One-Shot Randomized and Nonrandomized Partial Decoupling

Decoupling refers to the fact that we may destroy correlation between two quantum systems by applying an operation on one of the two subsystems. It has played significant roles in the development of quantum Shannon theory for a decade, particularly in proving the quantum capacity theorem [1], unifying various quantum coding theorems [2], analyzing a multipartite quantum communication task [3, 4] and in quantifying correlations in quantum states [5, 6]. It has also been applied to various fields of physics, such as the black hole information paradox [7], quantum many-body systems [8] and quantum thermodynamics [9, 10]. Dupuis et al. [11] provided one of the most general formulations of decoupling, which is often referred to as the decoupling theorem. The decoupling approach simplifies many problems of our interest, mostly due to the fact that any purification of a mixed quantum state is convertible to another reversibly [12].

All the above studies rely on the notion of random unitary, i.e., unitaries drawn at random from the set of all unitaries acting on the system, which leads to the full randomization over the whole Hilbert space. In various situations, however, the full randomization is a too strong demand. In the context of communication theory, for example, the full randomization leads to reliable transmission of quantum information, while we may be interested in sending classical information at the same time [13], for which the full randomization is more than necessary. In the context of quantum many-body physics, the random process caused by the complexity of dynamics is in general restricted by symmetry, and thus no randomization occurs among different values of conserved quantities. Hence, in order that the random-unitary-based method fits into broader context in quantum information theory and fundamental physics, it would be desirable to generalize the previous studies using the full-random unitary, to those based on random unitaries that are not fully random but with a proper structure.

As the first step toward this goal, we consider a scenario in which the unitaries take a simple form under the following direct-sum-product (DSP) decomposition of the Hilbert space:

Here, the superscripts l and r stand for “left” and “right”, respectively, and j is the index of the diagonal subspaces. This decomposition often appears in the context of quantum information theory, such as information-preserving structure [14, 15], the Koashi-Imoto decomposition [16], data compression of quantum mixed-state source [17], quantum Markov chains [18, 19] and simultaneous transmission of classical and quantum information [13]. Also, quantum systems with symmetry are represented by the Hilbert spaces decomposed into this form (see e.g. [20]), in which case j is the label of irreducible representations of a compact group G, \({\mathcal {H}}_j^l\) is the representation space and \({\mathcal {H}}_j^r\) is the multiplicity space for each j.

In this paper, we introduce and analyze a task that we call partial decoupling. We consider a scenario in which a bipartite quantum state \(\Psi \) on system AR is subject to a unitary operation U on A, followed by the action of a quantum channel (CP map) \({\mathcal {T}}:A\rightarrow E\). The unitary is assumed to be chosen at random, not from the set of all unitaries on A, but from the subset of unitaries that take a simple form under the DSP decomposition. Thus, partial decoupling is a generalization of the decoupling theorem [11] that incorporates the DSP decomposition. Along the similar line as [11], we analyze how close the final state \({\mathcal {T}}^{A \rightarrow E} (U^A \Psi ^{AR} U^{\dagger A})\) is, on average over the unitaries, to the averaged final state \({\mathbb {E}}_{U} [ {\mathcal {T}}^{A \rightarrow E}(U^A ( \Psi ^{AR} ) U^{\dagger A})]\).

The main result in this paper is that we derive upper and lower bounds on the average distance between the final state and the averaged one. The bounds are represented in terms of the smooth conditional entropies of quantum states involving the initial state, the channel and the decomposition. For a particular case where \(J=1\) and \(\dim {{\mathcal {H}}_j^{A_l}}=1\), the obtained formulae are equivalent to those given by the decoupling theorem [11].

The result in this paper is applicable for generalizing any problems within the scope of the decoupling theorem by incorporating the DSP structure. Some of the applications are investigated in our papers [21,22,23,24].

In Refs. [21,22,23], we investigate communication tasks between two parties in which the information to be transmitted has both classical and quantum components. In this case, the Hilbert space \({\mathcal {H}}_j^l\) in  (1) is assumed to be a one-dimensional space \({\mathbb {C}}\), and \({\mathcal {H}}_j^l\) to be the spaces with the same dimension for all j:

Here, \(j \in [1, J]\) and \(\mathcal{H}_j^r\) correspond to the degrees of freedom related to classical and quantum components of the information to be transmitted, respectively. We investigate the tasks of channel coding in [21, 22] and source coding in [23] in the one-shot regime. Based on the result in this paper, we obtain general trade-off relations among the resources of classical communication, quantum communication and entanglement for those tasks.

In Ref. [24], we apply the result of partial decoupling to investigate the information paradox of quantum black holes with symmetry. Our analysis is based on the framework of Hayden–Preskill model [7], where a decoupling technique is used under the postulate that the internal dynamics of the system is given by a fully random unitary. This postulate should be modified when the system has symmetry since the dynamics cannot be fully random due to a conserved quantity. By letting j be the labeling of the conserved quantity, the internal dynamics randomizes only the multiplicity spaces \(\{ {\mathcal {H}}_j^r\}\) and should be in the form of

where \(I_j^l\) is the identity on \({\mathcal {H}}_j^l\) and \(U_j^r\) is a random unitary on \({\mathcal {H}}_j^r\). Hence, this case is also in the scope of partial decoupling with a DSP decomposition given by the symmetry. Similarly, all physical phenomena investigated based on decoupling [7,8,9,10,11] can be lifted up by partial decoupling to the situation with symmetry. We think that further significant implications on various topics will be obtained beyond these examples.

This paper is organized as follows. In Sect. 2, we introduce notations and definitions. In Sect. 3, we present formulations of the problem and the main results. Before we prove our main results, we provide discussions about implementations of our protocols by quantum circuits in Sect. 4. Section 5 describes the structure of the proofs of the main results, and provides lemmas that will be used in the proofs. The detailed proofs of the main theorems are provided in Sects. 6–8. Conclusions are given in Sect. 9. Some technical lemmas and proofs are provided in Appendices.

We summarize notations and definitions that will be used throughout this paper. See also Appendix H for the list of notations.

2.1 Notations

We denote the set of linear operators and that of Hermitian operators on a Hilbert space \({\mathcal {H}}\) by \({\mathcal {L}}({\mathcal {H}})\) and \(\mathrm{Her}({\mathcal {H}})\), respectively. For positive semidefinite operators, density operators and sub-normalized density operators, we use the following notations, respectively:

A Hilbert space associated with a quantum system A is denoted by \({{\mathcal {H}}}^A\), and its dimension is denoted by \(d_A\). A system composed of two subsystems A and B is denoted by AB. When M and N are linear operators on \({{\mathcal {H}}}^A\) and \({{\mathcal {H}}}^B\), respectively, we denote \(M\otimes N\) as \(M^A\otimes N^B\) for clarity. In the case of pure states, we often abbreviate \(|\psi \rangle ^A\otimes |\phi \rangle ^B\) as \(|\psi \rangle ^A|\phi \rangle ^B\). For \(\rho ^{AB} \in {\mathcal {L}}({\mathcal {H}}^{AB})\), \(\rho ^{A}\) represents \(\mathrm{Tr}_B[\rho ^{AB}]\). We denote \(|\psi \rangle \!\langle \psi |\) simply by \(\psi \). The maximally entangled state between A and \(A'\), where \({\mathcal {H}}^{A} \cong {\mathcal {H}}^{A'}\), is denoted by \({|\Phi \rangle }^{AA'}\) or \(\Phi ^{AA'}\). The identity operator is denoted by I. We denote \((M^A\otimes I^B){|\psi \rangle }^{AB}\) as \(M^A{|\psi \rangle }^{AB}\), and \((M^A\otimes I^B)\rho ^{AB}(M^A\otimes I^B)^{\dagger }\) as \(M^A\rho ^{AB}M^{A\dagger }\).

When \({{\mathcal {E}}}\) is a supermap from \({\mathcal {L}}({\mathcal {H}}^{A})\) to \({\mathcal {L}}({\mathcal {H}}^{B})\), we denote it by \({\mathcal {E}}^{A \rightarrow B}\). When \(A = B\), we use \({\mathcal {E}}^{A}\) for short. We also denote \(({{\mathcal {E}}}^{A \rightarrow B} \otimes \mathrm{id}^C)(\rho ^{AC})\) by \({{\mathcal {E}}}^{A \rightarrow B} (\rho ^{AC})\). The set of linear completely-positive (CP) supermaps from A to B is denoted by \({\mathcal {C}}{\mathcal {P}}(A\rightarrow B)\), and the subset of trace non-increasing (resp. trace preserving) ones by \({\mathcal {C}}{\mathcal {P}}_\le (A\rightarrow B)\) (resp. \({\mathcal {C}}{\mathcal {P}}_=(A\rightarrow B)\)). When a supermap is given by a conjugation of a unitary \(U^A\) or an isometry \(W^{A \rightarrow B}\), we especially denote it by its calligraphic font such as

Let A be a quantum system such that the associated Hilbert space \({\mathcal {H}}^A\) is decomposed into the DSP form as

For the dimension of each subspace, we introduce the following notation:

We denote by \(\Pi _j^A\) the projection onto a subspace \({\mathcal {H}}_j^{A_l}\otimes {\mathcal {H}}_j^{A_r}\subseteq {\mathcal {H}}^A\) for each j. For any quantum system R and any \(X\in {\mathcal {L}}({\mathcal {H}}^A\otimes {\mathcal {H}}^R)\), we introduce a notation

which leads to \(X^{AR}=\sum _{j,k=1}^JX_{jk}^{AR}\).

2.2 Norms and distances

For a linear operator X, the trace norm is defined as \(|\! | X |\! |_1 = {\mathrm {Tr}}[ \sqrt{X^{\dagger }X}]\), and the Hilbert-Schmidt norm as \(|\! | X |\! |_2 = \sqrt{{\mathrm {Tr}}[ X^{\dagger }X]}\). The trace distance between two unnormalized states \(\rho ,\rho '\in {\mathcal {P}}({\mathcal {H}})\) is defined by \(\Vert \rho -\rho '\Vert _1\). For subnormalized states \(\rho ,\rho '\in {\mathcal {S}}_\le ({\mathcal {H}})\), the generalized fidelity and the purified distance are defined by

respectively [25]. The epsilon ball of a subnormalized state \(\rho \in {\mathcal {S}}_\le ({\mathcal {H}})\) is defined by

For a linear superoperator \({\mathcal {E}}^{A \rightarrow B}\), we define the DSP norm by

where the supremum is taken over all finite dimensional quantum systems C and all subnormalized states \(\xi \in {\mathcal {S}}_\le ({\mathcal {H}}^{AC})\) such that the reduced state on A is decomposed in the form of

Here, \(\{q_j\}_{j=1}^J\) is a probability distribution, \(\{\varpi _j\}_{j=1}^J\) is a set of subnormalized states on \({\mathcal {H}}_j^{A_l}\) and \(\pi _j^{A_r}\) is the maximally mixed state on \({\mathcal {H}}_j^{A_r}\). The epsilon ball of linear CP maps with respect to the DSP norm is defined by

For quantum systems V, W, a linear operator \(X\in {\mathcal {L}}({\mathcal {H}}^{VW})\) and a subnormalized state \(\varsigma \in {\mathcal {S}}_\le ({\mathcal {H}}^W)\), we introduce the following notation:

This includes the case where V is a trivial (one-dimensional) system, in which case \(X^{VW}=X^W\). We omit the superscript W for \(\varsigma \) when there is no fear of confusion.

2.3 One-shot entropies

For any subnormalized state \(\rho \in {\mathcal {S}}_\le ({\mathcal {H}}^{AB})\) and normalized state \(\varsigma \in {\mathcal {S}}_=({\mathcal {H}}^{B})\), define

The conditional min-, max- and collision entropies (see e.g. [26]) are defined by

respectively. The smoothed versions are of the key importance when we are interested in the one-shot scenario. We particularly use the smooth conditional min- and max-entropies:

for \(\epsilon \ge 0\). Note that Expressions (17)–(22) can be generalized to the case where \(\rho \in {\mathcal {P}}({\mathcal {H}})\).

