Appendix1.1 Pinching Maps

We begin with the following lemma, known as the pinching inequality by Hayashi [46], which presents a crucial inequality in quantum information and we frequently employ in our work.

Lemma 10

(Pinching inequality [46], see also [47]) Let \(\rho \) and \(\sigma \) be two quantum states and let \(}_\sigma \) be the pinching operation with respect to eigendecomposition of \(\sigma \). We have

$$\begin \rho \le \nu }_(\rho ), \end$$

where \(\nu \) is the number of distinct eigenvalues of the operator \(\sigma \).

Next, we define nested pinching maps on a single Hilbert space. Consider the following tripartite states:

$$\begin&\rho ^ = \sum _p(u,v)|u \rangle \!\langle u|\otimes |v \rangle \!\langle v|\otimes \rho ^_,\\&\rho ^ = \sum _p(u,v)|u \rangle \!\langle u|\otimes |v \rangle \!\langle v|\otimes \rho ^_,\\&\rho ^ = \sum _p(u,v)|u \rangle \!\langle u|\otimes |v \rangle \!\langle v|\otimes \rho ^_, \end$$

where U and V are classical, while B is quantum. Let us denote the dimension of systems U, V and B by \(d_U,d_V\), and \(d_B\), respectively. We further define three pinching maps as follows: A pinching map with respect to the eigenspace decomposition of \(\rho ^\), denoted by \(}^\). If we apply this pinching map to the state \(\rho ^\), we obtain

$$\begin }^(\rho ^) = \sum _p(u,v)|u \rangle \!\langle u|\otimes |v \rangle \!\langle v|\otimes }^(\rho ^_). \end$$

We are now interested in a pinching map with respect to the spectral decomposition of \(}^(\rho ^)\). Notice that now we need to define a pinching map with respect to a collection of states \(\}^(\rho ^_u)\}_u\) instead of one state. In other words, we have \(d_U\) different pinching maps rather than one. Hence, for every \(u\in }\), we define a pinching map with respect to the spectral decomposition of \(}^(\rho ^_u)\) and denote it by \(}_^\). In our nested pinching technique, this pinching map is extended to a pinching map on UB rather than a map on B alone, even though its action on U is trivial. Hence, we define the pinching map on UB with respect to the spectral decomposition of \(}^(\rho ^)\) as follows:

$$\begin }_^=\sum _|u \rangle \!\langle u|\otimes }_^. \end$$

Applied to an arbitrary state \(\sigma ^\), the pinching map \(}_^\) acts on \(\sigma _u^\). It is important to note that the pinching map \(}_1^\) is an operator on composite systems UB. For pedagogical purposes, let us consider the action of this map on the tensor product state \(\sum _q(u)|u \rangle \!\langle u|\otimes \sigma ^\) (notice that this is not a cq-state, as \(\sigma ^B\) does not depend on u). We see that the resulting pinched state \(}^\) is generically a cq-state on UB as follows:

$$\begin }^&= }_1^\Big (\sum _q(u)|u \rangle \!\langle u|\otimes \sigma ^\Big ) \\&=\sum _q(u)|u \rangle \!\langle u|\otimes }_^(\sigma ^)\\&= \sum _q(u)|u \rangle \!\langle u|\otimes \tilde^. \end$$

In other words, pinching map turned the product states into a cq-state. Let us consider pinching the state \(\rho ^\) with the operator \(}_1^\). Note that in the latter state, the system B is independent of U conditioned on V. However, after the application of the pinching map, we obtain a generic cq-state on UVB as follows:

$$\begin }^&=}_1^(\rho ^) \\&= \sum _p(u,v)|u \rangle \!\langle u|\otimes |v \rangle \!\langle v|\otimes }_^(\rho ^_)\\&= \sum _p(u,v)|u \rangle \!\langle u|\otimes |v \rangle \!\langle v|\otimes \tilde_^, \end$$

where for every \(u\in }\) and \(v\in }\), we define \(\tilde_^=}_^(\rho ^_)\). Now, we aim to define another pinching map with respect to the spectral decomposition of the state \(}^\). This pinching map is defined with respect to the spectral decomposition of the operators \(\tilde_^=}_^(\rho ^_)\) for every \((u,v)\in }\times }\). Let us denote the pinching map with respect to \(}_^(\rho ^_)\) by \(}_^\). This map is extended to a pinching map on UVB as follows:

$$\begin }_^ = \sum _|u \rangle \!\langle u|\otimes |v \rangle \!\langle v|\otimes }_^. \end$$

