \frac{1}{ℓ} \cdot \frac{1}{2} {∥ ∥ ∥ ∥ (O_{A^{'}}^{ξ} - α_{0} ℓ ⨂ i = 1 E_{A_{i}}^{r_{0}}) (ρ_{A^{'} R^{'}}) ∥ ∥ ∥ ∥}_{1}$

	$= \frac{1}{2} {∥ ∥ E_{A}^{r_{1}} - α_{0} E_{A}^{r_{0}} ∥ ∥}_{⋄} .$

Note that $ρ_{A^{'} R^{'}} = ({argmax}_{ρ_{A R}} ∥ (E_{A}^{r_{1}} - α_{0} E_{A}^{r_{0}}) (ρ_{A R}) ∥_{1})^{\otimes ℓ}$ satisfies the equality. Similarly, we obtain

θ_{⋄}^{1} (ℓ ⨂ i = 1 E_{A_{i}}^{r_{0}}) = \frac{1}{2} {∥ ∥ α_{1} E_{A}^{r_{1}} - E_{A}^{r_{0}} ∥ ∥}_{⋄} .

From $p_{e r r} (0) = 1 - 1 / ℓ$ , we obtain

	$p_{e r r} (n) \geq 1 - \frac{1}{ℓ}$	$- (n - k - 1 \sum i = 0 \frac{α_{0}^{i}}{2}) {∥ ∥ E_{A}^{r_{1}} - α_{0} E_{A}^{r_{0}} ∥ ∥}_{⋄}$
		$- (k - 1 \sum i = 0 \frac{α_{1}^{i}}{2}) {∥ ∥ α_{1} E_{A}^{r_{1}} - E_{A}^{r_{0}} ∥ ∥}_{⋄} .$

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., $p_{ξ} = 1 / ℓ (ξ \in Ξ)$ . The number of oracles is $ℓ = 3$ . In Fig. 1, the damping rates of the channels are $r_{0}$ and $r_{1} = r_{0} + 0.01$ . The parameters $α_{0}$ and $α_{1},$ are optimized to maximize the lower bound. In Fig. 1, the number of queries is $n = 15$ . In Fig. 2, the damping rates of the channels are $r_{0} = 0.10$ and $r_{1} = 0.11$ . Compared with a lower bound obtained by Zhuang and Pirandola [5], the lower bound given by Theorem 2 is better for all $r_{0}$ in Figs. 1 and for all $n$ in Fig. 2.

v.3 Application of Theorems 3 and 4: Discrimination of two amplitude damping channels

Next, we derive a lower bound on the error probability of the discrimination of two amplitude damping channels by using Theorem 3. For $QCDP({p0,p1},\allowbreak{Er0A,Er1A})$ , we obtain the following bound of $τ_{A}$ in Theorem 3 with a two-dimensional working system $R$ :

	$τ_{A}$	$= arccos min σ_{A} {∥ ∥ {T r}_{A} (U_{A \to A E}^{r_{1}} σ_{A} (U_{A \to A E}^{r_{0}})^{†}) ∥ ∥}_{1}^{†}$
		$\leq arccos min σ_{A} {∣ ∣ {T r}_{A E} (U_{A \to A E}^{r_{1}} σ_{A} (U_{A \to A E}^{r_{0}})^{†}) ∣ ∣}_{1}^{†}$
		$=arccosminσA∣∣\Braket0σA0A$
		$+ (\sqrt{}$

	$θ0⋄(ΨAR→BR)\coloneqqmax\|ϕ⟩AR∈S(AR)∑ξ∈Ξpξ$
	$\frac{1}{2} {∥ ∥ (O_{A \to B}^{ξ} - α_{0} Ψ_{A R \to B R}) (\| ϕ ⟩ ⟨ ϕ \|_{A R}) ∥ ∥}_{1},$
	$θ1⋄(ΨAR→BR)\coloneqqmax\|ϕ⟩AR∈S(AR)∑ξ∈Ξpξ$
	$\frac{1}{2} {∥ ∥ (α_{1} O_{A \to B}^{ξ} - Ψ_{A R \to B R}) (\| ϕ ⟩ ⟨ ϕ \|_{A R}) ∥ ∥}_{1} .$