2.4 Choi–Jamiołkowski representation

Let \({\mathcal {T}}^{A \rightarrow B}\) be a linear supermap from \({\mathcal {L}}({\mathcal {H}}^A)\) to \({\mathcal {L}}({\mathcal {H}}^B)\), and let \(\Phi ^{AA'}\) be the maximally entangled state between A and \(A'\). A linear operator \({\mathfrak {J}}({\mathcal {T}}^{A \rightarrow B})\in {\mathcal {L}}({\mathcal {H}}^{AB})\) defined by \({\mathfrak {J}}({\mathcal {T}}^{A \rightarrow B}) := {\mathcal {T}}^{A' \rightarrow B}(\Phi ^{AA'})\) is called the Choi–Jamiołkowski representation of \({\mathcal {T}}\) [27, 28]. The representation is an isomorphism. The inverse map is given by, for an operator \(X^{AB} \in {\mathcal {L}}({\mathcal {H}}^{AB})\),

where \(A^T\) denotes the transposition of A with respect to the Schmidt basis of \(\Phi ^{AA'}\). When \({\mathcal {T}}\) is completely positive, then \({\mathfrak {J}}({\mathcal {T}}^{A \rightarrow B})\) is an unnormalized state on AB and is called the Choi–Jamiołkowski state of \({\mathcal {T}}\).

Note that the Choi–Jamiołkowski representation depends on the choice of the maximally entangled state \(\Phi ^{AA'}\), i.e., the Schmidt basis thereof. When \({\mathcal {H}}^{A}\) is decomposed into the DSP form as (8), the isomorphic space \({\mathcal {H}}^{A'}\) is decomposed into the same form. In the rest of this paper, we fix the maximally entangled state \(\Phi ^{AA'}\), which is decomposed as

where \(\Phi _j^l\) and \(\Phi _j^r\) are fixed maximally entangled states on \({\mathcal {H}}^{A_l}_j\otimes {\mathcal {H}}^{A_l'}_j\) and \({\mathcal {H}}^{A_r}_j\otimes {\mathcal {H}}^{A_r'}_j\), respectively.

2.5 Random unitaries

Random unitaries play a crucial role in the analyses of one-shot decoupling. By using them, it can be shown that there exists at least one unitary that achieves the desired task. In particular, the Haar measure on the unitary group is often used. The Haar measure \(\mathsf{H}\) on the unitary group is the unique unitarily invariant provability measure, often called uniform distribution of the unitary group. When a random unitary U is chosen uniformly at random with respect to the Haar measure, it is referred to as a Haar random unitary and is denoted by \(U \sim \mathsf{H}\).

The most important property of the Haar measure is the left- and right-unitary invariance: for a Haar random unitary \(U \sim \mathsf{H}\) and any unitary V, the random unitaries VU and UV are both distributed uniformly with respect to the Haar measure. This property combined with the Schur–Weyl duality enables us to explicitly study the averages of many functions on the unitary group over the Haar measure. In the following, the average of a function f(U) on the unitary group over the Haar measure is denoted by \({\mathbb {E}}_{U \sim \mathsf{H}} [f]\).

In this paper, however, we are interested in the case where the Hilbert space is decomposed into the DSP form: \({\mathcal {H}}^A=\bigoplus _{j=1}^J{\mathcal {H}}_j^{A_l}\otimes {\mathcal {H}}_j^{A_r}\), and mainly consider the unitaries that act non-trivially only on \(\{ {\mathcal {H}}_j^{A_r} \}_{j=1}^{J}\) such as the untiary in the form of \(\bigoplus _{j=1}^J I_j^{A_l}\otimes U_j^{A_r}\), where \(U_j^{A_r}\) is a unitary on \({\mathcal {H}}_j^{A_r}\). In this case, we can naturally introduce a product \(\mathsf{H}_{\times }\) of the Haar measures by

where \(\mathsf{H}_{j}\) is the Haar measure on the unitary group on \({\mathcal {H}}_j^{A_r}\) for any j. Hence, when we write \(U \sim \mathsf{H}_{\times }\) below, it means that U is in the form of \(\bigoplus _{j=1}^J I_j^{A_l}\otimes U_j^{A_r}\) and \(U_j^{A_r} \sim \mathsf{H}_j\).

We consider two scenarios in which a bipartite quantum state \(\Psi ^{AR}\) is transformed by a unitary operation on A and then is subject to the action of a quantum channel (linear CP map) \({\mathcal {T}}^{A\rightarrow E}\). The unitary is chosen at uniformly random from the set of unitaries that take a simple form under the DSP decomposition (1).

In the first scenario, which we call non-randomized partial decoupling, the unitaries are such that they completely randomize the space \({\mathcal {H}}_j^{A_r}\) for each j, while having no effect on j or the space \({\mathcal {H}}_j^{A_l}\). This scenario may find applications when complex quantum many-body systems are investigated based on the decoupling approach, in which case the DSP decomposition is, for instance, induced by the symmetry the system has. In the second scenario, which we refer to as randomized partial decoupling, we assume that \(\mathrm{dim}{\mathcal {H}}_j^{A_l}=1\) and that \(\mathrm{dim}{\mathcal {H}}_j^{A_r}\) does not depend on j. The unitaries do not only completely randomize the space \({\mathcal {H}}^{A_r}\), but also randomly permute j. This scenario may fit to the communication problems. For instance, one of the applications may be classical-quantum hybrid communicational tasks, where the division of the classical and quantum information leads to the DSP decomposition.

For both scenarios, our concern is how close the final state is, after the action of the unitary and the quantum channel, to the averaged final state over all unitaries. It should be noted that the averaged final state is in the form of a block-wise decoupled state in general. This is in contrast to the decoupling theorem, in which the averaged final state is a fully decoupled state.

3.1 Non-randomized partial decoupling

Let us consider the situation where U has the DSP form: \(U:=\bigoplus _{j=1}^J I_j^{A_l} \otimes U_j^{A_r}\). For any state \(\Psi ^{AR}\), the averaged state obtained after the action of the random unitary \(U \sim \mathsf{H}_{\times }\) is given by

Here, \(\pi _j^{A_r}\) is the maximally mixed state on \({\mathcal {H}}_j^{A_r}\), and \(\Psi _{jj}^{A_lR}\) is an unnormalized state on \({\mathcal {H}}_j^{A_l}\otimes {\mathcal {H}}^R\) defined by

Our interest is on the average distance between the state \({\mathcal {T}}^{A \rightarrow E} (U^A \Psi ^{AR} U^{\dagger A}) \) and the averaged state \({\mathcal {T}}^{A \rightarrow E}(\Psi _{\mathrm{av}}^{AR})\) over all \(U \sim \mathsf{H}_{\times }\).

For expressing the upper bound on the average distance, we introduce a quantum system \(A^*\) represented by a Hilbert space

and a linear operator \(F^{A{\bar{A}}\rightarrow A^*}: {\mathcal {H}}^A\otimes {\mathcal {H}}^{{\bar{A}}} \rightarrow {\mathcal {H}}^{A^*}\) defined by

where \({\mathcal {H}}_j^{{\bar{A}}_l}\cong {\mathcal {H}}_j^{A_l}\), \({\mathcal {H}}_j^{{\bar{A}}_r}\cong {\mathcal {H}}_j^{A_r}\) and \({\mathcal {H}}^{{\bar{A}}}\cong {\mathcal {H}}^{A}\).

The following is our first main theorem about the upper bound:

Theorem 1

(Main result 1: One-shot non-randomized partial decoupling.) For any \(\epsilon ,\mu \ge 0\), any subnormalized state \(\Psi ^{AR} \in {\mathcal {S}}_\le ({\mathcal {H}}^{AR})\) and any linear CP map \({\mathcal {T}}^{A \rightarrow E}\), it holds that

Here, \(H_{\mathrm{min}}^{\epsilon ,\mu }(A^*|RE)_{{\Lambda }(\Psi ,{\mathcal {T}})}\) is the smooth conditional min-entropy for an unnormalized state \({\Lambda }(\Psi ,{\mathcal {T}})\), defined by \(F(\Psi ^{AR}\otimes \tau ^{{\bar{A}}E})F^\dagger \) with \(\tau ^{AE} = {\mathfrak {J}}({\mathcal {T}}^{A \rightarrow E})\) being the Choi–Jamiołkowski representation of \({\mathcal {T}}^{A \rightarrow E}\). It is explicitly given by

where \({\mathcal {B}}_{\mathrm{DSP}}^{\mu }({\mathcal {T}})\) is the set of \(\mu \)-neighbourhoods of \({\mathcal {T}}\), defined by (15).

In the literature of chaotic quantum many-body systems, it is often assumed that the dynamics is approximated well by a random unitary channel, which is sometimes called scrambling [7, 29, 30]. Despite the fact that a number of novel research topics have been opened based on the idea of scrambling, some of which are using the decoupling approach [7, 9, 10], symmetry of the physical systems has rarely been taken into account properly. When the system has symmetry, the associated Hilbert space is naturally decomposed into a DSP form as

where j is the label of irreducible representations of a compact group of the symmetry, \({\mathcal {H}}_j^{A_l}\) is the irreducible representation and \({\mathcal {H}}_j^{A_r}\) corresponds to the multiplicity for each j. Due to the conservation law, the scrambling dynamics in the system should be compatible with this decomposition and should be in the form of \(U^A=\bigoplus _{j=1}^J I_j^{A_l}\otimes U_j^{A_r}\). Hence, Theorem 1 is applicable to the study of complex physics in chaotic quantum many-body systems with symmetry.

Theorem 1 reduces to a simpler form when the symmetry is abelian. In this case, all the irreducible representation one-dimensional, i.e., \(\dim {\mathcal {H}}_j^{A_l} =1\). The averaged output state is explicitly calculated to be

The operator \(F^{A {\bar{A}} \rightarrow A^*}\) in (31) reduces to a direct sum of operators that are proportional to projectors, and the operator \(\Lambda (\Psi , \mathcal{T}) \in \mathcal{S}_{\le }(\mathcal{H}^{A^*RE})\) in Theorem 1 reduces to

Theorem 1 implies that, if the smooth conditional min-entropy of the unnormalized state \(\Lambda (\Psi , \mathcal{T})\) is sufficiently large, the final state \({\mathcal {T}}^{A \rightarrow E} \circ {\mathcal {U}}^A ( \Psi ^{AR} )\) is close to \({\mathcal {T}}^{A \rightarrow E}(\Psi _{\mathrm{av}}^{AR})\).

3.2 Randomized partial decoupling

Next we assume that

The Hilbert space \({\mathcal {H}}^A=\oplus _{j=1}^J{{\mathcal {H}}}_j^{A_r}\) is then isomorphic to a tensor product Hilbert space \({\mathcal {H}}^{A_c} \otimes {\mathcal {H}}^{A_r}\), i.e., \(A\cong A_cA_r\). Here, \({{\mathcal {H}}}^{A_c}\) is a J-dimensional Hilbert space with a fixed orthonormal basis \(\{|j\rangle \}_{j=1}^J\), and \({{\mathcal {H}}}^{A_r}\) is an r-dimensional Hilbert space. We consider a random unitary U on system A of the form

which we also denote by \(U \sim \mathsf{H}_{\times }\). In addition, let \({\mathbb {P}}\) be the permutation group on \([1,\ldots ,J]\), and \(\mathsf{P}\) be the uniform distribution on \({\mathbb {P}}\). We define a unitary \(G_\sigma \) for any \(\sigma \in {\mathbb {P}}\) by

We denote the supermap given by conjugation of \(G_\sigma \) by the calligraphic font as \({\mathcal {G}}_\sigma (\cdot )=G_\sigma (\cdot )G_\sigma ^\dagger \). For the initial state, we use the notion of classically coherent states, defined as follows:

Definition 2

(classically coherent states [31]) Let \(K_1\) and \(K_2\) be d-dimensional quantum systems with fixed orthonormal bases \(\{|k_1\rangle \}_{k_1=1}^d\) and \(\{|k_2\rangle \}_{k_2=1}^d\), respectively, and let W be a quantum system. An unnormalized state \(\varrho \in {\mathcal {P}}({\mathcal {H}}^{K_1K_2W})\) is said to be classically coherent in \(K_1K_2\) if it satisfies \(\varrho {|k\rangle }^{K_1}{|k'\rangle }^{K_2}=0\) for any \(k\ne k'\), or equivalently, if \(\varrho \) is in the form of

where \(\varrho _{kk'}\in {\mathcal {L}}({\mathcal {H}}^{W})\) for each k and \(k'\).