The number of distinct eigenvalues of an operator whose spectral decomposition gives rise to the pinching map plays a significant role in asymptotic analysis of our one-shot codes. In case of \(}^\), this number corresponds to the distinct eigenvalues of the operator \(\rho ^\). In case of \(}_1^\), we deal with operators \(\}^(\rho ^_u)\}_u\) and we are interested in the maximum number of distinct eigenvalues of these operators over \(u\in }\). Similarly, in case of \( }_^\), we deal with operators \(\}^_(\rho ^_v)\}_\) and we are interested in the maximum number of distinct eigenvalues of these operators for \((u,v)\in (},})\).

We can now proceed to construct analogous pinching maps when there are n identical copies of the aforementioned operators. But the important point is to show how the number of the distinct eigenvalues of the operators in n-fold space scales with n. More precisely, we want to find upper bounds on the maximum number of the distinct eigenvalues of the operators \((\rho ^)^\), \(\}^\left( \rho ^_\right) \}_\), and \(\}^_\left( \rho ^_\right) \}_\), where \(}^\) is the pinching map with respect to the spectral decomposition of \((\rho ^)^\), and \(}^_\) is the pinching map with respect to the spectral decomposition of \(\}^\left( \rho ^_\right) \}_\). We aim to demonstrate that the number of distinct eigenvalues in these scenarios is polynomial in the number of tensor product spaces.

The proof of these polynomial upper bounds on the number of distinct eigenvalues is based on the Schur–Weyl duality [48, Sec. 4.4.1], [49, Sec. 6.2.1]. Schur–Weyl duality is a relation between the finite-dimensional irreducible representations of the general linear group \(GL(}^)\) and the symmetric group \(S_n\). Any finite-dimensional irreducible representation of the general linear group \(GL(}^)\) has the same form as an irreducible representation of the special unitary group \(SU_d\). It states that for any \(n,d \in }\), the \((n \times d)\)-dimensional complex n-fold tensor product vector space \((}^)^\) decomposes into a direct sum of irreducible representations of the general linear group \(GL(}^)\) and the n-th symmetric group \(S_n\). These groups in fact identify each other via Young diagrams: When the vector of integers \(\lambda = (\lambda _1, \lambda _2, \ldots , \lambda _d)\) satisfies the condition \(\lambda _1 \ge \lambda _2 \ge \cdots \ge \lambda _d \ge 0\) and \(\sum _^ \lambda _i = n\), the vector \(\lambda \) is called a Young diagram (frame) of size n and depth d; the set of such vectors of size n and depth (at most) d is denoted as \(Y_^\). It can be shown that

$$\begin |Y_^|\le \le (n+1)^. \end$$

Lemma 11

(Schur–Weyl Duality) Using the Young diagram, the irreducible decomposition of \((}^)^\) can be described as follows:

$$\begin (}^)^ = \bigoplus _ }_(SU_d) \otimes }_(S_n), \end$$

where for an element \(\lambda \in Y_^\), \(}_(SU_d)\) is the irreducible representation space of the special unitary group \(SU_d\) characterized by \(\lambda \), and \(}_(S_n)\) is the irreducible representation space of the nth symmetric group \(S_n\) characterized also by \(\lambda \).

Schur–Weyl decomposition provides a profound insight into the structure of tensor product Hilbert spaces, revealing that any operator within this space can be expressed as irreducible representations of both \(SU_d\) and \(S_n\). Specifically, any tensor product state residing in \(B^\) takes the form:

$$\begin (\rho ^B)^ = \sum _^} p_\lambda \rho _\lambda \otimes \rho _, U_\lambda (S_)} \end$$

where \(\rho _\lambda \) is a state on the space \(}_(SU_d)\) and \(\rho _, U_\lambda (S_)}\) is the completely mixed state on \( }_(S_n)\), and \(p_\lambda \) is a probability distribution on \(Y_^\). In other words, \((\rho ^B)^\) can be written as

$$\begin (\rho ^B)^ = \sum _^} X_\lambda \otimes I_}_\lambda (S_)}, \end$$

(40)

where \(X_\lambda \) is an operator on \(}_\lambda (SU_)\) and \(I_}_\lambda (S_)}\) is the identity operator on \(}_\lambda (S_)\). We now present the polynomial upper bounds in the following lemma.

