Information theory can be defined as a science whose fundamental elements are the quantification, loading and transmission of information. Information theory is studied according to two scenarios, the classical and the quantum. In the classical scenario, basic elements of classical information theory are bits, while the transmission, processing and load of information obey the laws of classical mechanics [1]. In the quantum context, quantum information theory introduces a new paradigm for processing and transmitting information through quantum channels. Unlike classical systems, quantum information theory is governed by the principles of quantum mechanics, leveraging phenomena such as superposition and entanglement. This implies that information can be represented not only in the form of classical bits but also as quantum bits, commonly known as qubits. Whereas bits can only take on two values, qubits can exist in a superposition of both states simultaneously, providing greater flexibility and computational power in quantum information processing (see Fig. 1). However, the approach to concepts of information theory in the quantum setting is carried out with an analogous strategy to the classical setting [2], [3].

FIGURE 1.

(a) The classic bit has two well-defined values, either 0 or 1. (b) The qubit can be a superposition of the states $\left \vert {{0}}\right \rangle$ and $\left \vert {{1}}\right \rangle$ according to the following form $\left \vert {{\psi }}\right \rangle = \alpha \left \vert {{0}}\right \rangle + \beta \left \vert {{1}}\right \rangle$ , where $\alpha$ and $\beta$ are complex numbers and $|\alpha |^{2} + |\beta |^{2} = 1$ .

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The study of channel capacity is one of the applications of information theory and has a number of practical implications. In conceptual terms, the capacity of classical channels is a number that indicates the asymptotic rate whose information can be reliably transmitted through the channel. Due to features of quantum mechanics such as entanglement and superposition, it is possible to transmit information over quantum channels in a variety of ways. Naturally, the capacity of these channels can be defined in different ways, each depending on the type of information being transmitted, making it possible to transmit classical information or quantum states [4], [5]. Regarding the transmission of classical information, there are two examples of quantum channel capacity definitions. The quantum channel capacity was defined by Holevo-Schumacher-Westmoreland (HSW), as the maximum asymptotic rate in which classical information can be reliably transmitted using quantum encoding and decoding [6], [7]. Another example of definition for quantum channel capacity is the classical entanglement capacity $C_{E}$ , which is defined as the supreme of transmission rates of classical information through a quantum channel when an infinite amount of entanglement is available between transmitter and receiver [8].

The capacities mentioned previously are defined considering that the probability decreases exponentially as the code size increases. Shannon defined the classical zero-error capacity of channels as the maximum rate achieved when the probability of error is equal to zero [9]. Since its definition, zero-error capacity has been studied in various scenarios [10]. The zero-error capacity of quantum channels was defined by Medeiros and Assis [11] as the supreme of rates in which classical information can be transmitted by a quantum channel with an error probability exactly equal to zero. Thus, using quantum block coding, classical information is mapped into quantum states and these states are transmitted through a noisy quantum channel. As part of the process, quantum states are measured on reception and, in this case, the transmission needs to be carried out with exactly zero-error rate.

An important aspect in the study of zero-error capacity of quantum channels is to verify whether a quantum channel has positive zero-error capacity or not. It is well known that determining the HSW capacity of a quantum channel is NP-Complete, while determining the zero-error capacity of a quantum channel is QMA-Complete [12]. However, even determining whether a channel’s capacity is positive can be a challenging problem [13]. Three conditions are known in the literature to verify the zero-error capacity condition of quantum channels, which are found in Medeiros and Assis [11], [14] and Gupta et al. [15]. In addition to these zero-error capacity conditions, Oliveira et al. [16], proves another condition to verify the zero-error capacity of quantum channels, which takes into account whether Kraus operators representing the quantum channel have at least two common eigenstates.

The relation between eigenstates common to the Kraus operators and the zero-error capacity of quantum channels is useful to relate quantum zero-error information theory and Shemesh theorem. The Shemesh theorem establishes a criterion that guarantees the existence of a common subspace for two square matrices. In other words, the theorem states that two square matrices $A_{1}$ and $A_{2}$ in $M_{d}$ have common eigenstates if, and only if, the subspace

View Source\begin{equation*}\mathcal {M}= \cap _{r,s=1}^{d-1}\text {ker}\ [A_{1}^{r}, A_{2}^{s}]\end{equation*}

The main objective of this paper is to establish results concerning the relationship between Shemesh theorem and zero-error quantum information theory. The motivation behind this study stems from Shemesh theorem, which provides a criterion for linear operators to share a common eigenstate. Notably, every eigenstate common to the Kraus operators of a quantum channel is also a fixed point of the channel.

This paper is organized as follows: Section II presents the definition of a quantum channel and some basic properties. In addition, it discusses the zero-error capacity of a quantum channel, which is fundamental to understanding the main objective of this work. Section III presents the statement of Shemesh theorem along with its corresponding generalization. Section IV demonstrates results related to the connection between zero-error quantum information theory and Shemesh theorem. Finally, Section V presents the conclusions.

In this section we discuss the main definitions and known results on zero-error capacity of quantum channels is presented, which are important to facilitate the reading and understanding of this paper [11], [32].

A quantum channel $\mathcal {E}$ defined on $\mathcal {H}$ can be modeled by a completely positive and trace-preserving linear map of the density operators, $\mathcal {E} \equiv \{A_{i}\}$ , where $A_{i}$ are Kraus operators on $\mathcal {H}$ and satisfy the condition $\sum _{i}^{\kappa }A_{i}^{\dagger } A_{i}=\mathbb {I}$ and $\kappa \leq d^{2}$ .