We now provide our second main result:

Theorem 3

(Main result 2: One-shot randomized partial decoupling.) Let \(\epsilon ,\mu \ge 0, R\cong R_cR_r, \Psi ^{AR}\) be a subnormalized state that is classically coherent in \(A_cR_c\), and \({\mathcal {T}}^{A \rightarrow E}\) be a linear CP map such that the Choi–Jamiołkowski representation \(\tau ^{AE} = {\mathfrak {J}}({\mathcal {T}}^{A\rightarrow E})\) satisfies \(\mathrm{Tr}[\tau ]\le 1\). It holds that

where \(\Psi _{\mathrm{av}}^{AR}:={\mathbb {E}}_{U \sim \mathsf{H}_{\times }} [ {\mathcal {U}}^A ( \Psi ^{AR} ) ]\). The function \(\alpha (J)\) is 0 for \(J=1\) and \(\frac{1}{J-1}\) for \(J\ge 2\), and \(\beta (A_r)\) is 0 for \(\mathrm{dim}{\mathcal {H}}^{A_r}=1\) and 1 for \(\mathrm{dim}{\mathcal {H}}^{A_r}\ge 2\). The exponents \(H_I\) and \(H_{I\!I}\) are given by

Here, \({\mathcal {C}}\) is the completely dephasing channel on \(A_c\) with respect to the basis \(\{|j\rangle \}_{j=1}^J\), and \(\tau ^{AB} = {\mathfrak {J}}({\mathcal {T}}^{A\rightarrow B})\) is the Choi–Jamiołkowski representation of the complementary channel \({\mathcal {T}}^{A\rightarrow B}\) of \({\mathcal {T}}^{A\rightarrow E}\).

Note that, since the subnormalized state \(\Psi ^{AR}\) is classically coherent in \(A_cR_c\), the averaged state \(\Psi _{\mathrm{av}}^{AR}\) is explicitly given by

Small error for one-shot randomized partial decoupling implies that the third party having the purifying system of the final state may recover both classical and quantum parts of correlation in \(\Psi ^{AR}\). Thus, it will be applicable, e.g., for analyzing simultaneous transmission of classical and quantum information in the presence of quantum side information. In this context, \(H_I\) in the above expression quantifies how well the total correlation in \(\Psi ^{AR}\) can be transmitted by the channel \({\mathcal {T}}^{A\rightarrow B}\), whereas \(H_{I\!I}\) for only quantum part thereof (see [21,22,23]).

3.3 A converse bound

So far, we have presented achievabilities of non-randomized and randomized partial decoupling. At this point, we do not know whether the obtained bounds are “sufficiently tight”. To address this question, we prove a converse bound for partial decoupling. We assume the following two conditions for the converse:

Converse Condition 1:

\(\dim {\mathcal {H}}_j^l=1,\quad \dim {\mathcal {H}}_j^r=r \quad (j=1,\ldots , J)\).

Converse Condition 2:

The initial (normalized) state \(\Psi ^{AR}\) is classically coherent in \(A_cR_c, \text {where } R\cong R_cR_r\).

Throughout the paper, we refer to the conditions as CC1 and CC2, respectively. The two conditions are always satisfied in the case of randomized partial decoupling, but not necessarily satisfied in the case of non-randomize one. Consequently, the converse bound we prove below is directly applicable to randomized partial decoupling, but is not applicable to non-randomized partial decoupling in general.

The converse bound is stated by the following theorem.

Theorem 4

(Main result 3: Converse for partial decoupling.) Suppose that CC1 and CC2 are satisfied. Let \(|\Psi \rangle ^{ARD}\) be a purification of a normalized state \(\Psi ^{AR}\in {\mathcal {S}}_=({\mathcal {H}}^{AR})\), which is classically coherent in \(A_cR_c\) due to CC2, and \({\mathcal {T}}^{A \rightarrow E}\) be a trace preserving CP map with the complementary channel \({\mathcal {T}}^{A \rightarrow B}\). Suppose that, for \(\delta >0\), there exists a normalized state \(\Omega ^{ER}:=\sum _{j=1}^J\varsigma _j^E\otimes \Psi _{jj}^{R_r}\otimes {{|j\rangle }\!{\langle j|}}^{R_c}\), where \(\{\varsigma _j\}_{j=1}^J\) are normalized states on E, such that

Then, for any \(\upsilon \in [0,1/2)\) and \(\iota \in (0,1]\), it holds that

where \({\mathcal {C}}\) is the completely dephasing channel on \(A_c\), and the smoothing parameters \(\lambda \) and \(\lambda '\) are defined by

and \(x:=\sqrt{2}\root 4 \of {24\upsilon +2\delta }\).

Note that, when a quantum channel \({\mathcal {T}}^{A\rightarrow E}\) achieves partial decoupling for a state \(\Psi ^{AR}\) within a small error, it follows from the decomposition of \(\Psi _{\mathrm{av}}\) (see (43)) that

where \({\hat{\tau }}_j^E:={\mathcal {T}}^{A\rightarrow E}({{|j\rangle }\!{\langle j|}}^{A_c}\otimes \pi ^{A_r})\in \mathcal {S}_=({\mathcal {H}}^E)\). This is in the same form as the assumption of Theorem 4.

Let us compare the direct part of randomized partial decoupling (Theorem 3) and the converse bound presented above. In the case of \(J\ge 2\), the first term in the R.H.S. of the achievability bound (41) is calculated to be

On the other hand, the converse bound (45) yields

where \(\psi ^{AB}:={\mathcal {T}}^{A'\rightarrow B}(\Psi _p^{AA'})\), with \(|\Psi _p\rangle ^{AA'}\) being a purification of \(\Psi ^A\) and \({\mathcal {H}}^A\cong {\mathcal {H}}^{A'}\). Note that there exists a linear isometry from \(A'\) to RD that maps \(|\Psi _p\rangle \) to \(|\Psi \rangle \) [12], and that the conditional max entropy is invariant under local isometry (see Lemma 21 below). A similar argument also applies to the second term in (41) and (46). Thus, when \(\Psi ^A\) is the maximally mixed state, in which case \(|\Psi _p\rangle ^{AA'}=|\Phi \rangle ^{AA'}\) and thus \(\psi =\tau \), the gap between the two bounds is only due to the difference in values of smoothing parameters and types of conditional entropies. By the fully quantum asymptotic equipartition property [32], this gap vanishes in the limit of infinitely many copies. From this viewpoint, we conclude that the achievability bound of randomized partial decoupling and the converse bound are sufficiently tight.

3.4 Reduction to the existing results

We briefly show that the existing results on one-shot decoupling [11] and dequantization [31] are obtained from Theorems 1, 3 and 4 as corollaries, up to changes in smoothing parameters. Thus, our results are indeed generalizations of these two tasks.

First, by letting \(J=1\) in Theorem 3, we obtain the achievability of one-shot decoupling:

Corollary 5

(Achievability for one-shot decoupling: Theorem 3.1 in [11]) Let \(\epsilon ,\mu \ge 0\), \(\Psi ^{AR}\) be a subnormalized state, and \({\mathcal {T}}^{A \rightarrow E}\) be a linear CP map such that the Choi–Jamiołkowski representation \(\tau ^{AE} = {\mathfrak {J}}({\mathcal {T}}^{A\rightarrow E})\) satisfies \(\mathrm{Tr}[\tau ]\le 1\). Let \(U\sim \mathsf{H}\) be the Haar random unitary on \({\mathcal {H}}^A\). Then, it holds that

Note that the duality of the conditional min and max entropies ([25]: see also Lemma 24 in Sect. 5.2.2) implies \(H_{\mathrm{min}}^\mu (A|E)_{\tau }=-H_{\mathrm{max}}^\mu (A|B)_{\tau }\), with \(\tau ^{AB} = {\mathfrak {J}}({\mathcal {T}}^{A\rightarrow B})\) being the Choi–Jamiołkowski representation of the complementary channel \({\mathcal {T}}^{A\rightarrow B}\) of \({\mathcal {T}}^{A\rightarrow E}\). A similar bound is also obtained by letting \(J=1\) and \(\mathrm{dim}{\mathcal {H}}_j^{A_l}=1\) in Theorem 1. A converse bound for one-shot decoupling is obtained by letting \(J=1\) in Theorem 4, and by using the duality of the conditional entropies, as follows:

Corollary 6

(Converse for one-shot decoupling: Theorem 4.1 in [11] ) Consider a normalized state \(\Psi ^{AR}\in {\mathcal {S}}_=({\mathcal {H}}^{AR})\) and a trace preserving CP map \({\mathcal {T}}^{A \rightarrow E}\). Suppose that, for \(\delta >0\), there exists a normalized state \(\varsigma \in {\mathcal {S}}_=({\mathcal {H}}^E)\), such that \( \Vert {\mathcal {T}}^{A \rightarrow E} ( \Psi ^{AR} ) -\varsigma ^E\otimes \Psi ^R \Vert _1 \le \delta \). Then, for any \(\upsilon \in [0,1/2)\) and \(\iota \in (0,1]\), it holds that

where \(|\Psi _p\rangle ^{AA'}\) is a purification of \(\Psi ^A\), \({\mathcal {H}}^A\cong {\mathcal {H}}^{A'}\), and the smoothing parameter \(\lambda \) is defined by (47).

Next, we consider the opposite extreme for Theorem 3, i.e., we consider the case where \(\mathrm{dim}{\mathcal {H}}^{A_r}=1\). This case yields the dequantizing theorem:

Corollary 7

(Achievability for dequantization: Theorem 3.1 in [31]) Let A be a quantum system with a fixed basis \(\{|j\rangle \}_{j=1}^{d_A}\), \({\mathcal {H}}^R\cong {\mathcal {H}}^A\) and \(\epsilon ,\mu \ge 0\). Consider a subnormalized state \(\Psi ^{AR}\) that is classically coherent in AR, and a linear CP map \({\mathcal {T}}^{A \rightarrow E}\) such that the Choi–Jamiołkowski representation \(\tau ^{AE} = {\mathfrak {J}}({\mathcal {T}}^{A\rightarrow E})\) satisfies \(\mathrm{Tr}[\tau ]\le 1\). Let \(\sigma \) be the random permutation on \([1,\ldots ,d_A]\) with the associated unitary \(G_\sigma :=\sum _{j=1}^{d_A}{{|\sigma (j)\rangle }\!{\langle j|}}\). Then, it holds that

where \({\mathcal {C}}\) is the completely dephasing channel on A with respect to the basis \(\{|j\rangle \}_{j=1}^J\), and \(\tau ^{AB} = {\mathfrak {J}}({\mathcal {T}}^{A\rightarrow B})\) is the Choi–Jamiołkowski representation of the complementary channel \({\mathcal {T}}^{A\rightarrow B}\) of \({\mathcal {T}}^{A\rightarrow E}\).