Proposition 12

Consider the cq-states \(\rho ^,\rho ^,\rho ^\), and \(\rho ^\), classical on U, V, X and quantum on B. Let \(\nu ,\nu _1,\nu _2\) and \(\nu _3\) be defined as follows:

$$\begin \nu&= \left\\hspace(\rho ^B)^\right\} ,\\ \nu _1&= \max _}^n} \hspace }^\big (\rho ^_\big )\right\} },\\ \nu _2&= \max _}^n,v^n\in }^n} \hspace }^_\big (\rho ^_\big )\right\} },\\ \nu _3&= \max _ v^n\in }^n,x^n\in }^n\\ u^n\in }^n \end} \hspace }^_\big (\rho ^_\big )\right\} } \end$$

where \(\rho ^_= \rho _^\otimes \rho _^\otimes \cdots \otimes \rho _^\), \(\rho ^_= \rho _^\otimes \rho _^\otimes \cdots \otimes \rho _^\), \(\rho ^_=\rho _^B\otimes \cdots ,\otimes \rho _^B\), and \(}^\) and \(}^_\) are pinching maps corresponding to the spectral decomposition of the operators \((\rho ^B)^\) and \(}^\left( \rho ^_\right) \), respectively. We show that the number of distinct eigenvalues are polynomial in the number of tensor product Hilbert spaces. More precisely, we obtain:

$$\begin \nu&\le (n+1)^ \\ \nu _1&\le (n+1)^ \\ \nu _2&\le (n+1)^\\ \nu _3&\le (n+1)^ . \end$$

Proof

The values of \(\nu \) and \(\nu _1\) are determined in [37, Lemma 3]. However, the evaluations of \(\nu _2\) and \(\nu _3\) are not provided in the previous work. To find the upper bounds \(\nu _2\) and \(\nu _3\), we build on the techniques used to find \(\nu \) and \(\nu _1\). So we begin with the latter numbers. The number of eigenvalues of \((\rho ^)^\) is upper bounded by \(|Y_^n|\), the cardinality of the Young diagram of size n and depth at most \(d_B\). The latter can be bounded by the number of combinations of \(d_B-1\) objects from \(n+d_B-1\) elements, which itself is upper bounded by \((n+1)^\). This follows from counting the number of different type classes in information theory [50] (for further details between Schur duality and type classes see [51, Sec. 3], [49, Sec. 6.2.1] and [52, Chapter 6]). Next, our focus turns to \(\nu _1\). Suppose \(u^n\) takes the following form:

$$\begin u^n = (\underbrace_, \underbrace_, \ldots , \underbrace, \ldots , u_}_}). \end$$

(41)

Accordingly, we have \(\rho ^_=(\rho ^B_)^\otimes \cdots \otimes (\rho ^B_})^}\). For \(j=1, \ldots , d_U\), from Schur–Weyl duality given in Lemma 11, the tensor product Hilbert space \(B^\) is decomposed as

$$\begin \bigoplus _^} }}}_\lambda (SU_) \otimes }_\lambda (S_), \end$$

where \(Y_d^n\) is the set of Young indexes of size n and depth at most \(d_B\). Let \(}}}_j\) be the pinching map on \(B^\) with respect to the above Schur–Weyl decomposition. Hence, \(}_j\) is described by the following projectors:

$$\begin \}}}_\lambda (SU_) }\otimes I_}_\lambda (S_)} \}_^}, \end$$

(42)

where \(I_}}}_\lambda (SU_) }\) and \(I_}_\lambda (S_)}\) are the identity operators on the subspaces corresponding to \(}}}_\lambda (SU_)\) and \(}_\lambda (S_)\), respectively. Notice that this decomposition does not depend on \((\rho ^_)^\), nor on the eigenvectors of \(\rho ^B_\) (recall that Schur–Weyl duality is a statement generally on the tensor product space). Therefore, the pinching map \(}_j\) defined by operators in Eq. (42), commutes with all tensor product operators residing in \(B^\), such as \((\rho ^B_)^\) or \((\rho ^B)^\). Therefore, the pinching map \(}^ = }}}_1 \otimes \cdots \otimes }}}_\) on \(B^\) has no effect on \((\rho ^B)^\). To put it another way, if \(\_i\) and \(\_j\) are the pinching projectors with respect to \(}^\) and \(}^\), respectively, \(F_iE_j=E_jF_i\) for every i and j.