Let $\mathcal {X} \subset \mathcal {H}$ be a set of possible input states for the quantum channel $\mathcal {E}$ . If $\rho \in \mathcal {X}$ , which is denoted by $\sigma = \mathcal {E}(\rho)$ , the quantum state received when $\rho$ is transmitted over the quantum channel can be written as

View Source\begin{equation*} \mathcal {E}(\rho)=\sum _{i=1}^{\kappa }A_{i}\rho A_{i}^{\dagger }. \tag {1}\end{equation*}

When considering a quantum channel $\mathcal {E}$ , the communication protocol associated with the zero-error capacity can be summarized as follows. Define the finite subset $\mathcal {S}=\{\rho _{1},\ldots,\rho _{\ell } \} \subset \mathcal {X}$ of input states as the alphabet of the zero-error quantum code. The tensor product of any n states of $\mathcal {S}$ forms the quantum code words of length n, $\overline {\rho }_{i}=\rho _{i_{1}}\otimes \ldots \otimes \rho _{i_{n}}$ . Denote by $\mathcal {S}^{\otimes n}$ the set of all quantum code words of length n. Codewords are used to encode classic messages before they are sent over the channel. If Bob performs measurements using a POVM (Positive Operator-Valued Measurements) $\{M_{j}\}$ , where $\sum _{j}^{}M_{j}= \mathbb {I}$ , then $p(j|i)$ is defined as the probability of Bob measuring j given that Alice sent the state $\rho _{i}$ . Thus,

View Source\begin{equation*} p(j|i)=\text {tr}[\sigma _{i}M_{j}]=\text {tr} [\mathcal {E}(\rho _{i})M_{j}]. \tag {2}\end{equation*}

A quantum $(m,n)$ zero-error code for $\mathcal {E}$ is composed of:

1. a set of indices $\{1,\ldots,m\}$ , where each index is associated with a classical message;

2. a coding function

   $$\begin{equation*} f_{n}:\{1,\ldots,m\}\longrightarrow \mathcal {S}^{\otimes n} \tag {3}\end{equation*}$$ View Source\begin{equation*} f_{n}:\{1,\ldots,m\}\longrightarrow \mathcal {S}^{\otimes n} \tag {3}\end{equation*} leading to codewords $f_{n}(1)=\overline {\rho _{1}},\ldots, f_{n}(m)=\overline {\rho _{m}}$ , where each $\overline {\rho _{i}} \in \mathcal {S}^{\otimes n}$ ;

3. A decoding function

   $$\begin{equation*} g:\{1,\ldots,k\}\longrightarrow \{1,\ldots,m\} \tag {4}\end{equation*}$$ View Source\begin{equation*} g:\{1,\ldots,k\}\longrightarrow \{1,\ldots,m\} \tag {4}\end{equation*} which deterministically associates a message with one of the possible measurement $y \in \{1,\ldots,k\}$ performed by POVM $\{M_{i}\}_{i=1}^{k}$ . Furthermore, the decoding function has the following property: $$\begin{equation*} \text { Pr}[g(\mathcal {E}(f_{n}(i)))\neq i]=0 \tag {5}\end{equation*}$$ View Source\begin{equation*} \text { Pr}[g(\mathcal {E}(f_{n}(i)))\neq i]=0 \tag {5}\end{equation*} for all $i \in \{1,\ldots,m\}$ .

$$\begin{equation*} R=\frac {1}{n} \log \ m\ \text { bits/use.} \tag {6}\end{equation*}$$ View Source\begin{equation*} R=\frac {1}{n} \log \ m\ \text { bits/use.} \tag {6}\end{equation*}

Definition 1 Zero-Error Capacity of Quantum Channels[11]):

The quantum zero-error capacity of a quantum channel $\mathcal {E}(\cdot)$ , denoted by $C^{(0)}(\mathcal {E})$ , is the supreme of the achievable rates with decoding error probability equal to zero,

$$\begin{equation*} C^{(0)}(\mathcal {E})=\text {sup}_{\mathcal {S}}\text {sup}_{n}\ \frac {1}{n} \text { log}\ m \tag {7}\end{equation*}$$ View Source\begin{equation*} C^{(0)}(\mathcal {E})=\text {sup}_{\mathcal {S}}\text {sup}_{n}\ \frac {1}{n} \text { log}\ m \tag {7}\end{equation*} where m is the maximum number of classical messages the system can transmit without error when a zero-error quantum block code $(m,n)$ is used and the input alphabet is $\mathcal {S}$ .

The decoding error is related to the ability to distinguish between two states. Two quantum states are distinguishable if, and only if, the Hilbert spaces generated by the supports of these quantum states are orthogonal. Thus, two quantum states $\rho _{i},\rho _{j} \in \mathcal {S}$ , $i\neq j$ , then $\rho _{i}$ and $\rho _{j}$ are said to be non-adjacent (or distinguishable) at the output of the quantum channel $\mathcal {E}$ if $\mathcal {E}(\rho _{i})$ and $\mathcal {E}(\rho _{j})$ belong to orthogonal Hilbert subspaces. Otherwise, $\rho _{i}$ and $\rho _{j}$ are said to be adjacent (or indistinguishable) in the output of $\mathcal {E}$ .

Consequently, certain tests for determining whether a channel has positive zero-error capacity (e.g., [11, Proposition 1] and [14, Proposition 1]) assert that a quantum channel $\mathcal {E}$ possesses positive zero-error capacity if, and only if, there exist at least two non-adjacent quantum states. The zero-error capacity condition presented above is equivalently proved in [15, Lemma 4.4.1] since the zero-error capacity of the channel $\mathcal {E}$ is positive if, and only if, $| {\rho _{m}}{\rho _{m'}}|$ is orthogonal to the subspace

View Source\begin{equation*} S:=\text {span}\{A_{i}^{\dagger } A_{j}\} \tag {8}\end{equation*}

There is a relation between zero-error capacity and fixed point of the quantum channel $\mathcal {E}$ . Schauder’s fixed point theorem [33, Theorem 2.3.7] guarantees a quantum channel has at least one quantum state $\rho$ that is fixed point for the quantum channel $\mathcal {E}$ , i.e., $\mathcal {E} (\rho)=\rho$ .

