Quantum hypothesis testing for subnormalized states

Background

Many quantities in quantum information can be extended to the case where we consider positive semidefinite operators with trace smaller than or equal to 1. I do not understand how to think about such subnormalized states in certain settings.

Let us consider a POVM $mathcal{M}$ with POVM elements ${M_0, I – M_0}$. With a normalized state, it is clear that the measurement outcome for a state is either corresponds to $M_0$ and occurs with probability $text{tr}(M_0rho)$ or corresponds to $I – M_0$ and occurs with probability $text{tr}((I – M_0)rho)$. These probabilities sum to $text{tr}(rho)$ but it’s not clear to me what it means when $text{tr}(rho)<1$. Is there a third outcome I should consider?

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Specific setting

The exact setting I am looking at is distinguishing between quantum states, also known as asymmetric hypothesis testing. Alice gives Bob either $rho$ or $sigma$ and he can construct an optimal measurement $mathcal{M} = {M_0, I-M_0}$ with the aim of figuring out which state he got.

The null hypothesis is that the state is $rho$ and the alternative hypothesis is that the state is $sigma$. There are two types of errors: Type 1, where Bob accepts the alternative hypothesis and declares the state to be $sigma$ even though it is $rho$. Type 2 errors are when Bob accepts the null hypothesis and declares that the state is $rho$ when it is in fact $sigma$. We shall require that the type 1 error is no greater than $varepsilon$. Then we have

$$text{Tr}(rho M_0)geq 1-varepsilon.$$

With that constraint in place, Bob tries to minimize the type 2 error given by $text{Tr}(M_0sigma)$. The measurement $mathcal{M}$ is optimal if it minimizes the type 2 error.

1. Is it a sensible question to ask about assymetric hypothesis testing with $rho$ being subnormalized? That is, Bob is given either a subnormalized state $rho$ or a normalized state $sigma$ and asked to figure out which one he got. To bound type 1 errors, one imposes $text{Tr}(rho M_0)geq text{Tr}{rho}-varepsilon$ and the optimal $mathcal{M}$ minimizes $text{Tr}(M_0sigma)$.

2. Is the optimal POVM $mathcal{M}$ the same regardless of the normalization of $rho$?