In order to further understand the pinching map with respect to the spectral decomposition of the operator \(}^(\rho _^)\), we rewrite this operator as follows:

$$\begin }^(\rho _^)= }^}\left( (\rho _^B)^\right) \otimes }^}\left( (\rho _^B)^\right) \otimes \cdots \otimes }^}}\left( (\rho _}^B)^}\right) . \end$$

For any \(j=1,2,\ldots ,d_U\), based on Eq. (40), we can express

$$\begin (\rho _^B)^&=\sum _^} X_\lambda \otimes I_)},\\ (\rho ^B)^&=\sum _^} Z_\lambda \otimes I_)}, \end$$

where \(X_\lambda \) and \(Z_\lambda \) are operators acting on \(}_\lambda (SU_)\). Moreover, since the pinching map \(}_j\) commutes with both operators, we have \(}^}\bigl (}_j\big ((\rho _^B)^\big )\bigl )= }_j\bigl (}^}\big ((\rho _^B)^\big )\bigl )\). This means that in fact

$$\begin }^}\big ((\rho _^B)^\big )=\sum _^} T_\lambda \otimes I_)} \end$$

for an operator \(T_\lambda \) on \(}_\lambda (SU_)\). Now considering the tensor product operators, since

$$\begin }^\big (\rho _^}\big ) = }^ \bigl ((}}}_1 \otimes \cdots \otimes }_) \big (\rho _^}\big )\bigl ) = (}}}_1 \otimes \cdots \otimes }}}_) \big (}^\big (\rho _^}\big )\big ), \end$$

the operator \(}^\big (\rho _^}\big ) \) takes the form

$$\begin }^\big (\rho _^}\big )= \sum _^, \ldots , \lambda _ \in Y_^}} Q_} \otimes I_}}}_(S_)} \otimes \cdots \otimes I_}}}_}(S_})} \end$$

with an operator \(Q_}\) on \(}}_(SU_)\otimes \cdots \otimes }}_}(SU_)\). We can now find an upper bound on the number of the eigenvalues of the state. Note that the identity operators \(\}}}_}(S_})}\}_i\) do not increase the number of eigenvalues, therefore we focus on the operators \(Q_}\). First notice that the number of these operators in the summation equals the number of choices of \((\lambda _1, \ldots , \lambda _)\), which is upper bounded by \((n+1)^\). The dimension of \(}}_(SU_)\) is upper bounded by \((n_j+1)^\) [49, (6.16)]. Since \((n_j+1)^ \le (n+1)^\), the dimension of \(}}}_(SU)\otimes \cdots \otimes }}_}(SU)\) is upper bounded by \((n+1)^\). Bringing together these findings, the number of eigenvalues of \(}^\big (\rho _^}\big )\) is upper bound by \((n+1)^ \cdot (n+1)^ =(n+1)^ \). A general sequence \(}^n\) can be written as the application of a certain permutation \(g \in S_n\) to the above-mentioned sequence in Eq. (41). In this case, the finding for that particular choice of \(u^n\) also holds for a general \(}^n\) after application of \(g \in S_n\). Therefore, the pinching \(}^\) becomes \(U_g (}}}_1 \otimes \cdots \otimes }}}_) U_g^\dagger \). That is, this pinching depends only on an element of \(d_U\)-nomial combinatorics among n elements.

We are now in a position to find an upper bound on \(\nu _2\). We again assume that \(u^n\) has the particular form given by Eq. (41), which we repeat here for clarity:

$$\begin u^n = (\underbrace_, \underbrace_, \ldots , \underbrace, \ldots , u_}_}). \end$$

(43)