The proof of this result can be found in [11, Proposition 2].

Next, it is proved every eigenstate common to all Kraus operators representing the quantum channel is a fixed point of this channel. Based on this result, one can conclude that zero-error capacity of quantum channel is positive if all the Kraus operators have at least two common eigenstates.

Lemma 3[16]:

Let a quantum channel be $\mathcal {E}: M_{d}\longrightarrow M_{d}$ with Kraus operators $A_{1},\ldots,A_{\kappa }$ . If $\left \vert {{\psi }}\right \rangle$ is a common eigenstate of the operators $A_{i}$ , then it is a fixed point to $\mathcal {E}$ .

Proof:

We just need to show that $\mathcal{E}(|\psi\rangle\langle\psi|)=|\psi\rangle\langle\psi|$ . In general, consider that $A_{i}=\lambda _{i} \left \vert {{\psi }}\right \rangle$ can be associated with different eigenvalues to different Kraus operators. Then,

$$\begin{align*} \mathcal{E}(|\psi\rangle\langle\psi|) & \stackrel{(\mathrm{a})}{=} \sum_{i=1}^\kappa A_i|\psi\rangle\langle\psi| A_i^{\dagger} \stackrel{(\mathrm{b})}{=} \sum_{i=1}^\kappa \lambda_i|\psi\rangle \lambda_i^*\langle\psi| \tag{9}\\ & \stackrel{(\mathrm{c})}{=}|\psi\rangle\langle\psi| \sum_{i=1}^\kappa\left|\lambda_i\right|^2 \stackrel{(\mathrm{~d})}{=}|\psi\rangle\langle\psi|, \tag{10}\end{align*}$$ View Source\begin{align*} \mathcal{E}(|\psi\rangle\langle\psi|) & \stackrel{(\mathrm{a})}{=} \sum_{i=1}^\kappa A_i|\psi\rangle\langle\psi| A_i^{\dagger} \stackrel{(\mathrm{b})}{=} \sum_{i=1}^\kappa \lambda_i|\psi\rangle \lambda_i^*\langle\psi| \tag{9}\\ & \stackrel{(\mathrm{c})}{=}|\psi\rangle\langle\psi| \sum_{i=1}^\kappa\left|\lambda_i\right|^2 \stackrel{(\mathrm{~d})}{=}|\psi\rangle\langle\psi|, \tag{10}\end{align*} where in (a) we used Kraus representation of the quantum channel, in (b) the fact that $\left \vert {{\psi }}\right \rangle$ is an eigenstate of every $A_{i}$ (by hypothesis) and (d) because $\mathcal {E}$ is trace preserving.□

Define $N_{\mathcal {E}} = \{ \left \vert {{\psi }}\right \rangle \left \langle {{\psi }}\right \vert \in \mathcal {S}\ :\ A_{i}\left \vert {{\psi }}\right \rangle =\lambda _{i}\left \vert {{\psi }}\right \rangle,\ i=1,\ldots,\kappa \}$ as the set of eigenstates the are common to every Kraus operator for the quantum channel $\mathcal {E}$ . Denote by $|N_{\mathcal {E}}|$ the cardinality of $N_{\mathcal {E}}$ .

The condition for positive zero-error capacity of a quantum channel established in Theorem 4 is highly practical. Given the d possible distinct eigenstates in the Hilbert space, it is sufficient to verify that at least two of these eigenstates are common to the Kraus operators representing the channel.

In order to show the relation between Shemesh theorem and quantum zero-error information theory, in this section we will present Shemesh theorem in the non-generalized version with proof, and state the generalized theorem.

Theorem 5 (Shemesh Theorem[17]):

Let $A_{1}$ and $A_{2}$ be $M_{d}$ . Then $A_{1}$ and $A_{2}$ have common eigenstates if, and only if, the subspace

$$\begin{equation*} \mathcal {M}= \cap _{r,s=1}^{d-1}\text {ker}\ [A_{1}^{r}, A_{2}^{s}] \tag {11}\end{equation*}$$ View Source\begin{equation*} \mathcal {M}= \cap _{r,s=1}^{d-1}\text {ker}\ [A_{1}^{r}, A_{2}^{s}] \tag {11}\end{equation*} is a nontrivial subspace, that is, some $\left \vert {{\varphi }}\right \rangle \in \mathcal {M}$ with $0 \neq \left \vert {{\varphi }}\right \rangle \in \mathbb {C}^{d}$ . The symbol $[\cdot, \cdot]$ denotes the matrix commutator and $r,s \in \{1,2,\ldots,d-1\}$ , while $A_{1}^{r}$ and $A_{2}^{s}$ denote the r-th and s-th powers of $A_{1}$ and $A_{2}$ , respectively.

The demonstration of Shemesh theorem can be found in [17].

Observation 6:

The subspace of Shemesh theorem is contained in $\mathbb {C}^{d}$ , i.e.,

$$\begin{equation*} \mathcal {M}= \cap _{r,s=1}^{d-1}\text {ker}[A_{1}^{r}, A_{2}^{s}] \subset \mathbb {C}^{d}. \tag {12}\end{equation*}$$ View Source\begin{equation*} \mathcal {M}= \cap _{r,s=1}^{d-1}\text {ker}[A_{1}^{r}, A_{2}^{s}] \subset \mathbb {C}^{d}. \tag {12}\end{equation*}

The Theorem 5 can be generalized to a quantity s of matrices and give conditions for them to have common eigenstates. This generalization occurs due to the use of the standard polynomial concept and the Amitsur-Levitzki theorem. The generalized Shemesh Theorem is stated below and it can be found in [18, Theorem 2.4].