In the same extreme, Theorem 4 provides a converse bound for dequantization, which has not been known so far:

Corollary 8

(Converse for dequantization.) Consider the same setting as in Corollary 7, and assume that \(\Psi ^{AR}\) is normalized, and that \({\mathcal {T}}^{A \rightarrow E}\) is trace preserving. Let \(|\Psi \rangle ^{ARD}\) be a purification of \(\Psi ^{AR}\). Suppose that, for \(\delta >0\), there exists a normalized state \(\Omega ^{ER}:=\sum _{j=1}^Jp_j\varsigma _j^E\otimes {{|j\rangle }\!{\langle j|}}^{R}\), where \(\{p_j,\varsigma _j\}_{j=1}^J\) is an ensemble of normalized states on E, such that \( \Vert {\mathcal {T}}^{A \rightarrow E} ( \Psi ^{AR} ) -\Omega ^{ER} \Vert _1 \le \delta \). Then, for any \(\upsilon \in [0,1/2)\) and \(\iota \in (0,1]\), it holds that

where the smoothing parameter \(\lambda \) is defined by (47).

Before we proceed to the proofs, we here briefly discuss how the random unitaries \(U \sim \mathsf{H}_{\times }\) that respect the DSP form can be implemented by quantum circuits. Since Haar random unitaries are in general hard to implement, unitary t-designs, mimicking the t-th statistical moments of the Haar measure on average [33,34,35], have been exploited in many cases. Since the decoupling method makes use of the second statistical moments of the Haar measure, we could use the unitary 2-designs instead of the Haar measure for our tasks. Although a number of efficient implementations of unitary 2-designs have been discovered [33,34,35,36,37,38,39,40,41], and it is also shown that decoupling can be achieved using unitaries less random than unitary 2-designs [42, 43], we here need unitary designs in a given DSP form, which we refer to as the DSP unitary designs. Thus, we cannot directly use the existing constructions, posing a new problem about efficient implementations of DSP unitary designs. Although this problem is out of the scope in this paper, we will briefly discuss possible directions toward the solution.

One possible way is to simply modify the constructions of unitary designs known so far. This could be done by regarding each Hilbert space \({\mathcal {H}}^{A_r}_j\), on which each random unitary \(U_j^{A_r} \sim \mathsf{H}_j\) acts, as the Hilbert space of “virtual” qubits. The complexity of the implementation, i.e. the number of quantum gates, is then determined by how complicated the unitary is that transforms the basis in each \({\mathcal {H}}^{A_r}_j\) into the standard basis of the virtual qubits. Another way is to use the implementation of designs on one qudit [44], where it was shown that alternate applications of random diagonal unitaries in two complementary bases achieves unitary designs. This implementation would be suited in quantum many-body systems because we can choose two natural bases, position and momentum bases, and just repeat switching random potentials in those bases under the condition that the potentials satisfy the DSP form. Finally, when the symmetry-induced DSP form is our concern, unitary designs with symmetry may possibly be implementable by applying random quantum gates that respects the symmetry.

In any case, the implementations of DSP unitary designs, or the symmetric unitary designs, and their efficiency are left fully open. Further analyses are desired.

Outline of our proofs. The PD stands for the partial decoupling. For the smoothed randomized partial decoupling and the converse bound, we first assume two conditions, WA 1 and WA 2, but will remove them later to complete the proof. The details of the conditions are given in the main text

Full size image

In the rest of the paper, we prove the three main theorems, Theorems 1, 3 and 4 in Sects. 6, 7 and 8, respectively. For the sake of clarity, we sketch the outline of the proofs in Sect. 5.1 (see also Fig. 1). We then list useful lemmas in Sect. 5.2. See also Appendix H for the list of notations used in the proofs.

5.1 Key lemmas and the structure of the proofs

For the achievability statements (Theorems 1 and 3), the key technical lemma is the twisted twirling, which can be seen as a generalization of the twirling method often used in quantum information science. See Appendix A for the proof.

Lemma 9

(Twisted Twirling) Let \({\mathcal {H}}_j^{A_r}\) be a \(r_j\)-dimensional subspace of \({\mathcal {H}}^{A_r}\), and \(\Pi _j^{A_r}\) be the projector onto \({\mathcal {H}}_j^{A_r}\subset {\mathcal {H}}^{A_r}\) for each of \(j=1,\ldots ,J\). Let \({\mathbb {I}}^{A_rA_r'}\) be \(I^{A_r} \otimes I^{A_r'}\), and \({\mathbb {F}}^{A_rA_r'} \in {\mathcal {L}}({\mathcal {H}}^{A_rA_r'})\) be the swap operator defined by \(\sum _{a,b} |a\rangle \langle b|^{A_r} \otimes |b\rangle \langle a|^{A_r'}\) for any orthonormal basis \(\{ {|a\rangle } \}\) in \({\mathcal {H}}^{A_r}\) and \({\mathcal {H}}^{A_r'}\). In addition, let \({\mathbb {I}}_{jk}^{A_rA_r'}\) and \({\mathbb {F}}_{jk}^{A_rA_r'}\) be \(\Pi _j^{A_r} \otimes \Pi _k^{A_r'}\) and \(( \Pi _j^{A_r} \otimes \Pi _k^{A_r'}){\mathbb {F}}^{A_rA_r'}\), respectively. For any \(M^{A_rA_r'BB'}\in {\mathcal {L}}({\mathcal {H}}^{A_rA_r'BB'})\), define

Then, it holds that, for \(j \ne k\),

Moreover,

Otherwise, \({\mathbb {E}}_{U_j,U_k,U_m,U_n} \bigl [ (U_j^{A_r} \otimes U_k^{A_r'}) M^{A_rA_r'BB'} (U_m^{A_r} \otimes U_n^{A_r'})^{\dagger } \bigr ]=0.\)

The twisted twirling enables us to show the following lemma (see Appendix B).

Lemma 10

For any \(\varsigma ^{ER} \in {\mathcal {S}}_=({\mathcal {H}}^{ER})\) and any \(X\in \mathrm{Her}({\mathcal {H}}^{AR})\) such that \(X_{jj}^{A_lR}=0\), the following inequality holds for any possible permutation \(\sigma \in {\mathbb {P}}\):

Here, \(A_l^T\) denotes the transposition of \(A_l\) with respect to the Schmidt basis of the maximally entangled state \(|\Phi _j^l\rangle ^{A_lA_l'}\) in (26), and the norm in the R.H.S. is defined by (16).

Based on this lemma, we can prove the non-smoothed versions of Theorems 1 and 3 in Sects. 6.1 and 7.1, respectively.

To complete the proofs of Theorems 1 and 3, smoothing the statements is needed, which is done in Sects. 6.2 and 7.2 based on the following lemma proven in Appendix C.

Lemma 11

Consider arbitrary unnormalized states \(\Psi ^{AR},{\hat{\Psi }}^{AR}\in {\mathcal {P}}({\mathcal {H}}^{AR})\) and arbitrary CP maps \({\mathcal {T}},\hat{{\mathcal {T}}}:A\rightarrow E\). Let \({\mathcal {D}}_+^{A \rightarrow E}\) and \({\mathcal {D}}_-^{A \rightarrow E}\) be arbitrary CP maps such that \({\mathcal {T}}-\hat{{\mathcal {T}}}={\mathcal {D}}_+-{\mathcal {D}}_-\). Let \(\delta _+^{AR}\) and \(\delta _-^{AR}\) be linear operators on \({\mathcal {H}}^A\otimes {\mathcal {H}}^{R}\), such that

and that

The following inequality holds for any possible permutation \(\sigma \in {\mathbb {P}}\) and for both \({\Psi }_*={\Psi }_{\mathrm{av}}\) and \({\Psi }_*={\mathcal {C}}^A(\Psi )\):

Here, \({\hat{\Psi }}_*={\mathbb {E}}_{U\sim \mathsf{H}_\times }[{\mathcal {U}}^A({\hat{\Psi }}^{AR})]\) for \({\Psi }_*={\Psi }_{\mathrm{av}}\) and \({\hat{\Psi }}_*={\mathcal {C}}^A({\hat{\Psi }})\) for \({\Psi }_*={\mathcal {C}}^A(\Psi )\).

The converse statements are proved independently in Sect. 8.

When we prove the one-shot randomized decoupling theorem (Theorem 3) and the converse (Theorem 4), we first put the following two working assumptions:

WA 1:

\(E\cong E_cE_r\), where \(E_c\) is a quantum system of dimension J.

WA 2:

The CP map \({\mathcal {T}}^{A \rightarrow E}\) is decomposed into

$$\begin{aligned} {\mathcal {T}}^{A \rightarrow E}(X)=\sum _{j,k=1}^J{{|j\rangle }\!{\langle k|}}^{E_c}\otimes {\mathcal {T}}_{jk}^{A_r \rightarrow E_r}(X_{jk}), \end{aligned}$$

(64)

in which \({\mathcal {T}}_{jk}\) is a linear supermap from \({\mathcal {L}}({\mathcal {H}}^{A_r})\) to \({\mathcal {L}}({\mathcal {H}}^{E_r})\) defined by \({\mathcal {T}}_{jk}(\zeta )={\mathcal {T}}({{|j\rangle }\!{\langle k|}}\otimes \zeta )\) for each j, k.

These assumptions are finally dropped in Sects. 7.3 and 8.3 using the following lemma (see Appendix D for a proof).

Lemma 12

Let \({\mathcal {T}}^{A\rightarrow E}\) be a linear CP map that does not necessarily satisfies WA 1 and WA 2. By introducing a quantum system \(E_c\) with dimension J, define an isometry \(Y^{A_c \rightarrow A_c E_c}:=\sum _{j}{|jj\rangle }^{A_cE_c}{\langle j|}^{A_c}\), and a linear map \(\check{{\mathcal {T}}}^{A \rightarrow EE_c}\) by \({\mathcal {T}}^{A \rightarrow E} \circ {\mathcal {Y}}^{A_c \rightarrow A_c E_c}\). Then, \(\check{{\mathcal {T}}}^{A \rightarrow EE_c}\) is a linear CP map and, for any \(\Psi ^{AR}\) that is classically coherent in \(A_cR_c\), the following equalities hold:

5.2 List of useful lemmas

We here provide several useful lemmas, some of which are in common with those in the proof of the one-shot decoupling theorem [11]. Proofs of Lemmas 16–20 and 29 –35 will be provided in Appendix E.

5.2.1 Properties of norms and distances

Lemma 13

(Lemma 3.6 in [11]) For any \(\xi ^{AB} \in \mathrm{Her}({\mathcal {H}}^{AB})\), \(|\!|\xi ^{AB}|\!|_2 \le \sqrt{d_A}|\!|\xi ^B|\!|_2\).

Lemma 14

(Lemma 3.7 in [11]) For any \(X\in \mathrm{Her}({\mathcal {H}})\) and \(\gamma \in {\mathcal {P}}({\mathcal {H}})\), it holds that

Lemma 15

(Sec. II in [25]) The purified distance defined by (11) satisfies the following properties:

1. 1.

   triangle inequality: For any \(\rho ,\varsigma ,\tau \in {\mathcal {S}}_\le ({\mathcal {H}})\), it holds that \(P(\rho ,\varsigma )\le P(\rho ,\tau )+P(\tau ,\varsigma )\).

2. 2.

   monotonicity: For any \(\rho ,\varsigma \in {\mathcal {S}}_\le ({\mathcal {H}})\) and trace-nonincreasing CP map \({\mathcal {E}}\), it holds that \(P(\rho ,\varsigma )\ge P({\mathcal {E}}(\rho ),{\mathcal {E}}(\varsigma ))\).

3. 3.

   Uhlmann’s theorem: For any \(\rho ,\varsigma \in {\mathcal {S}}_\le ({\mathcal {H}})\) and any purification \(|\varphi _\rho \rangle \in {\mathcal {H}}\otimes {\mathcal {H}}'\) of \(\rho \), where \({\mathcal {H}}'\cong {\mathcal {H}}\), there exists a purification \(|\varphi _\varsigma \rangle \in {\mathcal {H}}\otimes {\mathcal {H}}'\) of \(\varsigma \) such that \(P(\rho ,\varsigma )=P(\varphi _\rho ,\varphi _\varsigma )\).

Lemma 16

The purified distance defined by (11) satisfies the following properties:

1. 1.

   pure states: For any subnormalized pure state \(|\psi \rangle \in {\mathcal {H}}\) and any normalized pure state \(|\phi \rangle \in {\mathcal {H}}\), \(P(\psi ,\phi )=\sqrt{1-|\langle \psi |\phi \rangle |^2}\).