We next make an assumption about the structure of the sequence \(v^n\). We suppose the first \(n_1\) elements of \(v^n\) are arranged as follows:

$$\begin (\underbrace_}, \underbrace_}, \ldots , \underbrace, \ldots , v_}_}), \end$$

such that \(\sum _^n_=n_1\). We assume a similar structure for the subsequent \(n_2\) elements, followed by the next \(n_3\) elements, and so forth, up to the final \(n_\) elements. So the sequence \(v^n\) is written as follows:

$$\begin&\big ( \overbrace_}, \underbrace_}, \ldots , \underbrace, \ldots , v_}_}}^n_1\text }, \overbrace_}, \ldots , \underbrace, \ldots , v_}_}}^n_2\text },\nonumber \\&\quad \ldots ,\overbrace_}, \ldots , \underbrace, \ldots , v_}_}}^n_\text }\big ). \end$$

(44)

Consider the first \(n_\) elements of both \(u^n\) and \(v^n\). Let \(}_\) be the pinching map with respect to the Schur–Weyl decomposition of the tensor product space \(B^}\). Analogous to Eq. (42), \(}_\) is described by the following projectors:

$$\begin \}}}_\lambda (SU_) }\otimes I_}_\lambda (S_})} \}_^}}, \end$$

where \(I_}}}_\lambda (SU_) }\) and \(I_}_\lambda (S_})}\) are the identity operators on the subspaces corresponding to \(}}}_\lambda (SU_)\) and \(}_\lambda (S_})\), respectively. If we denote the pinching map with respect to the spectral decomposition of \(}^}}\big (\rho _}}^}}}\big )\) by \(}^}}_^}}\) (which is the notation we have introduced before), from the structure of the operators \(}^}}_^}}\equiv }^}}\big (\rho _}}^}}}\big )\) and \(\rho _^}}^}\), we know that

$$\begin }^}}_^}}\circ }_ (\rho _^}}^}) = }_\circ }^}}_^}}(\rho _^}}^}). \end$$

We proceed with this procedure for each subset of elements within \(v^n\), as indicated by the underbraces in Eq. (44). Let \(}_\) denote the pinching map according to the Schur–Weyl decomposition of the tensor product Hilbert space \(B^}\), so a pinching map on \(B^\) is defined as \(}}}_ \otimes }}}_ \otimes \cdots \otimes }}}_ \). We have:

$$\begin }^_\big (\rho _^}\big )&= }^_\circ ( }}}_ \otimes }}}_ \otimes \cdots \otimes }}}_ )\big (\rho _^}\big )\\&= ( }}}_ \otimes }}}_ \otimes \cdots \otimes }}}_ )\circ }^_ \big (\rho _^}\big ). \end$$

We can now apply the same evaluation as for \(\nu _1\). We find that the number of eigenvalues of \(}^_\big (\rho _^}\big )\) is upper bound by \((n+1)^ \).

Next, we obtain an upper bound on \(\nu _3\), the maximum number of distinct eigenvalues of the operator \(}^_\big (\rho ^_\big )\). We again assume that the sequences \(u^n\) and \(v^n\) have the structure given by Eqs. (43) and (44). Now, we assume that the first \(n_\) the sequence \(x^n\) have the following structure:

$$\begin (\underbrace_}, \underbrace_}, \ldots , \underbrace, \ldots , x_}_}), \end$$

such that \(\sum _^n_=n_\). Therefore, similar to Eq. (44), we can write \(x^n\) as follows:

$$\begin&\overbrace_}, \underbrace_}, \ldots , \underbrace, \ldots , x_}_}}^n_\text v^n}, \overbrace_}, \ldots , \underbrace, \ldots , x_}_}}^n_\text v^n },\ldots , \overbrace_}, \ldots , \underbrace, \ldots , x_}_}}^n_\text n_}}^n_1\text u^n} \end$$

where \(\sum _^n_=n_\), \(\sum _^\sum _^n_=n_\), and \(\sum _^n_=n\). Now, considering the first \(n_\) elements of \(x^n\), we need to find an upper bound on the distinct number of eigenvalues of the operator \(}_}}\left( \rho ^}}_^}}\right) \). By using the lessons we learned in the previous evaluations, we obtain

$$\begin (n_+1)^\times (n_+1)^&\le (n+1)^\times (n+1)^\\&=(n+1)^. \end$$

This will be repeated for the next \((d_X-1)\) sub-sequences corresponding to \((x_2,\ldots ,x_2),\ldots ,(x_,\ldots ,x_)\), so that for the first \(n_\) elements of \(v^n\), the number of distinct eigenvalues are upper bounded by \((n+1)^\). This will again be repeated for the subsequent \(n_\) elements of \(v^n\), and so forth, yielding \((n+1)^\). So far we have considered the first \(n_1\) elements of \(u^n\); eventually by taking into account the rest of \(d_U-1\) sub-sequences of \(u^n\), we obtain the upper bound \((n+1)^\). This concludes the proof. \(\square \)