Theorem 7 (Generalized Shemesh Theorem[18]):

The matrices $A_{1},\ldots, A_{\kappa } \in M_{d}$ have common eigenstates if, and only if, the subspace

$$\begin{equation*} \mathcal {M}= \cap _{\alpha _{i},l_{i}=1}^{d-1}\text {ker}\ [A_{1}^{\alpha _{1}}\ldots \ A_{\kappa }^{\alpha _{\kappa }}, A_{1}^{l_{1}}\ldots \ A_{\kappa }^{l_{\kappa }}] \tag {13}\end{equation*}$$ View Source\begin{equation*} \mathcal {M}= \cap _{\alpha _{i},l_{i}=1}^{d-1}\text {ker}\ [A_{1}^{\alpha _{1}}\ldots \ A_{\kappa }^{\alpha _{\kappa }}, A_{1}^{l_{1}}\ldots \ A_{\kappa }^{l_{\kappa }}] \tag {13}\end{equation*} is a nontrivial subspace, or in other words, for some $x \in \mathcal {M}$ with $0\neq x \in \mathbb {C}^{d}$ and $\sum _{i}^{}\alpha _{i}\neq 0$ and $\sum _{i}^{}l_{i}\neq 0$ .

Note that the sufficient and necessary condition for matrices $A_{1},\ldots, A_{\kappa } \in M_{d}$ to have a common eigenstate is that Shemesh subspace $\mathcal {M}$ is nontrivial. Initially, the nonzero vectors belonging to $\mathcal {M}$ are not necessarily common eigenstates of $A_{i}$ . In the following section it is proved that if $\mathcal {M}$ is nontrivial, then the eigenstates common to $A_{i}$ belong to $\mathcal {M}$ .

In this section, results are presented so that we can construct the relation between Shemesh theorem and quantum zero-error information theory.

Lemma 8:

Let $A \in M_{d}$ and an associated eigenstate $\left \vert {{\varphi }}\right \rangle$ . Then for $m\gt 0$ , $\left \vert {{\varphi }}\right \rangle$ is eigenstate of $A^{m}$ associated with the eigenvalue $\lambda ^{m}$ . In other words, $A^{m}\left \vert {{\varphi }}\right \rangle =\lambda ^{m}\left \vert {{\varphi }}\right \rangle$ .

Proof:

By hypothesis $\left \vert {{\varphi }}\right \rangle$ is eigenstate associated to $\alpha$ , so for $m=2$ we have

$$\begin{equation*} A^{2}\left \vert {{\varphi }}\right \rangle = A(A \left \vert {{\varphi }}\right \rangle)= A(\lambda \left \vert {{\varphi }}\right \rangle)=\lambda A\left \vert {{\varphi }}\right \rangle =\lambda ^{2}\left \vert {{\varphi }}\right \rangle. \tag {14}\end{equation*}$$ View Source\begin{equation*} A^{2}\left \vert {{\varphi }}\right \rangle = A(A \left \vert {{\varphi }}\right \rangle)= A(\lambda \left \vert {{\varphi }}\right \rangle)=\lambda A\left \vert {{\varphi }}\right \rangle =\lambda ^{2}\left \vert {{\varphi }}\right \rangle. \tag {14}\end{equation*} By induction hypothesis suppose that the result holds for k, then $$\begin{equation*} A^{k}\left \vert {{\varphi }}\right \rangle =\lambda ^{k}\left \vert {{\varphi }}\right \rangle. \tag {15}\end{equation*}$$ View Source\begin{equation*} A^{k}\left \vert {{\varphi }}\right \rangle =\lambda ^{k}\left \vert {{\varphi }}\right \rangle. \tag {15}\end{equation*} Thus, $$\begin{align*} A^{k+1}\left \vert {{\varphi }}\right \rangle =A(A^{k}\left \vert {{\varphi }}\right \rangle)=A(\lambda ^{k}\left \vert {{\varphi }}\right \rangle)=\lambda ^{k}\lambda \left \vert {{\varphi }}\right \rangle =\lambda ^{k+1}\left \vert {{\varphi }}\right \rangle. \tag {16}\end{align*}$$ View Source\begin{align*} A^{k+1}\left \vert {{\varphi }}\right \rangle =A(A^{k}\left \vert {{\varphi }}\right \rangle)=A(\lambda ^{k}\left \vert {{\varphi }}\right \rangle)=\lambda ^{k}\lambda \left \vert {{\varphi }}\right \rangle =\lambda ^{k+1}\left \vert {{\varphi }}\right \rangle. \tag {16}\end{align*} Therefore, the result holds for all $m\gt 0$ .□

Proposition 9:

Let be a quantum channel $\mathcal {E}: M_{d}\longrightarrow M_{d}$ with Kraus operators $A_{1},\ldots,A_{\kappa }$ . Then, every eigenstate of $N_{\mathcal {E}}$ belongs to the subspace

$$\begin{equation*} \mathcal {M}= \cap _{\alpha _{i},l_{i}=1}^{d-1}\text {ker}\ [A_{1}^{\alpha _{1}}\ldots A_{\kappa }^{\alpha _{\kappa }}, A_{1}^{l_{1}}\ldots A_{\kappa }^{l_{\kappa }}] \tag {17}\end{equation*}$$ View Source\begin{equation*} \mathcal {M}= \cap _{\alpha _{i},l_{i}=1}^{d-1}\text {ker}\ [A_{1}^{\alpha _{1}}\ldots A_{\kappa }^{\alpha _{\kappa }}, A_{1}^{l_{1}}\ldots A_{\kappa }^{l_{\kappa }}] \tag {17}\end{equation*} with $\sum _{i}^{}\alpha _{i}\neq 0$ and $\sum _{i}^{}l_{i}\neq 0$ .

