Balancing the minimum error rate and minimum copy consumption in quantum state discrimination

1.   Introduction

Multiple-copy quantum state discrimination is a fundamental concept with wide-ranging implications for various quantum tasks, including quantum computation^[1–3] and quantum communication^[4–6]. Perfect discrimination between two non-orthogonal quantum states is unattainable when only a finite number of copies are provided, thus research in this domain predominantly focuses on maximizing efficiency^[7–22]. Some studies aim to minimize the error rate probability by measuring a fixed number of copies^{[7–13, 18, 20, 21]}, while others seek to minimize the average copy consumption to meet a specific error rate requirement^{[14–16, 22]}. Although it is recognized that optimal strategies differ for these distinct scenarios, the precise constraints between them remain elusive.

In this study, we introduce a state discrimination model to elucidate the inherent relationship between the average error rate and average copy consumption. Consider a scenario where $n$ copies of a quantum state are available. Prior to measurement, it is known that the state is either $|\psi_0\rangle=\cos{\dfrac{x}{2}}|0\rangle+\sin{\dfrac{x}{2}}|1\rangle$ or $|\psi_1\rangle=\cos{\dfrac{x}{2}}|0\rangle-\sin{\dfrac{x}{2}}|1\rangle$, with a prior probability $q$ for $|\psi_0\rangle$. Without loss of generality, we consider a specific case with $x=30^\circ$ and $q=0.5$ in this work. The task is to measure these copies and discriminate between them, resulting in two possible scenarios. In one scenario, perfect discrimination is achieved (the error rate probability $p_{\mathrm{e}}=0$) after measuring a subset of the N copies, while the remaining copies can be saved for other tasks. In the other scenario, perfect discrimination is not attained before all copies are measured, resulting in a judgment with an error rate $p_{\mathrm{e}} \geqslant 0$. The central challenge is to determine how to minimize the score, represented as

where $\langle n\rangle$ signifies the expected number of copy consumed when we finish the discrimination process, and $\langle p_{\mathrm{e}}\rangle$ denotes the expected error rate when finishing the discrimination process, with the parameter $w$ ranging from 0 to 1, meaning the requirement to minimize the weighted sum of the two parameters. This work aims to unravel the intricate interplay between these fundamental metrics, offering valuable insights into the optimization of quantum state discrimination in diverse quantum applications.

2.   Materials and methods

2.1   Extreme situation

We first consider the extreme situation, if $w=0$, we need to minimize the average error rate, thus the optimal strategy is local optimal local measurement^[8] which can achieve the minimum average error rate

where $x$ has been defined in the introduction part, representing the angle of the states. Another extreme situation is $w=1$ which requires us to minimize the average copy consumption. In this situation, the optimal strategy is doing three-element POVM (positive operator-valued measure) with three measurement angles 0, $-\dfrac{x}{2}+\dfrac{\pi}{2}$, and $\dfrac{x}{2}+\dfrac{\pi}{2}$ on each copy^[22]. The corresponding minimum average copy consumption is

2.2   General situation

In the general context, we begin by introducing the overarching expression for a one-way local strategy. All local adaptive strategies can be effectively described by a POVM with rank-one measurement elements, given as^[22]:

where $\theta \in [0, \pi)$ and is contingent upon the prior probability $q$ for $|\psi_0\rangle$. The function $F_q(\theta)$ uniquely defines the POVM and must adhere to completeness and semidefinite positivity with the following four specific conditions^[22]:

The strategies that satisfy these four constraints form the set denoted as $D$.

Then we can calculate the minimum value of the average score $\langle s\rangle_{\min }$ by dynamic program to find the globally optimal adaptive local strategy^{[8, 10, 22]}. According to this method, we can define $n$ functions $s_j\left(q\right)$, where $j=1,2,3, \cdots, n.\;\; s_j\left(q\right)$ means the minimum average score when we has measured $j-1$ copies and leave a prior probability $q$ for $\left|\psi_1\right\rangle$ before we measure the $j$th copy. $s_j\left(q\right)$ will satisfy

where $\rho = q|\psi_0\rangle\langle\psi_0| + (1-q)|\psi_1\rangle\langle\psi_1|$ and $q_\theta = \dfrac{q\operatorname{tr}[M(\theta)|\psi_0\rangle\langle\psi_0|]}{\operatorname{tr} [M(\theta)\rho]}$ denotes the posterior probability. This is the minimum average score since it consider all the possible measurement forms and measurement outcomes. The method to minimize the integral has been shown in Ref. [22] and $s_{n+1}(q)=wn+ (1-w) \min{\{q,1-q\}}$, and Ref. [22] has proven that the measurement which minimize the integral (e.g., the optimal measurement) must be a projective measurement or a three element POVM, and the exact measurement form is determined by the prior probability and the measurement round $j$ (As a comparison, when $w=0$ or $1$, the measurement form only depends on the prior probability). Therefore, we can calculate $s_{n}$, $s_{n-1}$, and so on. And $s_1(0.5)$ will be the calculation result.

3.   Results and discussion

To unify the unit, here we define a new score $S= w \dfrac{\langle n\rangle}{\langle n\rangle_{\min }}+(1-w) \dfrac{\langle p_{\mathrm{e}}\rangle}{\langle p_{\mathrm{e}}\rangle_{\min}}$. This score function is still a linear combination and it satisfies the iterative relationship similar to Eq. (6), only replacing $w$ by $\dfrac{w}{\langle n\rangle_{\min }}$ and $s_j$ by $S_j$.

As what we are more concerned about is the constraint between $\langle n\rangle$ and $\langle p_\mathrm{e}\rangle$, and $\forall w \in[0,1]$,

If $\dfrac{\langle n\rangle}{\langle n\rangle_{\text {min }}}$ has been determined, we can directly calculate the minimum value of the average error rate $\langle p_{\mathrm{e}}\rangle$:

Then we can obtain the constraints between the average error rate and average copy consumption.

Through the iterative method described above, we have obtained the minimum average score, denoted as S (as depicted in Fig. 1a), and unveiled the constraints between $\langle n\rangle$ and $\langle p_{\mathrm{e}}\rangle$ (as illustrated in Fig. 1b). Remarkably, in Fig. 1a, S exceeds 1 when the parameter $w$ does not equal 0 or 1, signifying that simultaneous minimization of the average error rate and average copy consumption is unattainable. Moreover, we can find that when $w\in(0,1)$, S(w) is positively correlated with N, showcasing that this trade-off relation will be more strong when the value of N increases.

Figure  1.  The constraints between the average copy consumption and average error rate. Here $\displaystyle x=\pi/6$, we calculate the constraint when N = 5, 10, or 15. The average score $S$ is shown in (a) and the corresponding constraint is shown in (b).

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The constraints depicted in Fig. 1b underscore that optimizing one of the two indices comes at the cost of diminishing the performance of the other. This trade-off reveals valuable insights into the delicate balance required when seeking to optimize quantum state discrimination tasks.

4.   Conclusions

Our research uncovers the disparity between minimum-error discrimination and minimum-consumption discrimination, shedding light on the inherent constraints that exist between these two objectives. The crucial observation that minimizing one of the metrics, either $\langle n\rangle$ or $\langle p_{\mathrm{e}}\rangle$, significantly enhances the other, emphasizes the inherent trade-off. In certain instances, a balanced approach that assigns a specific weight to the average cost and the reduction of the error rate may prove to be a more effective strategy, simultaneously achieving the goals of conserving quantum resources and reducing error rates.