2. 2.

   relation to the trace distance: For any \(\rho ,\varsigma \in {\mathcal {S}}_\le ({\mathcal {H}})\), \( \frac{1}{2}\Vert \rho -\varsigma \Vert _1 \le P(\rho ,\varsigma ) \le \sqrt{2\Vert \rho -\varsigma \Vert _1}\).

3. 3.

   inequality for subnormalized pure states: For any subnormalized pure states \({|\psi \rangle },{|\phi \rangle }\in {\mathcal {H}}\), \(P(\psi ,\phi ) \le \sqrt{1-|{\left\langle \psi |\phi \right\rangle }|^2} + \sqrt{1-{\left\langle \phi |\phi \right\rangle }}\).

Lemma 17

Let \(\{p_k\}_k\) be a normalized probability distribution, \(\{\rho _k\}_k\) be a set of normalized states on AB, and \(\{{\hat{\rho }}_k\}_k\) be that of subnormalized ones. For \(\rho ^{ABK}:=\sum _kp_k\rho _k^{AB}\otimes {{|k\rangle }\!{\langle k|}}^K\) and \({\hat{\rho }}^{ABK}:=\sum _kp_k{\hat{\rho }}_k^{AB}\otimes {{|k\rangle }\!{\langle k|}}^K\), the purified distance satisfies

Lemma 18

Let \(\{p_k\}_k\) and \(\{q_k\}_k\) be subnormalized probability distributions, and \(\{\rho _k\}_k\) and \(\{\varsigma _k\}_k\) be sets of normalized states on A. For \(\rho ^{AK}:=\sum _kp_k\rho _k^{A}\otimes {{|k\rangle }\!{\langle k|}}^K\) and \(\varsigma ^{AK}:=\sum _kq_k\varsigma _k^{A}\otimes {{|k\rangle }\!{\langle k|}}^K\), it holds that

Lemma 19

The DSP norm defined by (13) satisfies the triangle inequality, i.e., for any superoperators \({\mathcal {E}}\) and \({\mathcal {F}}\) from \({\mathcal {L}}({\mathcal {H}}^A)\) to \({\mathcal {L}}({\mathcal {H}}^B)\), \(\Vert {\mathcal {E}}+{\mathcal {F}}\Vert _{\mathrm{DSP}} \le \Vert {\mathcal {E}}\Vert _{\mathrm{DSP}} + \Vert {\mathcal {F}}\Vert _{\mathrm{DSP}}\).

Lemma 20

Let \(\{\Pi _j\}_j\) be a set of orthogonal projectors on \({\mathcal {H}}\) such that \(\sum _j\Pi _j=I\). For any \(\varrho \in {\mathcal {P}}({\mathcal {H}})\), \(\left\| \varrho \right\| _2^2=\sum _{j,k}\left\| \Pi _j\varrho \Pi _k\right\| _2^2\).

5.2.2 Properties of conditional entropies

Lemma 21

(Corollary of Lemma 13 in [25]) For any \(\epsilon \ge 0\), \(\rho ^{AB} \in {\mathcal {S}}_\le ({\mathcal {H}}^{AB})\) and any linear isometry \(V:A\rightarrow C\), \(H_{\mathrm{min}}^\epsilon (A|B)_\rho =H_{\mathrm{min}}^\epsilon (C|B)_{{\mathcal {V}}(\rho )}\).

Lemma 22

(Corollary of Lemma 15 in [25]) For any \(\epsilon \ge 0\), \(\rho ^{AB} \in {\mathcal {S}}_\le ({\mathcal {H}}^{AB})\) and any linear isometry \(W:B\rightarrow D\), \(H_{\mathrm{max}}^\epsilon (A|B)_\rho =H_{\mathrm{max}}^\epsilon (A|D)_{{\mathcal {W}}(\rho )}\).

Lemma 23

(Lemma A.1 in [11]) For any \(\rho ^{AB} \in {\mathcal {S}}_\le ({\mathcal {H}}^{AB})\) and \(\varsigma ^{B} \in {\mathcal {S}}_=({\mathcal {H}}^{B})\), it holds that

Lemma 24

(Definition 14, Equality (6) and Lemma 16 in [25]) For any subnormalized pure state \(|\psi \rangle \) on system ABC, and for any \(\epsilon >0\), \(H_{\mathrm{max}}^\epsilon (A|B)_\psi = - H_{\mathrm{min}}^\epsilon (A|C)_\psi \).

Lemma 25

(Lemma B.2 in [11]) Let \(\psi ^{ABC}\in {\mathcal {S}}_\le ({\mathcal {H}}^{ABC})\) be a subnormalized pure state. For any full-rank state \(\varsigma ^B\in {\mathcal {S}}_=({\mathcal {H}}^B)\), it holds that \(\psi ^{ABC}\le Z^{AB}\otimes I^C\), where

Lemma 26

(Lemma A.5 in [11]) For any state \(\rho ^{ABK} \in {\mathcal {S}}_=({\mathcal {H}}^{ABK})\) in the form of

where \(\rho _k \in {\mathcal {S}}_=({\mathcal {H}}^{AB})\), \(\langle k|k'\rangle =\delta _{k,k'}\) and \(\{p_k\}_k\) is a normalized probability distribution, it holds that

(It is straightforward to show that the above equalities also hold for \(\rho ^{ABK} \in {\mathcal {S}}_\le ({\mathcal {H}}^{ABK})\) and \(\rho _k \in {\mathcal {S}}_\le ({\mathcal {H}}^{AB})\), by noting that \(H_{\mathrm{min}}(A|BK)_\rho =H_{\mathrm{min}}(A|BK)_{\rho /\mathrm{Tr}[\rho ]}-\log {\mathrm{Tr}[\rho ]}\) and that \(H_{\mathrm{max}}(A|BK)_\rho =H_{\mathrm{max}}(A|BK)_{\rho /\mathrm{Tr}[\rho ]}+\log {\mathrm{Tr}[\rho ]}\).)

Lemma 27

(Lemma A.7 in [11]) For any state \(\rho ^{ABK_1K_2} \in {\mathcal {S}}_\le ({\mathcal {H}}^{ABK_1K_2})\) in the form of

where the notations are the same as in Lemma 26, and for any \(\epsilon \ge 0\) it holds that

(Note that, although Lemma A.7 in [11] assumes that \(\rho ^{ABK_1K_2}\) is normalized, the condition is not used in the proof thereof.)

Lemma 28

(Lemma A.1 in [31]) Let \(\rho \in {\mathcal {S}}_\le ({\mathcal {H}}^{K_1K_2AB})\) be a subnormalized state that is classically coherent in \(K_1K_2\). For any \(\epsilon \ge 0\), there exists \({\hat{\rho }}\in {\mathcal {B}}^\epsilon (\rho )\) that is classically coherent in \(K_1K_2\), and \(\varsigma \in {\mathcal {S}}_=({\mathcal {H}}^{K_2B})\) that is decomposed as \(\varsigma =\sum _k{{|k\rangle }\!{\langle k|}}^{K_2}\otimes \varsigma _k^B\), such that

Lemma 29

In the same setting as in Lemma 27, it holds that

Lemma 30

Let \(\rho \in {\mathcal {S}}_\le ({\mathcal {H}}^{K_1K_2AB})\) be a subnormalized state that is classically coherent in \(K_1K_2\). For any \(\epsilon \ge 0\), there exists \({\hat{\rho }}\in {\mathcal {B}}^\epsilon (\rho )\) that is classically coherent in \(K_1K_2\), such that

If \(\rho \) is also diagonal in \(K_1K_2\) (i.e., if \(\rho \) is in the form of (75)), there exists \({\hat{\rho }}\), satisfying the above conditions, that is diagonal in \(K_1K_2\).

Lemma 31

Consider the same setting as in Lemma 26. For any \(\{\epsilon _k\}_k\) such that \(\epsilon _k\ge 0\), it holds that

where \(\varepsilon :=\sum _kp_k\epsilon _k\).

5.2.3 Other technical lemmas

Lemma 32

Consider two linear operators \(X,Y:{\mathcal {H}}^A\rightarrow {\mathcal {H}}^B\) and assume that \(A\cong A'\), \(B\cong B'\). Let \({|\Phi \rangle }^{AA'}\) and \({|\Phi \rangle }^{BB'}\) be maximally entangled states between A and \(A'\), and B and \(B'\), respectively. Then, \(\mathrm{Tr}[X^TY]=\sqrt{d_Ad_B}{\langle \Phi |}^{BB'}(X\otimes Y){|\Phi \rangle }^{AA'}\), where \(d_A:=\dim {\mathcal {H}}^A\), \(d_B:=\dim {\mathcal {H}}^B\) and the transposition is taken with respect to the Schmidt bases of \({|\Phi \rangle }^{AA'}\) and \({|\Phi \rangle }^{BB'}\).

Lemma 33

If \(\varrho ^2\) is classically coherent in XY for a positive semidefinite operator \(\varrho \in {\mathcal {P}}({\mathcal {H}}^{AXY})\), so is \(\varrho \).

Lemma 34

Let \(\pi \) be the maximally mixed state on system A, and let \({\mathcal {C}}\) be the completely dephasing operation on A with respect to a fixed basis \(\{|i\rangle \}_{i=1}^{d_A}\). For any \(\rho \in {\mathcal {P}}({\mathcal {H}}^{AB})\), it holds that

Lemma 35

For subnormalized pure states \({|\psi \rangle },{|\phi \rangle }\in {\mathcal {H}}\) and a real number \(c>0\), suppose that there exists a normalized pure state \({|e\rangle }\in {\mathcal {H}}\) that satisfies \({\left\langle e|\psi \right\rangle }\ge c\) and \({\left\langle e|\phi \right\rangle }\ge c\). Then, \(|{\left\langle \psi |\phi \right\rangle }|\ge 2c^2-1\).

Lemma 36

(Lemma 35 in [45]) Let \(c\in (0,\infty )\) be a constant, \(f:[0,c]\rightarrow {{\mathbb {R}}}\) be a monotonically nondecreasing function that satisfies \(f(c)<\infty \), and \(\{p_k\}_{k\in {{\mathbb {K}}}}\) be a probability distribution on a countable set \({{\mathbb {K}}}\). Suppose \(\epsilon _k\,(k\in {{\mathbb {K}}})\) satisfies \(\epsilon _k\in [0,c]\), and \(\sum _{k\in {{\mathbb {K}}}}p_k\epsilon _k\le \epsilon \) for a given \(\epsilon \in (0,c^2]\). Then we have

We now prove the non-randomized partial decoupling (Theorem 1). As sketched in Sect. 5.1, we proceed the proof in two steps: showing the non-smoothed version in Sect. 6.1, and then smoothing it in Sect. 6.2.

6.1 Proof of the non-smoothed non-randomized partial decoupling

The non-smoothed version of Theorem 1 is given by

where \(\Psi _{\mathrm{av}}^{AR} = \bigoplus _{j=1}^J \Psi _{jj}^{A_lR}\otimes \pi _j^{A_r}\). Note that, due to the definition of the conditional collision entropy (19), (22) and its relation to the conditional min-entropy (see Lemma 23), we have

for a proper choice of \(\varsigma ^{ER}\in {\mathcal {S}}_=({\mathcal {H}}^{ER})\). In addition, it holds that

We first show this relation.

Let \(\Pi _j^{A^*}\) be the projection onto a subspace \({\mathcal {H}}_j^{A_r}\otimes {\mathcal {H}}_j^{{\bar{A}}_r}\subset {\mathcal {H}}^{A^*}\) for each j. Due to the definition of \(F^{A{\bar{A}}\rightarrow A^*}\) given by (31), it holds that

Using the property of the Hilbert–Schmidt norm (Lemma 20), we have

Using Eq. (87) and the explicit form of \({\Lambda }(\Psi ,{\mathcal {T}})\), i.e. \({\Lambda }(\Psi ,{\mathcal {T}}):=F(\Psi ^{AR}\otimes \tau ^{{\bar{A}}E})F^\dagger \), each term in the summand is given by

where the last line follows from Lemma 32. Thus, we obtain (86).

