State Exclusion

State Exclusion Overview¶

In a state exclusion task, a referee has a set of states $\{\rho_{x} \}^{N}, ~ x \in \{1, \ldots ,N\}$ and sends one state from the set, with probability $p_{x}$, to a player. The player then performs a general $N$ outcome measurement on the state, as described by a positive operator-valued measure (POVM) $\{T_{g}\}_{g=1}^{N}$, where $T_g\ge0$ $\forall\,g$ and $\sum_gT_g=\mathbb{I}$. From this measurement, the player outputs a label $g \in \{1, \ldots , N\}$. The player wins the task if $g \neq x$ and fail if $g=x$. Namely, they win if they successfully exclude the state by outputting a label that was not associated to the sent state; the player fails if they output the label associated to the sent state (Caves et al. (2002), Pusey et al. (2012), Bandyopadhyay et al. (2014)).

1-State Exclusion¶

If the player outputs a single label $g$ such that $g \neq x$ with certainty, this is conclusive 1-state exclusion. This occurs if the player is able to find a POVM such that

If the player gets the measurement outcome associated to $T_{g}$, they output $g$ knowing with certainty the referee could not have sent $\rho_{g}$.

In general, a POVM may not exists that can perform conclusive state exclusion. Instead, state exclusion can be performed with some error, as given by the following SPD:

where $P_{err}$ is the probability of making an error when performing exclusion, meaning the probability of outputting the index $g=x$.

k-State Exclusion¶

If the player outputs a set of $k$ labels, $\{ g_{i} \}^{k}$, such that $g_{i} \neq x ~ \forall ~ g_{i} \in \{g_{i}\}^{k}$ with certainty, this is conclusive $k$-state exclusion. There are $N \choose k$ different sets of $k$ labels the player could exclude, corresponding to all the different subsets of $\{1,\ldots,N\}$ of length $k$. Therefore, when performing $k$-state exclusion the player aims to find a POVM with $N \choose k$ elements such that each measurement outcome allows the player to exclude a subset of states from $\{\rho_{x} \}^{N}$ of length $k$.

k-State Exclusion as 1-State Exclusion¶

All $k$-state exclusion tasks can be recast as 1-state exclusion tasks by reformulating the set $\{\rho_{x}\}^{N}$ Bandyopadhyay et al. (2014). Conceptually, this means all $k$-state exclusion tasks have a 1-state exclusion task that they are dual to, allowing all state exclusion tasks to be studied under the 1-state exclusion framework.

Let $Y_{(N, k)}$ be the set of all subsets of the integers $\{1,2,...,N\}$ of length $k$. The player aims to measure a POVM on a state $\sigma \in \{ \rho_{x} \}^{N}$, given to them by the referee, and output a set of labels $Y \in Y_{(N,k)}$ such that $\sigma \notin \{ \rho_{y} \}_{y \in Y}$. Such a measurement will be a POVM with $a = {N \choose k}$ terms as there are $a$ subsets of $\{1,2,...,N\}$ of length $k$. For each elements of this POVM, $S = \{S_{l}\}^{a}_{l=1}$, the following equation must hold,

By defining

the $k$-state exclusion task can be expressed as the following set of equations

This is now the same form as the 1-state exclusion task conditions. This is the dual 1-state exclusion task to the $k$-state exclusion task.

Using the dual 1-state exclusion task, $k$-state exclusion can also now be formulated as an SPD:

where $P^{k}_{err} > 0$ if no POVM to conclusively perform $k$-state exclusion exists.

Sub-Channel Exclusion¶

A closely related task to state exclusion is sub-channel exclusion. Consider a collection of completely-positive trace non-increasing linear maps, $\Psi = \{\Psi_{x}\}_{x=1}^N$, such that $\sum_{x=1}^N \Psi_{x}$ is a channel. This collection is called a quantum instrument, and each map $\Psi_{x}$ is called a sub-channel.

In sub-channel exclusion, a player has a reference state $\rho$ that they send to the referee. The referee then measures $\rho$ using the instrument and returns the post-measurement state to the player. The player measures a POVM on the state and outputs a label $g \in \{1, \ldots ,N\}$. They succeed if they output a label of a sub-channel that was not applied. As before, the player can output the label of a sub-channel not applied with certainty, they can output $k$ labels, $\{ g_{i} \}^{k}_{i=1}$, or they can output $k$ labels with certainty.