Proof:

Suppose that $\left \vert {{\varphi }}\right \rangle \in N_{\mathcal {E}}$ , then $\left \vert {{\varphi }}\right \rangle$ is common eigenstate to $A_{1},\ldots,A_{\kappa }$ , that is,

$$\begin{align*} A_{1} \left \vert {{\varphi }}\right \rangle = \lambda _{1} \left \vert {{\varphi }}\right \rangle,\ A_{2} \left \vert {{\varphi }}\right \rangle = \lambda _{2} \left \vert {{\varphi }}\right \rangle,\ldots,A_{\kappa }\left \vert {{\varphi }}\right \rangle = \lambda _{\kappa }\left \vert {{\varphi }}\right \rangle. \tag {18}\end{align*}$$ View Source\begin{align*} A_{1} \left \vert {{\varphi }}\right \rangle = \lambda _{1} \left \vert {{\varphi }}\right \rangle,\ A_{2} \left \vert {{\varphi }}\right \rangle = \lambda _{2} \left \vert {{\varphi }}\right \rangle,\ldots,A_{\kappa }\left \vert {{\varphi }}\right \rangle = \lambda _{\kappa }\left \vert {{\varphi }}\right \rangle. \tag {18}\end{align*} By using the Theorem 7 we have that the subspace $\mathcal {M} \neq 0$ , so suppose $a_{1},\ldots,a_{\kappa } \in \{1,2,\ldots,d-1\}$ and $b_{1},\ldots,b_{\kappa } \in \{1,2,\ldots,d-1\}$ and consider the commutator $$\begin{align*}& [A_{1}^{a_{1}}\ldots A_{\kappa }^{a_{\kappa }},A_{1}^{b_{1}}\ldots A_{\kappa }^{b_{\kappa }}] \\ & = (A_{1}^{a_{1}}\ldots A_{\kappa }^{a_{\kappa }})(A_{1}^{b_{1}}\ldots A_{\kappa }^{b_{\kappa }})-(A_{1}^{b_{1}}\ldots A_{\kappa }^{b_{\kappa }})(A_{1}^{a_{1}}\ldots A_{\kappa }^{a_{\kappa }}). \tag {19}\end{align*}$$ View Source\begin{align*}& [A_{1}^{a_{1}}\ldots A_{\kappa }^{a_{\kappa }},A_{1}^{b_{1}}\ldots A_{\kappa }^{b_{\kappa }}] \\ & = (A_{1}^{a_{1}}\ldots A_{\kappa }^{a_{\kappa }})(A_{1}^{b_{1}}\ldots A_{\kappa }^{b_{\kappa }})-(A_{1}^{b_{1}}\ldots A_{\kappa }^{b_{\kappa }})(A_{1}^{a_{1}}\ldots A_{\kappa }^{a_{\kappa }}). \tag {19}\end{align*}