From Eqs. (85) and (86), it suffices to prove that

for any \(\varsigma ^{ER}\in {\mathcal {S}}_=({\mathcal {H}}^{ER})\). In the following, we denote the L.H.S. of Ineq. (90) by \(\kappa \). Due to Lemma 14, for any \(\varsigma \in {\mathcal {S}}_=({\mathcal {H}}^{ER})\), we have

Using this and Jensen’s inequality, we obtain

Noting that \(\Psi _{jj}^{A_lR} = \mathrm{Tr}_{A_r}[\Psi _{jj}^{A_lA_rR}] = \mathrm{Tr}_{A_r}[\Psi _{\mathrm{av}, jj}^{A_lR}\otimes \pi _j^{A_r}]=\Psi _{\mathrm{av}, jj}^{A_lR}\), we can apply Lemma 10 for \(X^{AR}=\Psi ^{AR}-\Psi _{\mathrm{av}}^{AR}\) and \(\sigma =\mathrm{id}\). This yields

where the second line follows from the fact that \(\Psi _{\mathrm{av},jk}^{A_lA_rR}=0\) for \(j\ne k\). To calculate the first term in (93), note that

and that

Thus, we simply apply Lemma 34 to obtain

for each j. Substituting this to (93), we arrive at Ineq. (90). \(\square \)

6.2 Proof of the smoothed non-randomized partial decoupling

We now smoothen the conditional min-entropy to complete the proof of Theorem 1. To this end, fix \({\hat{\Psi }}\in {\mathcal {B}}^\epsilon (\Psi )\) and \(\hat{{\mathcal {T}}}\in {\mathcal {B}}_{\mathrm{DSP}}^\mu ({\mathcal {T}})\) so that

Let \(|\Psi _{p,\mathrm{av}}\rangle ^{AA'}\) be a purification of \(\Psi _{\mathrm{av}}^A\). Noting that \(\Psi _{\mathrm{av}}\) is decomposed in the form of (28), by properly choosing a DSP decomposition for \(A'\), it holds that

where \(q_j:={\mathrm {Tr}}{\Psi _{jj}}\) and \(\varpi _j\) is a purification of \(\Psi _{jj}^{A_l}/q_j\) for each j. Let \(\Delta _+^{A'E}\) and \(\Delta _-^{A'E}\) be linear operators on \({\mathcal {H}}^E\otimes {\mathcal {H}}^{A'}\) such that \(\Delta _+^{A'E}\ge 0,\; \Delta _-^{A'E}\ge 0, \; \mathrm{supp}[\Delta _+^{A'E}]\perp \mathrm{supp}[\Delta _-^{A'E}]\) and that

In addition, let \({\mathcal {D}}_+^{A\rightarrow E}\) and \({\mathcal {D}}_-^{A\rightarrow E}\) be superoperators such that

which yields \({\mathcal {T}}-\hat{{\mathcal {T}}}={\mathcal {D}}_{+}-{\mathcal {D}}_{-}\). Note that, in general, it does not necessarily imply that \({\mathcal {D}}_{+}={\mathcal {T}}\) and \({\mathcal {D}}_{-}=\hat{{\mathcal {T}}}\).

We now apply Lemma 11 for the case where \(\sigma =\mathrm{id}\). To obtain the explicit forms, we compute

where we have used the properties of \(\Psi _{p,\mathrm{av}}^{AA'}\), \(\Delta _{\pm }^{A'E}\), and \({\mathcal {D}}_{\pm }^{A \rightarrow E}\) described above. The last line follows from the definition of the DSP norm. Furthermore, introducing a notation \(\bar{{\mathcal {U}}}(\cdot ):={\mathbb {E}}_{U \sim \mathsf{H}_{\times }}[\,{\mathcal {U}}(\cdot )]\), we also have (see Lemma 11 for the definition and properties of \(\delta _{\pm }^{AR}\))

where the fourth line follows from the definition of the DSP norm (13), and the seventh line from the triangle inequality for the DSP norm (Lemma 19). Applying the non-smoothed version of the non-randomized partial decoupling (Ineq. (84)) to a state \({\hat{\Psi }}\) and a CP map \(\hat{{\mathcal {T}}}\), we have

All together, Ineq. (63) in Lemma 11 leads to

which, together with (97), concludes the proof of Theorem 1. \(\square \)

We here show Theorem 3. We first put the following two assumptions, which simplify the proof:

WA 1:

\(E\cong E_cE_r\), where \(E_c\) is a quantum system of dimension J with a fixed orthonormal basis \({\{|j\rangle \}^{J}_{j=1}}\).

WA 2:

The CP map \({\mathcal {T}}^{A \rightarrow E}\) is decomposed into

$$\begin{aligned} {\mathcal {T}}^{A \rightarrow E}(X)=\sum _{j,k=1}^J{{|j\rangle }\!{\langle k|}}^{E_c}\otimes {\mathcal {T}}_{jk}^{A_r \rightarrow E_r}(X_{jk}), \end{aligned}$$

(105)

in which \({\mathcal {T}}_{jk}\) is a linear supermap from \({\mathcal {L}}({\mathcal {H}}^{A_r})\) to \({\mathcal {L}}({\mathcal {H}}^{E_r})\) defined by \({\mathcal {T}}_{jk}(\zeta )={\mathcal {T}}({{|j\rangle }\!{\langle k|}}\otimes \zeta )\) for each j, k.

We show the non-smoothed version in Sect. 7.1 and the smoothed version in Sect. 7.2. The above assumptions are then dropped in Sect. 7.3.

7.1 Proof of the non-smoothed randomized partial decoupling under WA 1 and WA 2

Under the assumptions WA 1 and WA 2, the non-smoothed version of the randomized partial decoupling is given by

Note that, as we will describe in Sect. 7.3 for general cases, the min entropies \(H_{\mathrm{min}}(A|E)_{\tau }\) and \(H_{\mathrm{min}}(A|E)_{{\mathcal {C}}(\tau )}\) are equal to the max entropies \(-H_{\mathrm{max}}(A|B)_{{\mathcal {C}}(\tau )}\) and \(-H_{\mathrm{max}}(A_r|BA_c)_{{\mathcal {C}}(\tau )}\), respectively, due to the duality of the conditional entropies for pure states (Lemma 24). The proof of this inequality will be divided into three steps.

7.1.1 Upper bound on the average trace norm

To prove Ineq. (106), we first introduce the following lemma that relates the average trace norm of an operator \({\mathcal {T}}^{A \rightarrow E} \circ {\mathcal {G}}_\sigma ^A \circ {\mathcal {U}}^A ( X^{AR})\) to the average Hilbert–Schmidt norm.

Lemma 37

Let \(X^{AR}\) be an arbitrary Hermitian operator such that \(X^{AR}=\sum _{j,k=1}^J{{|j\rangle }\!{\langle k|}}^{A_c}\otimes X_{jk}^{A_rR_r}\otimes {{|j\rangle }\!{\langle k|}}^{R_c}\), and let \(\zeta \in {\mathcal {S}}_=({\mathcal {H}}^{E})\) and \(\xi \in {\mathcal {S}}_=({\mathcal {H}}^{R})\) be arbitrary states that are decomposed as \(\zeta ^{E}\!=\!\sum _j{{|j\rangle }\!{\langle j|}}^{E_c}\!\otimes \zeta _j^{E_r}\), \(\xi ^{R}\!=\!\sum _j{{|j\rangle }\!{\langle j|}}^{R_c}\!\otimes \xi _j^{R_r}\), respectively. Then it holds that

where the norm in the R.H.S. is defined by (16).

It should be noted that Lemma 37 provides a stronger inequality than that obtained simply using Lemma 14.

Proof

We exploit techniques developed in [31]. Recall that U is in the form of \(\sum _{j=1}^J{{|j\rangle }\!{\langle j|}}^{A_c} \otimes U_j^{A_r}\), and \(G_\sigma \) is defined by \(G_\sigma :=\sum _{j=1}^J{{|\sigma (j)\rangle }\!{\langle j|}}^{A_c} \otimes I^{A_r}\) for any \(\sigma \in {\mathbb {P}}\).

We define a subnormalized state \(\gamma _\sigma \in {\mathcal {S}}_\le ({\mathcal {H}}^{ER})\) for each \(\sigma \) by \(\gamma _\sigma ^{ER}:=\sum _{j=1}^J{{|\sigma (j)\rangle }\!{\langle \sigma (j)|}}^{E_c}\otimes \zeta _{\sigma (j)}^{E_r}\otimes \xi _j^{R_r}\otimes {{|j\rangle }\!{\langle j|}}^{R_c}\). Further, by letting P be a quantum system with an orthonormal basis \(\{|\sigma \rangle \}_{\sigma \in {\mathbb {P}}}\), we define a subnromalized state \(\gamma \in {\mathcal {S}}_\le ({\mathcal {H}}^{PER})\) by

Using Lemma 14 and Jensen’s inequality, we obtain

In the last line, we used the following relation:

which can be observed from the fact that, due to the decomposition of \({\mathcal {T}}^{A\rightarrow E}\) from WA 2,

Due to the fact that

for all j, we obtain

Substituting this to (109), and by using Jensen’s inequality, we arrive at the desired result. \(\square \)

7.1.2 Generalization of the dequantizing theorem

Our second step to prove the non-smoothed randomized partial decoupling is to generalize the non-smoothed version of the dequantizing theorem (Proposition 3.5 in [31]).

Lemma 38

In the same setting as in Theorem 3, it holds that

where we have defined \(\Psi _{\mathrm{dp}}^{AR}:={\mathcal {C}}^A(\Psi ^{AR})=\sum _{j=1}^J{{|j\rangle }\!{\langle j|}}^{A_c}\otimes \Psi _{jj}^{A_rR}\).

Note that \(\alpha (J)\) is 0 for \(J=1\) and \(\frac{1}{J-1}\) for \(J\ge 2\).

Proof

Since \(\Psi ^{AR}\) and \(\Psi _{\mathrm{av}}^{AR}\) are classically coherent in \(A_cR_c\) by assumption, we can apply Lemma 37 for \(X^{AR}=\Psi ^{AR}-\Psi _{\mathrm{dp}}^{AR}\) to obtain

Noting that \(\Psi _{jj}^{AR}-\Psi _{\mathrm{dp},jj}^{AR}=0\), we can also apply Lemma 10 under the assumption that \(A_l\) is a one-dimensional system, \(r_j=r\) and \(\varsigma ^{ER}=\zeta ^E\otimes \xi ^R\). Then, we obtain, for any \(\sigma \in {\mathbb {P}}\),

where we have used \(d_A=rJ\) in the last line. Taking the case of \(J=1\) into account, and noting that \({\mathbb {E}}_{\sigma }[g(\sigma )]={\mathbb {E}}_{\sigma }[g(\sigma ^{-1})]\) for any function g, it follows that

Here, we have used the definitions \(\Psi _{\mathrm{dp}}^{AR}:={\mathcal {C}}^A(\Psi ^{AR})\) and \(\tau _{\mathrm{dp}}^{AE}:={\mathcal {C}}^A(\tau ^{AE})\) in the sixth line, and Lemma 34 in the seventh line. Due the relation between the conditional collision entropy and the conditional min-entropy (Lemma 23), it is further bounded from above by \(2^{-H_{\mathrm{min}}(A|R)_{\Psi |\xi }-H_{\mathrm{min}}(A|E)_{\tau |\zeta }}\).