1.2 Binary Hypothesis-Testing

We next study a binary hypothesis-testing problem between a null state \(\rho \) and an alternative state \(\sigma \), where the corresponding two-outcome POVM is constructed using pinching maps. For two Hermitian operators T and O, consider the operator \((T-O)\) with the following eigenspace decomposition

$$\begin T - O=\sum _g_i P_i, \end$$

where \(P_i\) is the projector onto the eigenspace corresponding to the eigenvalue \(g_i\). We define the operator \(\\) as

$$\begin \:=\sum _ P_i. \end$$

We use this definition to define the projector \(\Pi :=\}_(\rho )\ge M\sigma \}\), where \(}_\) is the pinching map with respect to the spectral decomposition of the state \(\sigma \) and M is a positive number. Projectors of this kind are naturally encountered in the derivation of coding theorems. In particular, when we employ Hayashi–Nagaoka inequality Lemma 1 in the coding theorem, two terms emerge. The first term denotes the probability of making an error in detecting the true state (type I error), while the second term encompasses all other codewords, represented collectively by \(\sigma \) due to random coding, are erroneously identified as true (type II error). We now find an upper bound on combination of these errors using the aforementioned projector.

Lemma 13

Consider the binary POVM \( \\) constructed from the projector \(\Pi :=\}_(\rho )\ge M\sigma \}\), where \(}_\) is the pinching map with respect to the spectral decomposition of the state \(\sigma \) and M is a positive number. The following holds for \(\alpha \in (0,1)\):

$$\begin \textrm\,(I-\Pi )\rho +M\textrm\,\Pi \sigma&\le M^2^(}_(\rho )\Vert \sigma )}. \end$$

Proof

We find an upper bound on each of the terms on the left-hand-side as follows: Note that we adopt the notation \(}_(\rho )^=\left( }_(\rho )\right) ^\). For the first term we obtain:

$$\begin \textrm\,(I-\Pi )\rho&\limits ^}}\textrm\,\}_\sigma (I-\Pi )\big )\rho \}\\&\limits ^}}\textrm\,(I-\Pi )}_\sigma (\rho )\\&=\textrm\,(I-\Pi )}_\sigma (\rho )^}_\sigma (\rho )^\\&\limits ^}} \textrm\,(I-\Pi )}_\sigma (\rho )^(M\sigma )^, \end$$

where (a) follows from the fact the projector \(\Pi \) commutes with the operator \(\sigma \), thus the pinching map \(}_\sigma \) has no effect on the projector, (b) uses the cyclicity of the pinching map inside the trace, and (c) follows from the definition of the pinching projector as well as the operator monotonicity of the function \(x^\) for \(\alpha \in (0,1)\). For the second term, we obtain:

$$\begin \textrm\,\Pi \sigma&=\textrm\,\Pi \sigma ^\sigma ^\alpha \\&\le \textrm\,\Pi (M^}_\sigma (\rho ))^\sigma ^\alpha , \end$$

where the inequality follows from the definition of the projector and the monotonicity of the function \(x^\). Summing up, we obtain:

$$\begin \textrm\,(I-\Pi )\rho +M\textrm\,\Pi \sigma \le M^\alpha \textrm\,}_\sigma (\rho )^\sigma ^\alpha =M^2^(}_\sigma (\rho )\Vert \sigma )}. \end$$

\(\square \)

In the previous lemma, the error probability is upper bounded in terms of the Petz’s Rényi entropy where the first argument involves a pinching map. The following lemma demonstrates that removing the pinching from the first argument is possible, but the Petz’s Rényi entropy transforms to the Sandwich Rényi entropy.