Note that

$$\begin{align*}& ((A_{1}^{a_{1}}\ldots A_{\kappa }^{a_{\kappa }})(A_{1}^{b_{1}}\ldots A_{\kappa }^{b_{\kappa }})-(A_{1}^{b_{1}}\ldots A_{\kappa }^{b_{\kappa }})(A_{1}^{a_{1}}\ldots A_{\kappa }^{a_{\kappa }}))\left \vert {{\varphi }}\right \rangle \tag {20}\\ & =(A_{1}^{a_{1}}\ldots A_{\kappa }^{a_{\kappa }})(A_{1}^{b_{1}}\ldots A_{\kappa -1}^{b_{\kappa -1}})A_{\kappa }^{b_{\kappa }}\left \vert {{\varphi }}\right \rangle \\ & \quad -(A_{1}^{b_{1}}\ldots A_{\kappa }^{b_{\kappa }})(A_{1}^{a_{1}}\ldots A_{\kappa -1}^{a_{\kappa -1}})A_{\kappa }^{a_{\kappa }}\left \vert {{\varphi }}\right \rangle \tag {21}\\ & =(A_{1}^{a_{1}}\ldots A_{\kappa }^{a_{\kappa }})(A_{1}^{b_{1}}\ldots A_{\kappa -1}^{b_{\kappa -1}})\lambda _{\kappa }^{b_{\kappa }}\left \vert {{\varphi }}\right \rangle \\ & \quad -(A_{1}^{b_{1}}\ldots A_{\kappa }^{b_{\kappa }})(A_{1}^{a_{1}}\ldots A_{\kappa -1}^{a_{\kappa -1}})\lambda _{\kappa }^{a_{\kappa }}\left \vert {{\varphi }}\right \rangle \tag {22}\\ & =\lambda _{\kappa }^{b_{\kappa }}(A_{1}^{a_{1}}\ldots A_{\kappa }^{a_{\kappa }})(A_{1}^{b_{1}}\ldots A_{\kappa -1}^{b_{\kappa -1}})\left \vert {{\varphi }}\right \rangle \\ & \quad -\lambda _{\kappa }^{a_{\kappa }}(A_{1}^{b_{1}}\ldots A_{\kappa }^{b_{\kappa }})(A_{1}^{a_{1}}\ldots A_{\kappa -1}^{a_{\kappa -1}})\left \vert {{\varphi }}\right \rangle \tag {23}\\ & =\lambda _{\kappa }^{b_{\kappa }}(A_{1}^{a_{1}}\ldots A_{\kappa }^{a_{\kappa }})(A_{1}^{b_{1}}\ldots A_{\kappa -2}^{b_{\kappa -2}})A_{\kappa -1}^{b_{\kappa -1}}\left \vert {{\varphi }}\right \rangle \\ & \quad -\lambda _{\kappa }^{a_{\kappa }}(A_{1}^{b_{1}}\ldots A_{\kappa }^{b_{\kappa }})(A_{1}^{a_{1}}\ldots A_{\kappa -2}^{a_{\kappa -2}})A_{\kappa -1}^{a_{\kappa -1}}\left \vert {{\varphi }}\right \rangle \tag {24}\\ & =\lambda _{\kappa }^{b_{\kappa }}\lambda _{\kappa -1}^{b_{\kappa -1}}(A_{1}^{a_{1}}\ldots A_{\kappa }^{a_{\kappa }})(A_{1}^{b_{1}}\ldots A_{\kappa -2}^{b_{\kappa -2}})\left \vert {{\varphi }}\right \rangle \\ & \quad -\lambda _{\kappa }^{a_{\kappa }}\lambda _{\kappa -1}^{a_{\kappa -1}}(A_{1}^{b_{1}}\ldots A_{\kappa }^{b_{\kappa }})(A_{1}^{a_{1}}\ldots A_{\kappa -2}^{a_{\kappa -2}})\left \vert {{\varphi }}\right \rangle \\ & \quad \vdots \hspace {2cm} \vdots \hspace {2cm} \vdots \hspace {2cm} \vdots \tag {25}\\ & =\lambda _{\kappa }^{b_{\kappa }}\lambda _{\kappa -1}^{b_{\kappa -1}}\ldots \lambda _{1}^{b_{1}} \lambda _{\kappa }^{a_{\kappa }}\lambda _{\kappa -1}^{a_{\kappa -1}}\ldots \lambda _{1}^{a_{1}}\left \vert {{\varphi }}\right \rangle \\ & \quad -\lambda _{\kappa }^{a_{\kappa }}\lambda _{\kappa -1}^{a_{\kappa -1}}\ldots \lambda _{1}^{a_{1}}\lambda _{\kappa }^{b_{\kappa }}\lambda _{\kappa -1}^{b_{\kappa -1}}\ldots \lambda _{1}^{b_{1}}\left \vert {{\varphi }}\right \rangle \tag {26}\\ & =(\lambda _{\kappa }^{b_{\kappa }+a_{s}}\lambda _{\kappa -1}^{b_{\kappa -1}+a_{\kappa -1}}\ldots \lambda _{1}^{b_{1}+a_{1}} \\ & \quad -\lambda _{\kappa }^{a_{\kappa }+b_{\kappa }}\lambda _{\kappa -1}^{a_{\kappa -1}+b_{\kappa -1}}\ldots \lambda _{1}^{a_{1}+b_{1}})\left \vert {{\varphi }}\right \rangle \tag {27}\\ & = 0 \cdot \left \vert {{\varphi }}\right \rangle =0. \hspace {5.74cm} \tag {28}\end{align*}$$ View Source\begin{align*}& ((A_{1}^{a_{1}}\ldots A_{\kappa }^{a_{\kappa }})(A_{1}^{b_{1}}\ldots A_{\kappa }^{b_{\kappa }})-(A_{1}^{b_{1}}\ldots A_{\kappa }^{b_{\kappa }})(A_{1}^{a_{1}}\ldots A_{\kappa }^{a_{\kappa }}))\left \vert {{\varphi }}\right \rangle \tag {20}\\ & =(A_{1}^{a_{1}}\ldots A_{\kappa }^{a_{\kappa }})(A_{1}^{b_{1}}\ldots A_{\kappa -1}^{b_{\kappa -1}})A_{\kappa }^{b_{\kappa }}\left \vert {{\varphi }}\right \rangle \\ & \quad -(A_{1}^{b_{1}}\ldots A_{\kappa }^{b_{\kappa }})(A_{1}^{a_{1}}\ldots A_{\kappa -1}^{a_{\kappa -1}})A_{\kappa }^{a_{\kappa }}\left \vert {{\varphi }}\right \rangle \tag {21}\\ & =(A_{1}^{a_{1}}\ldots A_{\kappa }^{a_{\kappa }})(A_{1}^{b_{1}}\ldots A_{\kappa -1}^{b_{\kappa -1}})\lambda _{\kappa }^{b_{\kappa }}\left \vert {{\varphi }}\right \rangle \\ & \quad -(A_{1}^{b_{1}}\ldots A_{\kappa }^{b_{\kappa }})(A_{1}^{a_{1}}\ldots A_{\kappa -1}^{a_{\kappa -1}})\lambda _{\kappa }^{a_{\kappa }}\left \vert {{\varphi }}\right \rangle \tag {22}\\ & =\lambda _{\kappa }^{b_{\kappa }}(A_{1}^{a_{1}}\ldots A_{\kappa }^{a_{\kappa }})(A_{1}^{b_{1}}\ldots A_{\kappa -1}^{b_{\kappa -1}})\left \vert {{\varphi }}\right \rangle \\ & \quad -\lambda _{\kappa }^{a_{\kappa }}(A_{1}^{b_{1}}\ldots A_{\kappa }^{b_{\kappa }})(A_{1}^{a_{1}}\ldots A_{\kappa -1}^{a_{\kappa -1}})\left \vert {{\varphi }}\right \rangle \tag {23}\\ & =\lambda _{\kappa }^{b_{\kappa }}(A_{1}^{a_{1}}\ldots A_{\kappa }^{a_{\kappa }})(A_{1}^{b_{1}}\ldots A_{\kappa -2}^{b_{\kappa -2}})A_{\kappa -1}^{b_{\kappa -1}}\left \vert {{\varphi }}\right \rangle \\ & \quad -\lambda _{\kappa }^{a_{\kappa }}(A_{1}^{b_{1}}\ldots A_{\kappa }^{b_{\kappa }})(A_{1}^{a_{1}}\ldots A_{\kappa -2}^{a_{\kappa -2}})A_{\kappa -1}^{a_{\kappa -1}}\left \vert {{\varphi }}\right \rangle \tag {24}\\ & =\lambda _{\kappa }^{b_{\kappa }}\lambda _{\kappa -1}^{b_{\kappa -1}}(A_{1}^{a_{1}}\ldots A_{\kappa }^{a_{\kappa }})(A_{1}^{b_{1}}\ldots A_{\kappa -2}^{b_{\kappa -2}})\left \vert {{\varphi }}\right \rangle \\ & \quad -\lambda _{\kappa }^{a_{\kappa }}\lambda _{\kappa -1}^{a_{\kappa -1}}(A_{1}^{b_{1}}\ldots A_{\kappa }^{b_{\kappa }})(A_{1}^{a_{1}}\ldots A_{\kappa -2}^{a_{\kappa -2}})\left \vert {{\varphi }}\right \rangle \\ & \quad \vdots \hspace {2cm} \vdots \hspace {2cm} \vdots \hspace {2cm} \vdots \tag {25}\\ & =\lambda _{\kappa }^{b_{\kappa }}\lambda _{\kappa -1}^{b_{\kappa -1}}\ldots \lambda _{1}^{b_{1}} \lambda _{\kappa }^{a_{\kappa }}\lambda _{\kappa -1}^{a_{\kappa -1}}\ldots \lambda _{1}^{a_{1}}\left \vert {{\varphi }}\right \rangle \\ & \quad -\lambda _{\kappa }^{a_{\kappa }}\lambda _{\kappa -1}^{a_{\kappa -1}}\ldots \lambda _{1}^{a_{1}}\lambda _{\kappa }^{b_{\kappa }}\lambda _{\kappa -1}^{b_{\kappa -1}}\ldots \lambda _{1}^{b_{1}}\left \vert {{\varphi }}\right \rangle \tag {26}\\ & =(\lambda _{\kappa }^{b_{\kappa }+a_{s}}\lambda _{\kappa -1}^{b_{\kappa -1}+a_{\kappa -1}}\ldots \lambda _{1}^{b_{1}+a_{1}} \\ & \quad -\lambda _{\kappa }^{a_{\kappa }+b_{\kappa }}\lambda _{\kappa -1}^{a_{\kappa -1}+b_{\kappa -1}}\ldots \lambda _{1}^{a_{1}+b_{1}})\left \vert {{\varphi }}\right \rangle \tag {27}\\ & = 0 \cdot \left \vert {{\varphi }}\right \rangle =0. \hspace {5.74cm} \tag {28}\end{align*} In other words, the eigenstate $\left \vert {{\varphi }}\right \rangle$ belongs to the subspace $$\begin{equation*} \mathcal {M}= \cap _{\alpha _{i},l_{i}=1}^{d-1}\text {ker}\ [A_{1}^{\alpha _{1}}\ldots A_{\kappa }^{\alpha _{\kappa }}, A_{1}^{l_{1}}\ldots A_{\kappa }^{l_{\kappa }}]. \tag {29}\end{equation*}$$ View Source\begin{equation*} \mathcal {M}= \cap _{\alpha _{i},l_{i}=1}^{d-1}\text {ker}\ [A_{1}^{\alpha _{1}}\ldots A_{\kappa }^{\alpha _{\kappa }}, A_{1}^{l_{1}}\ldots A_{\kappa }^{l_{\kappa }}]. \tag {29}\end{equation*} Therefore, every eigenstate of $N_{\mathcal {E}}$ belongs to the subspace $\mathcal {M}$ .□