Finally, we use the property of the the conditional min-entropy (Lemma 28). There exist normalized states \(\xi \) and \(\zeta \) in the form of

such that \(H_{\mathrm{min}}(A|R)_{\Psi |\xi }=H_{\mathrm{min}}(A|R)_{\Psi }\) and \(H_{\mathrm{min}}(A|E)_{\tau |\zeta }=H_{\mathrm{min}}(A|E)_{\tau }\). Thus, we obtain

which, together with Ineq. (115), complete the proof of Lemma 38. \(\square \)

7.1.3 Proof of the non-smoothed randomized partial decoupling

We now prove the non-smoothed randomized partial decoupling, i.e.,

under the assumptions WA 1 and WA 2. Note that \(\beta (A_r)\) is 0 for \(\mathrm{dim}{\mathcal {H}}^{A_r}=1\) and 1 for \(\mathrm{dim}{\mathcal {H}}^{A_r}\ge 2\). By the triangle inequality, we have

where we have used the fact that the unitary invariance of the Haar measure implies \({\mathcal {U}}^A(\Psi _{\mathrm{av}}^{AR})=\Psi _{\mathrm{av}}^{AR}\) for any unitary U. The first term is bounded by simply using Lemma 38.

To bound the second term in (121), we use Lemma 37, leading to

Since \(\Psi _{\mathrm{dp},jj}^{R}=\Psi _{\mathrm{av},jj}^{R}\) by definition, we can apply Lemma 10 for \(X^{AR}=\Psi ^{AR}_{\mathrm{dp}}-\Psi _{\mathrm{av}}^{AR}\). Noting that \(\Psi _{\mathrm{dp},jk}^{ A_r R}=\Psi _{\mathrm{av},jk}^{ A_r R}=0\) for \(j\ne k\), this yields

Thus, similarly to the derivation around Eq. (117), we obtain

Substituting this into Ineq. (122), and noting that \(\Psi _{\mathrm{dp}}^{AR} - \Psi _{\mathrm{av}}^{AR}=0\) if \(\mathrm{dim}{\mathcal {H}}^{A_r}=1\), we obtain an upper bound on the second term of the R.H.S. in Ineq. (121).

All together, we obtain Ineq. (120) as desired. \(\square \)

7.2 Proof of the randomized partial decoupling under the conditions WA 1 and WA 2

We now show, under the conditions WA 1 and WA 2, the randomized partial decoupling:

where \(\Psi _{\mathrm{av}}^{AR}:={\mathbb {E}}_{U \sim \mathsf{H}_{\times }} [ {\mathcal {U}}^A ( \Psi ^{AR} ) ]\). The function \(\alpha (J)\) is 0 for \(J=1\) and \(\frac{1}{J-1}\) for \(J\ge 2\), and \(\beta (A_r)\) is 0 for \(\mathrm{dim}{\mathcal {H}}^{A_r}=1\) and 1 for \(\mathrm{dim}{\mathcal {H}}^{A_r}\ge 2\). The exponents \({\tilde{H}}_I\) and \({\tilde{H}}_{I\!I}\) are given by

Note that, the duality of the conditional smooth entropies for pure states (Lemma 24) implies \(H_{\mathrm{min}}^\mu (A|E)_{\tau }=-H_{\mathrm{max}}^\mu (A|B)_{{\mathcal {C}}(\tau )}\) and \(H_{\mathrm{min}}^\mu (A|E)_{{\mathcal {C}}(\tau )}=-H_{\mathrm{max}}^\mu (A_r|BA_c)_{{\mathcal {C}}(\tau )}\) (see Sect. 7.3 for the detail).

To prove the statement, we again start with the triangle inequaltiy: By the triangle inequality, we have

Below, we derive upper bounds on the two terms in the R.H.S. separately.

For an upper bound on the first term, fix \({\hat{\Psi }}\in {\mathcal {B}}^\epsilon (\Psi )\) and \({\hat{\tau }}\in {\mathcal {B}}^\mu (\tau )\) so that we have \(H_{\mathrm{min}}(A|R)_{{{\hat{\Psi }}}}=H_{\mathrm{min}}^\epsilon (A|R)_{\Psi }\) and \(H_{\mathrm{min}}(A|E)_{{{\hat{\tau }}}}=H_{\mathrm{min}}^\epsilon (A|E)_{\tau }\). Let \(\Delta _+^{A'E}\) and \(\Delta _-^{A'E}\) be linear operators on \({\mathcal {H}}^{A'}\otimes {\mathcal {H}}^E\) such that

and that

Let \({\mathcal {D}}_+^{A\rightarrow E}\) and \({\mathcal {D}}_-^{A\rightarrow E}\) be superoperators such that

which yields \({\mathcal {T}}-\hat{{\mathcal {T}}}={\mathcal {D}}_+-{\mathcal {D}}_-\). From Lemma 11, the CP map \(\hat{{\mathcal {T}}}^{A \rightarrow E}\) having the Choi–Jamiołkowski state \({\hat{\tau }}^{AE}\) satisfies

Due to Lemma 38, the first term in the R.H.S. of the above inequality is bounded as

Similarly to (101) and (102), using (128) and (129), it turns out that the second and the third terms are bounded from above by

and

respectively. Hence, we obtain

In the same way, we also have

Substituting these inequalities into Eq. (127), we obtain the desired result (Ineq. (125)).

\(\square \)

7.3 Dropping working assumptions WA 1 and WA 2

We now drop the working assumptions WA 1 and WA 2, and show that Theorem 3 holds in general. To remind the working assumptions, we write them down here again:

WA 1:

\(E\cong E_cE_r\), where \(E_c\) is a quantum system of dimension J

WA 2:

The CP map \({\mathcal {T}}^{A \rightarrow E}\) is decomposed into

$$\begin{aligned} {\mathcal {T}}^{A \rightarrow E}(X)=\sum _{j,k=1}^J{{|j\rangle }\!{\langle k|}}^{E_c}\otimes {\mathcal {T}}_{jk}^{A_r \rightarrow E_r}(X_{jk}), \end{aligned}$$

(136)

in which \({\mathcal {T}}_{jk}\) is a linear supermap from \({\mathcal {L}}({\mathcal {H}}^{A_r})\) to \({\mathcal {L}}({\mathcal {H}}^{E_r})\) defined by \({\mathcal {T}}_{jk}(\zeta )={\mathcal {T}}({{|j\rangle }\!{\langle k|}}\otimes \zeta )\) for each j, k,

To drop these assumptions, we use Lemma 12. Using the linear isometry \(Y^{A_c\rightarrow A_cE_c}\), given by \(Y=\sum _{j}{|jj\rangle }^{A_cE_c}{\langle j|}^{A_c}\), we define a new CP map \(\check{{\mathcal {T}}}^{A \rightarrow EE_c}\) by \({\mathcal {T}}^{A \rightarrow E} \circ {\mathcal {Y}}^{A_c \rightarrow A_c E_c}\). Lemma 12 states that

Let \(\check{\tau }^{AEE_c}\) be the Choi–Jamiołkowski state of \({\check{{\mathcal {T}}}}^{A \rightarrow EE_c}\), i.e., \(\check{\tau }^{AEE_c}:={\mathfrak {J}}(\check{{\mathcal {T}}}^{A \rightarrow EE_c})\). We denote by \(|\tau \rangle ^{ABE}\) a purification of \(\tau ^{AE}\) such that the reduced state \(\tau ^{AB}\) is equal to \({\mathfrak {J}}({\mathcal {T}}^{A\rightarrow B})\), where \({\mathcal {T}}^{A\rightarrow B}\) is the complementary map of \({\mathcal {T}}^{A\rightarrow E}\). Then, it is clear that \(\check{\tau }^{AEE_c}={\mathcal {Y}}(\tau ^{AE})\), which implies that a purification \(|\check{\tau }\rangle ^{ABEE_c}\) of \(\check{\tau }^{AEE_c}\) is given by \(|\check{\tau }\rangle ^{ABEE_c}=Y|\tau \rangle ^{ABE}\). It is also straightforward to verify that \(\check{\tau }^{AB}={\mathcal {C}}(\tau ^{AB})\).

The new CP map \(\check{{\mathcal {T}}}^{A \rightarrow EE_c}\) clearly satisfies WA 1 and WA 2. Hence, using Eq. (137) and achievability of the randomized partial decoupling under those assumptions (Ineq. (125)), we obtain

Due to the duality of conditional smooth entropies (Lemma 24), we have

Using the property of the conditional smooth entropy for classical-quantum states (Lemma 27), and noting that \(\check{\tau }^{AEE_c}\) is classically coherent in \(A_cE_c\), we also have

Substituting these into (138), and noting that \({\mathrm {Tr}}[\check{\tau }]={\mathrm {Tr}}[\tau ]\le 1\) by assumption, we obtain Theorem 3. \(\square \)

We provide the proof of Theorem 4 under Converse Conditions 1 and 2, which are

CC 1:

\(\dim {\mathcal {H}}_j^l=1,\quad \dim {\mathcal {H}}_j^r=r \quad (j=1,\ldots , J)\).

CC 2:

The initial (normalized) state \(\Psi ^{AR}\) is classically coherent in \(A_cR_c\).

The proof proceeds along the similar line as the proof of the converse part of the one-shot decoupling theorem (see Section 4 in [11]). Suppose that there exists a normalized state \(\Omega ^{ER}:=\sum _{j=1}^J\varsigma _j^E\otimes \Psi _{jj}^{R_r}\otimes {{|j\rangle }\!{\langle j|}}^{R_c}\), where \(\{\varsigma _j\}_{j=1}^J\) are normalized states on E, such that, for \(\delta >0\),

We separately prove that, in this case, the following inequalities hold for any \(\upsilon \in [0,1/2)\) and \(\iota \in (0,1]\):

Here, \(\lambda \) and \(\lambda '\) are given by

and \(x:=\sqrt{2}\root 4 \of {24\upsilon +2\delta }\).

First, we prove these relations based on the working assumptions WA 1 and WA 2 in Sects. 8.1 and 8.2. We complete the proof of Theorem 4 by dropping these assumptions in Sect. 8.3.

8.1 Proof of Ineq. (142) under WA 1 and WA 2

To prove Ineq. (142), we introduce the following notations:

• \({|\Psi \rangle }^{ARD}\): A purification of \(\Psi ^{AR}\).

• \(V^{A\rightarrow BE}\): A Stinespring dilation of \({\mathcal {T}}^{A\rightarrow E}\).

• \({|\Theta \rangle }^{BERD}\): A pure state on BERD defined by \({|\Theta \rangle }:=V{|\Psi \rangle }\).

• \({|\theta \rangle }^{BERD}\): A subnormalized pure state on BERD such that

  $$\begin{aligned} H_{\mathrm{max}}(RD|E)_\theta =H_{\mathrm{max}}^{\upsilon }(RD|E)_{\Theta }, \quad P(\theta ^{BERD},\Theta ^{BERD})\le \upsilon , \end{aligned}$$

  (146)

  which is classically coherent in \(E_cR_c\).

Note that the existence of \({|\theta \rangle }\) satisfying the above condition follows from Lemma 30 about the property of the conditional max-entropy for classically coherent states. From the definition of the conditional max-entropy, and from the definitions of \(\theta \) and \(\Theta \), we have

The proof of Ineq. (142) proceeds as follows. First, we prove that for any \(X\in {\mathcal {P}}({\mathcal {H}}^{ER})\), we can construct a subnormalized pure state \({|\theta _X\rangle }^{BERD}\) from \(\theta \) and X such that

Second, we prove that if \(X^{ER}\) satisfies certain conditions, the \(\theta _X\) satisfies

Third, we prove that for a proper choice of \(X^{ER}\) satisfying the conditions for (149), Ineq. (148) implies

Combining (147), (149) and (150), we arrive at (142).

Before we start, we remark that the partial decoupling condition (141) is used in the proof of (149), particularly when we evaluate the smoothing parameter \(\lambda \).