Lemma 14

Let \(}_\) be the pinching map with respect to the spectral decomposition of the operator \(\sigma \). The following inequality holds:

$$\begin }_ (\rho \Vert \sigma ) \le \log \nu + D_(}_(\rho )\Vert \sigma ), \end$$

where \(\nu \) is the number of the distinct eigenvalues of the operator \(\sigma \). It is useful to write this inequality equivalently as follows:

$$\begin 2^(}_(\rho )\Vert \sigma )}\le \nu ^ 2^}_ (\rho \Vert \sigma ) }. \end$$

Proof

From the pinching inequality in Lemma 10, \(\rho \le \nu }_(\rho )\), we have

$$\begin \sigma ^} \rho \sigma ^}\le \nu \sigma ^} }_(\rho ) \sigma ^}. \end$$

Since \(x \mapsto -x^\) is operator monotone for \(\alpha \in (0,1)\), we have

$$\begin \Big (\nu \sigma ^} }_\sigma (\rho ) \sigma ^}\Big )^ \le \Big (\sigma ^}\rho \sigma ^}\Big )^. \end$$

(45)

Similar to Eq. (65) of [37], we have

$$\begin 2^ (}_\sigma (\rho )\Vert \sigma ) }&=\textrm\,}_\sigma (\rho )^ \sigma ^\alpha \\&= \textrm\,\Big (\sigma ^} }_\sigma (\rho ) \sigma ^} \Big )^ \\&= \textrm\,\Big (\sigma ^} }_\sigma (\rho ) \sigma ^} \Big ) \Big (\sigma ^} }_\sigma (\rho ) \sigma ^} \Big )^ \\&\limits ^a)}} \textrm\,\Big (}_\sigma \Big (\sigma ^} \rho \sigma ^} \Big )\Big ) \Big (\sigma ^} }_\sigma (\rho ) \sigma ^} \Big )^ \\&\limits ^b)}} \textrm\,\Big (\sigma ^} \rho \sigma ^} \Big ) \Big (\sigma ^} }_\sigma (\rho ) \sigma ^} \Big )^ \\&\limits ^c)}} \nu ^ \textrm\,\Big (\sigma ^} \rho \sigma ^} \Big ) \Big (\sigma ^} \rho \sigma ^} \Big )^ \\&=\nu ^ \textrm\,\Big (\sigma ^} \rho \sigma ^} \Big )^\\&= \nu ^ 2^}_ (\rho \Vert \sigma ) }, \end$$

where (a) follows from \( \sigma ^} }_\sigma (\rho ) \sigma ^}=}_\sigma \left( \sigma ^} \rho \sigma ^} \right) \), (b) follows from the cyclicity of pinching operator inside trace, and (c) follows from (45). \(\square \)

Proposition 15

Consider the following four cq-states

$$\begin \rho ^&= \sum _p(u,v,x)|u \rangle \!\langle u|\otimes |v \rangle \!\langle v|\otimes |x \rangle \!\langle x|\otimes \rho ^_,\\ \rho ^&= \sum _p(u,v,x)|u \rangle \!\langle u|\otimes |v \rangle \!\langle v|\otimes |x \rangle \!\langle x|\otimes \rho ^_,\\ \rho ^&= \sum _p(u,v,x)|u \rangle \!\langle u|\otimes |v \rangle \!\langle v|\otimes |x \rangle \!\langle x|\otimes \rho ^_,\\ \rho ^&= \sum _p(u,v,x)|u \rangle \!\langle u|\otimes |v \rangle \!\langle v|\otimes |x \rangle \!\langle x|\otimes \rho ^_. \end$$

Let \(}^\), \(}^_1\), \(}^_2\), and \(}^_3\) be the pinching maps with respect to the spectral decompositions of the operators \(\rho ^\), \(}^\left( \rho ^\right) \), and \(}^_1\left( \rho ^\right) \), and \(}^_2(\rho ^)\), respectively. The following inequalities hold:

$$\begin 2^ \left( }}}_2^(\rho ^)\Vert }_1^(\rho ^)\right) }&\le \nu _2^\alpha 2^}_ \left( \rho ^\Vert }_1^(\rho ^)\right) } \\ 2^ \left( }_3^(\rho ^)\Vert }_2^(\rho ^)\right) }&\le \nu _3^\alpha 2^}_ \left( \rho ^\Vert }_2^( \rho ^)\right) }, \end$$

where \(\nu _2\) and \(\nu _3\) are the number of the distinct eigenvalues of the operators \(}^_1\left( \rho ^\right) \) and \(}^_2(\rho ^)\), respectively.

Proof

The proofs resemble those of Lemma 14. We defer them to the interested reader. \(\square \)