The Proposition 9 shows that if $\left \vert {{\varphi }}\right \rangle \left \langle {{\varphi }}\right \vert \in N_{\mathcal {E}}$ , then $\left \vert {{\varphi }}\right \rangle \left \langle {{\varphi }}\right \vert \in \mathcal {M}$ , i.e., $N_{\mathcal {E}} \subseteq \mathcal {M}$ . In addition, consider the following set

View Source\begin{equation*}{{\mathcal {M}}_{\mathcal {E}}=\{ \left \vert {{\varphi }}\right \rangle \left \langle {{\varphi }}\right \vert \in \mathcal {M}: \mathcal {E}(\left \vert {{\varphi }}\right \rangle \left \langle {{\varphi }}\right \vert)=\left \vert {{\varphi }}\right \rangle \left \langle {{\varphi }}\right \vert \}},\end{equation*}

The result of Proposition 11 establishes a connection between Shemesh theorem and the zero-error capacity of the quantum channel.

Example 12:

Let $\mathcal {E}$ be the quantum channel with Kraus operators $A_{1}$ and $A_{2}$ , where $p \in (0,1)$ :

$$\begin{align*}A_{1}& = \sqrt {p}\left ({{ \begin{array}{ccccc} 1 & \quad 0 & \quad 0 & \quad 0 \\[7pt] 0 & \quad \dfrac {1}{2} & \quad \dfrac {\sqrt {3}}{2} & \quad 0 \\[7pt] 0 & \quad \dfrac {\sqrt {3}}{2} & \quad -\dfrac {1}{2} & \quad 0 \\[7pt] 0 & \quad 0 & \quad 0 & \quad 1 \\ \end{array} }}\right), \\ A_{2}& = \sqrt {1-p}\left ({{ \begin{array}{ccccc} 1 & \quad 0 & \quad 0 & \quad 0 \\[7pt] 0 & \quad \dfrac {1}{2} & \quad - \dfrac {\sqrt {3}}{2} & \quad 0 \\[7pt] 0 & \quad -\dfrac {\sqrt {3}}{2} & \quad -\dfrac {1}{2} & \quad 0 \\[7pt] 0 & \quad 0 & \quad 0 & \quad 1 \\ \end{array} }}\right).\end{align*}$$ View Source\begin{align*}A_{1}& = \sqrt {p}\left ({{ \begin{array}{ccccc} 1 & \quad 0 & \quad 0 & \quad 0 \\[7pt] 0 & \quad \dfrac {1}{2} & \quad \dfrac {\sqrt {3}}{2} & \quad 0 \\[7pt] 0 & \quad \dfrac {\sqrt {3}}{2} & \quad -\dfrac {1}{2} & \quad 0 \\[7pt] 0 & \quad 0 & \quad 0 & \quad 1 \\ \end{array} }}\right), \\ A_{2}& = \sqrt {1-p}\left ({{ \begin{array}{ccccc} 1 & \quad 0 & \quad 0 & \quad 0 \\[7pt] 0 & \quad \dfrac {1}{2} & \quad - \dfrac {\sqrt {3}}{2} & \quad 0 \\[7pt] 0 & \quad -\dfrac {\sqrt {3}}{2} & \quad -\dfrac {1}{2} & \quad 0 \\[7pt] 0 & \quad 0 & \quad 0 & \quad 1 \\ \end{array} }}\right).\end{align*} For $p=\frac {1}{4}$ , we have that $A_{1}$ and $A_{2}$ have the form: $$\begin{align*}A_{1}= \left ({{ \begin{array}{ccccc} \dfrac {1}{2} & \quad 0 & \quad 0 & \quad 0 \\[7pt] 0 & \quad \dfrac {1}{4} & \quad \dfrac {\sqrt {3}}{4} & \quad 0 \\[7pt] 0 & \quad \dfrac {\sqrt {3}}{4} & \quad -\dfrac {1}{4} & \quad 0 \\[7pt] 0 & \quad 0 & \quad 0 & \quad \dfrac {1}{2} \\ \end{array} }}\right), \\ A_{2}= \left ({{ \begin{array}{ccccc} \dfrac {\sqrt {3}}{2} & \quad 0 & \quad 0 & \quad 0 \\[7pt] 0 & \quad \dfrac {\sqrt {3}}{4} & \quad - \dfrac {3}{4} & \quad 0 \\[7pt] 0 & \quad -\dfrac {3}{4} & \quad -\dfrac {\sqrt {3}}{4} & \quad 0 \\[7pt] 0 & \quad 0 & \quad 0 & \quad \dfrac {\sqrt {3}}{2} \\ \end{array} }}\right).\end{align*}$$ View Source\begin{align*}A_{1}= \left ({{ \begin{array}{ccccc} \dfrac {1}{2} & \quad 0 & \quad 0 & \quad 0 \\[7pt] 0 & \quad \dfrac {1}{4} & \quad \dfrac {\sqrt {3}}{4} & \quad 0 \\[7pt] 0 & \quad \dfrac {\sqrt {3}}{4} & \quad -\dfrac {1}{4} & \quad 0 \\[7pt] 0 & \quad 0 & \quad 0 & \quad \dfrac {1}{2} \\ \end{array} }}\right), \\ A_{2}= \left ({{ \begin{array}{ccccc} \dfrac {\sqrt {3}}{2} & \quad 0 & \quad 0 & \quad 0 \\[7pt] 0 & \quad \dfrac {\sqrt {3}}{4} & \quad - \dfrac {3}{4} & \quad 0 \\[7pt] 0 & \quad -\dfrac {3}{4} & \quad -\dfrac {\sqrt {3}}{4} & \quad 0 \\[7pt] 0 & \quad 0 & \quad 0 & \quad \dfrac {\sqrt {3}}{2} \\ \end{array} }}\right).\end{align*} The eigenstates $\left \vert {{\psi }}\right \rangle =(1,0,0,0)$ and $\left \vert {{\varphi }}\right \rangle =(0,0,0,1)$ are common to $A_{1}$ and $A_{2}$ , so by the lemma 3 the eigenstates $\left \vert {{\psi }}\right \rangle$ and $\left \vert {{\varphi }}\right \rangle$ are fixed points of the quantum channel. Therefore, $\left \vert {{\psi }}\right \rangle \left \langle {{\psi }}\right \vert, \left \vert {{\varphi }}\right \rangle \left \langle {{\varphi }}\right \vert \in {\mathcal {M}}_{\mathcal {E}}$ , i.e., $|{\mathcal {M}}_{\mathcal {E}}| \geq 2$ and by the Proposition 11 we have that $C^{(0)}(\mathcal {E})\geq \log 2$ .

In this paper, we established a relationship between Shemesh theorem and zero-error quantum information theory. We first defined a quantum channel using Kraus operators on a finite-dimensional Hilbert space and introduced the concept of zero-error capacity. We particularly emphasized a test for zero-error capacity based on the eigenstates common to the Kraus operators. We then analyzed Shemesh theorem and demonstrated its connection to zero-error quantum information theory by proving that every eigenstate common to the Kraus operators, which serves as a fixed point of the channel, belongs to the Shemesh subspace. It is also important to recognize that there may be additional fixed points within the Shemesh subspace beyond those eigenstates common to the Kraus operators. Given that a lower bound for the zero-error capacity of a quantum channel can be associated with the number of its fixed points, we highlight the connection between Shemesh theorem and zero-error quantum information theory, particularly in the context of zero-error capacity.