8.1.1 Proof of Ineq. (148)

Define \( Y^{ERD}:= 2^{-\frac{1}{2}H_{\mathrm{max}}(RD|E)_{\theta |\theta }} \cdot (\theta ^E)^{-\frac{1}{2}} \sqrt{ (\theta ^E)^{\frac{1}{2}} \theta ^{ERD}(\theta ^E)^{\frac{1}{2}} } (\theta ^E)^{-\frac{1}{2}}. \) Due to Lemma 25, it holds that \( \theta ^{BERD} \le 2^{H_{\mathrm{max}}(RD|E)_{\theta |\theta }} \cdot I^B \otimes Y^{ERD} \) and thus

Let \(X\in {\mathcal {P}}({\mathcal {H}}^{ER})\) be an arbitrary positive semidefinite operator, and define

and \({|\theta _X\rangle }^{BERD}:=\Gamma _X^{ER}{|\theta \rangle }^{BERD}\). From (151), \(X\ge 0\) and the assumption that \(\iota \le 1\), it follows that

and consequently,

8.1.2 Proof of Ineq. (149)

Define a subnormalized probability distribution \(\bigl \{q_k :=\Vert {\langle k|}^{R_c}{|\theta \rangle }\Vert _1^2 \bigr \}_{k=1}^J\), and normalized pure states \({|\theta _k\rangle }^{E_rR_r}\) by \({|\theta _k\rangle }^{E_rR_r}:=q_k^{-1/2}{\langle k|}^{E_c}{\langle k|}^{R_c}{|\theta \rangle }\) for k such that \(q_k>0\). Let \(\omega \in {\mathcal {S}}_\le ({\mathcal {H}}^{ER})\) be a subnormalized state defined by

where \(\theta _k^{E_r}\) and \(\theta _k^{R_r}\) are reduced states of \({|\theta _k\rangle }\) on \(E_r\) and \(R_r\), respectively. Consider an arbitrary \(X\in {\mathcal {P}}({\mathcal {H}}^{ER})\) so that

and

As we prove in Appendix F, for any such X, the state \({|\theta _X\rangle }\) is a subnormalized pure state, and the partial decoupling condition (141) implies

where \(\lambda \) is defined by (144). Due to the definition of \(\Theta \) and the invariance of min-entropy under local isometry (Lemma 21), we obtain

8.1.3 Proof of Ineq. (150)

We choose a proper \(X^{ER}\) satisfying Conditions (156) and (157), and prove Ineq. (150) from (148). Define a normalized state

where \(J':=|\{k|1\le k\le J,\,q_k>0\}|\), and \(X^{ER}:=J'\cdot I^E\otimes {\hat{\theta }}^R\). Noting that \(\theta \) is classically coherent in \(E_cR_c\), it is straightforward to verify that

Consequently, \(X^{ER}\) satisfies Conditions (156) and (157).

Using Ineq. (148), we have

which implies, together from the definition of the conditional min-entropy and \(J'\le J\), that

8.2 Proof of Ineq. (143) under WA 1 and WA 2

We prove (143), that is,

under the assumptions WA 1 and WA 2. To show this, we introduce the following notations:

• \({|\Psi \rangle }^{ARD}\): A purification of \(\Psi ^{AR}\), in the same way as in the previous subsection.

• \({\mathcal {T}}_{{\mathcal {C}}}^{A\rightarrow E}\): A trace preserving CP map defined by \({\mathcal {T}}_{{\mathcal {C}}}^{A\rightarrow E}:={\mathcal {T}}^{A\rightarrow E}\circ {\mathcal {C}}^A\).

• \(\Theta _{{\mathcal {C}}}^{ERD}\): A normalized state on ERD defined by \(\Theta _{{\mathcal {C}}}^{ERD}:={\mathcal {T}}^{A\rightarrow E}\circ {\mathcal {C}}^A(\Psi ^{ARD})\).

• \(\theta _{{\mathcal {C}}}^{ERD}\): A subnormalized state on ERD such that \(H_{\mathrm{max}}(RD|E)_{\theta _{{\mathcal {C}}}}=H_{\mathrm{max}}^{\upsilon }(RD|E)_{\Theta _{{\mathcal {C}}}}\) and \(P(\theta _{{\mathcal {C}}},\Theta _{{\mathcal {C}}})\le \upsilon \), which is classically coherent and diagonal in \(E_cR_c\).

• \({\hat{\theta }}_{{\mathcal {C}}}^{ERD}\): A normalized state on ERD defined by \({\hat{\theta }}_{{\mathcal {C}}}^{ERD}:=\theta _{{\mathcal {C}}}^{ERD}/\mathrm{Tr}[{\theta _{{\mathcal {C}}}}]\).

The assumptions WA 1 and WA 2 imply that \(\Theta _{{\mathcal {C}}}^{ERD}\) is classically coherent and diagonal in \(E_cR_c\). Thus, the existence of \(\theta _{{\mathcal {C}}}\) satisfying the above condition follows from Lemma 30. By definition, we have

The proof of Ineq. (164) proceeds as follows. First, we introduce a quantum state \({\hat{\Psi }}^{ARD}\) and a quantum channel \(\hat{{\mathcal {T}}}_{{\mathcal {C}}}^{A\rightarrow E}\), such that \(\hat{{\mathcal {T}}}_{{\mathcal {C}}}^{A\rightarrow E}({\hat{\Psi }}^{ARD})\) is close to the state \({\mathcal {T}}_{{\mathcal {C}}}^{A\rightarrow E}(\Psi ^{ARD})\). Second, we apply the converse inequality (142) to the channel \(\hat{{\mathcal {T}}}_{{\mathcal {C}},k}^{A_r\rightarrow E}\) and the state \({\hat{\Psi }}_{k}^{A_rR_r}\), which are obtained by restricting \(\hat{{\mathcal {T}}}_{{\mathcal {C}}}^{A\rightarrow E}\) and \({\hat{\Psi }}^{ARD}\) to the k-th subspace. The obtained inequalities are then averaged over all k. Finally, by using the properties of the smooth entropies, we obtain Ineq. (164).

To explicitly define \({\hat{\Psi }}^{ARD}\) and \(\hat{{\mathcal {T}}}_{{\mathcal {C}}}^{A\rightarrow E}\), observe that, since \(\Theta _{{\mathcal {C}}}\) is a normalized state, we have

Thus, due to Uhlmann’s theorem, and noting that \(\Theta _{{\mathcal {C}}}^{RD}=\Psi ^{RD}\), there exists a normalized pure state \(|{\hat{\Psi }}\rangle ^{ARD}\) such that \(P(\Psi ^{ARD},{\hat{\Psi }}^{ARD})\le \upsilon \) and \({\hat{\Psi }}^{RD}={\hat{\theta }}_{{\mathcal {C}}}^{RD}\). It follows from the latter equality that there exists a trace preserving CP map \(\hat{{\mathcal {T}}}_{{\mathcal {C}}}^{A\rightarrow E}\) satisfying \({\hat{\theta }}_{{\mathcal {C}}}^{ERD}=\hat{{\mathcal {T}}}_{{\mathcal {C}}}^{A\rightarrow E}({\hat{\Psi }}^{ARD})\).

8.2.1 Block-wise application of the converse inequality (142)

Define a normalized probability distribution \(\{r_k:=\Vert {\langle k|}^{R_c}{|{\hat{\Psi }}\rangle }\Vert _1^2\}_{k=1}^J\), and let \( |{\hat{\Psi }}_{k}\rangle ^{A_rR_rD}:=r_k^{-1/2}{\langle k|}^{E_c}{\langle k|}^{R_c}{|{\hat{\Psi }}\rangle } \) for k such that \(r_k>0\). Since \({\hat{\Psi }}\) is classically coherent in \(E_cR_c\), the \({\hat{\Psi }}_{k}\) are normalized states. Define also a CP map \(\hat{{\mathcal {T}}}_{{\mathcal {C}},k}^{A_r\rightarrow E}\) by

which is trace preserving due to the assumptions WA1 and WA2. We apply the converse inequality (142) for \({\hat{\Psi }}_{k}\) and \(\hat{{\mathcal {T}}}_{{\mathcal {C}},k}^{A_r\rightarrow E}\) for each k, by letting \(J=1\). We particularly choose \(\upsilon =0\), in which case Ineq. (142) leads to

The smoothing parameter \(\lambda _k\) is given by

A simple calculation yields

8.2.2 Calculation of averaged entropies

Using the fact that \({\hat{\theta }}_{{\mathcal {C}}}\) is classically coherent and diagonal in \(E_cR_c\), it is straightforward to verify that \( {\hat{\theta }}_{{\mathcal {C}}}^{ERD}=\hat{{\mathcal {T}}}_{{\mathcal {C}}}^{A\rightarrow E}({\hat{\Psi }}^{ARD})=\sum _kr_k\hat{{\mathcal {T}}}_{{\mathcal {C}},k}^{A_r\rightarrow E}({\hat{\Psi }}_{k}^{A_rR_rD})\otimes {{|k\rangle }\!{\langle k|}}^{R_c} \). Thus, by using the property of the smooth conditional entropies (Lemmas 26 and 31 ) and \(P(\Psi ,{\hat{\Psi }})\le \upsilon \), both sides of Ineq. (170) are calculated to be

where \({\bar{\lambda }}:=\sum _kr_k\lambda _k\). Combining these all together with Eq. (165), we obtain

As we prove in Appendix G, the partial decoupling condition (141) implies

where \(\lambda (\iota ,x):=2\sqrt{\iota +2x}+\sqrt{x}+2x\). A simple calculation then yields

whose right-hand side is exactly \(\lambda '\) given in (145). In addition, noting that \(\Theta _{{\mathcal {C}}}\) is normalized, and by using the relation between the purified distance and the trace distance (Property 2 in Lemma 16), the last term in the R.H.S. of (173) is calculated to be

Combining these all together, we arrive at

8.3 Dropping the working assumptions WA 1 and WA 2

We here show that the working assumptions WA 1 and WA 2 can be dropped. The proof is based on Lemma 12. Since the CP map \(\check{{\mathcal {T}}}^{A \rightarrow E E_c}\), defined in Lemma 12, satisfy both conditions, it satisfies Ineq. (142), which is

Let \(V^{A\rightarrow BE}\) be a Stinespring dilation of \({\mathcal {T}}^{A\rightarrow E}\), and let \(Z^{R_c\rightarrow R_cE_c}\) be a linear isometry defined by \(Z:=\sum _{j}{|jj\rangle }^{R_cE_c}{\langle j|}^{R_c}\). A purification \(|\vartheta \rangle ^{BRDEE_c}\) of \(\check{{\mathcal {T}}}^{A\rightarrow EE_c}(\Psi ^{ARD})\) is given by \(|\vartheta \rangle ^{BRDEE_c}=(V^{A\rightarrow BE}\otimes Z^{R_c\rightarrow R_cE_c})|\Psi \rangle ^{ARD}\), and satisfies \(\vartheta ^{BRD}={\mathcal {T}}^{A\rightarrow B}\circ {\mathcal {C}}^A(\Psi ^{ARD})\). Hence, due to the duality for the conditional smooth entropy (Lemma 24), it holds that

Combining this with (178), we conclude

The map \(\check{{\mathcal {T}}}^{A \rightarrow E E_c}\) also satisfies Ineq. (143):

Similarly to (179) and (140), by using the property of the conditional max entropy for classical-quantum states (Lemma 29), we have

which leads to

This concludes the proof of Theorem 4 for any trace preserving CP map \({\mathcal {T}}^{A \rightarrow E}\). \(\square \)

In this paper, we have proposed and analyzed a task that we call partial decoupling. We have presented two different formulations of partial decoupling, and derived lower and upper bounds on how precisely partial decoupling can be achieved. The bounds are represented in terms of the smooth conditional entropies of quantum states involving the initial state, the channel and the decomposition of the Hilbert space. Thereby we provided a generalization of the decoupling theorem in the version of [11], by incorporating the direct-sum-product decomposition of the Hilbert space. Applications of our result to quantum communication tasks and black hole information paradox are provided in Refs. [21,22,23] and [24], respectively. A future direction is to apply the result to various scenarios that have been analyzed in terms of the decoupling theorem, such as relative thermalization [10] and area laws [8] in the foundation of statistical mechanics.