	$θ_{A} ({i d}_{A})$	$= arccos min \| ϕ ⟩_{A R} \sum ξ \in Ξ \frac{1}{\| Ξ \|} F (O_{A}^{ξ} (\| ϕ ⟩ ⟨ ϕ \|_{A R}), \| ϕ ⟩ ⟨ ϕ \|_{A R})$
		$=arccosmin\|ϕ⟩AR∑ξ∈Ξ1\|Ξ\|∣∣\BraketϕOξAϕAR∣∣$
		$≤arccosmin\|ϕ⟩AR∣∣ ∣∣∑ξ∈Ξ1\|Ξ\|\BraketϕOξAϕAR∣∣ ∣∣$
		$= arccos (1 - \frac{2 k}{\| H \|})$
		$= 2 arcsin \sqrt{\frac{k}{\| H \|}} .$

	$θ_{0}$	$= arccos \sqrt{1 - max η \in H ⎛ ⎝ \sum ξ \in C_{η} \frac{}{1} \| Ξ \| ⎞ ⎠}$
		$= arcsin \sqrt{max η \in H ⎛ ⎝ \sum ξ \in C_{η} \frac{1}{\| Ξ \|} ⎞ ⎠}$
		$= arcsin \sqrt{(\frac{\| H \| - 1}{k - 1}) {(\frac{\| H \|}{k})}^{- 1}}$
		$= arcsin \sqrt{\frac{k}{\| H \|}} .$

Lower bounds on the error probability of multiple quantum channel discrimination by the Bures angle and the trace distance

Local lower bounds on characteristics of quantum and classical systems

Lower Bounds on Learning Pauli Channels

Security of Quantum Key Distribution from Attacker's View

Time-Space Lower Bounds for Simulating Proof Systems with Quantum and Randomized Verifiers

Quantum Lower Bounds for Tripartite Versions of the Hidden Shift and the Set Equality Problems

Secret key distillation over satellite-to-satellite free-space optics channel with a limited-sized aperture eavesdropper in the same plane of the legitimate receiver

Discriminative Learning via Adaptive Questioning

I Introduction

Ii The trace distance and the Bures angle

Definition 1 (Distance measure for quantum states).

Definition 2 (Trace distance).

Definition 3 (Fidelity).

Definition 4 (Bures angle).

Iii Quantum channel discrimination problems

Definition 5 (Quantum channel discrimination problem).

Definition 6 (Quantum channel group discrimination problem).

Definition 7 (Discrimination algorithm).

Definition 8 (The minimum error probability of a QCGDP).

Proposition 1 (Holevo–Helstrom theorem [6, 7, 15]).

Proposition 2 ([15]).

Definition 9 (Diamond norm).

Iv Main results

Theorem 1.

Theorem 2.

Theorem 3.

Theorem 4.

V Applications

v.1 Application of Theorem 1: Grover’s search problem

v.2 Application of Theorem 2: Channel position finding for amplitude damping channels

Definition 10.

v.3 Application of Theorems 3 and 4: Discrimination of two amplitude damping channels

Lower bounds on the error probability of multiple quantum channel discrimination by the Bures angle and the trace distance

Related Research

Local lower bounds on characteristics of quantum and classical systems

Lower Bounds on Learning Pauli Channels

Security of Quantum Key Distribution from Attacker's View

Time-Space Lower Bounds for Simulating Proof Systems with Quantum and Randomized Verifiers

Quantum Lower Bounds for Tripartite Versions of the Hidden Shift and the Set Equality Problems

Secret key distillation over satellite-to-satellite free-space optics channel with a limited-sized aperture eavesdropper in the same plane of the legitimate receiver

Discriminative Learning via Adaptive Questioning

I Introduction

Ii The trace distance and the Bures angle

Definition 1 (Distance measure for quantum states).

Definition 2 (Trace distance).

Definition 3 (Fidelity).

Definition 4 (Bures angle).

Iii Quantum channel discrimination problems

Definition 5 (Quantum channel discrimination problem).

Definition 6 (Quantum channel group discrimination problem).

Definition 7 (Discrimination algorithm).

Definition 8 (The minimum error probability of a QCGDP).

Proposition 1 (Holevo–Helstrom theorem [6, 7, 15]).

Proposition 2 ([15]).

Definition 9 (Diamond norm).

Iv Main results

Theorem 1.

Theorem 2.

Theorem 3.

Theorem 4.

V Applications

v.1 Application of Theorem 1: Grover’s search problem

v.2 Application of Theorem 2: Channel position finding for amplitude damping channels

Definition 10.

v.3 Application of Theorems 3 and 4: Discrimination of two amplitude damping channels