Introduction

Q uantum Multiple Access Channel (QMAC) was first introduced by Winter 1, which takes two or more messages (classical or quantum) as inputs and produces one output.

Similar to the classical world, decoding messages over a QMAC is based on two main techniques: successive decoding and simultaneous decoding. In 1, the author employs the successive decoding technique. A quantum broadcast channel (QBC) has a sender and two or more receivers, in which the sender wishes to transmit two or more messages (classical or quantum) over the channel to the receivers. The QBC In recent decades, with the development of quantum data processing and its applications, the necessity to study the security of quantum channels has increased. In this regard, the quantum wiretap channel (QWTC) was first introduced in 2, and 3. Then, the secrecy constraints are extended to multi-user quantum channels such as quantum interference channel (QIC) 45, and quantum multiple access channel (QMAC) 678910.

There are two bottlenecks in studying the security of quantum channels. The first is decoding three or more messages simultaneously (reliability), and the second is about how to securely decode two or more messages (security). The first bottleneck arises from the nonexistence of a general quantum joint typicality lemma. However, this problem has been solved in some cases, such as the min-entropy case and QMACs with commutative output 11. Therefore, in i.i.d. case, successive decoding combined with time-sharing techniques should be used. In this setting, transmitters are allowed to transmit their messages using the channel only once. Sen proved a joint typicality lemma which helps to decode any number of messages simultaneously in the one-shot case 11. Obtaining secrecy against the eavesdropper by Wyner's approach 12 of randomizing over a block becomes problematic in the quantum setting. Wyner's technique has been shown to work for point-to-point quantum channels by Devetak 3 and explained further in 13. However, there are no easy generalizations to multiple senders for a quantum channel. This issue is discussed in detail in 13.

In this paper, we want to investigate the secrecy problem of quantum multiple access channel (QMAC) with classical inputs under one-shot setting. Also, we have investigated some bottlenecks connected to the decoding process for CQ-MA-WTC. The achievement of this paper is about analyzing bottlenecks in the decoding process and providing solutions to overcome them.

Also, we present two techniques for quantum multiple access wiretap channel with classical inputs(CQ-MA-WTC). The first approach is based on the method presented in 11, and another technique is the simultaneous position-based decoder. From 14, we know that the simultaneous position-based decoder tends to a multiple quantum hypothesis testing problem solvable in a special case. Also, from 15, we know that the convex split lemma could not be used to analyze the privacy of multiple messages in simultaneous decoding.

The paper is organized as follows: We have explained related works, and our motivations in Section 2, and Section 3, respectively. In Section 4, some seminal definitions are presented. In Section 5, the main channel and information processing tasks are presented. In Section 6, the results and main theorems are presented. In Section 7 we have compared our method and results with other methods. Section 8 is dedicated to discussion and future works.

Related Works

The security problem of QMACs has received a lot of attention in recent years, both in i.i.d. and one-shot regimes. The problem of secure communication over QMACs with classical inputs was first investigated by the authors in the i.i.d. regime using successive decoding 6. After that, we studied the main channel under the one-shot setting [10] using Sen's one-shot joint typicality lemma 11 and convex splitting 16. Also, the main channel under the one-shot setting and entanglement assistance is studied by the authors in 10 using the simultaneous position-based decoder. As we know from the quantum information theory and its computational bottlenecks, the study of the security problem of multipartite quantum channels faces some computational limits. Some papers are written in order to overcome or bypass these bottlenecks 51013, which we will explain in the following sections. In 13, the authors suggested a new approach to securely decode C-QMA-WTC under the one-shot setting instead of using convex splitting. Also, the paper 17 studies a degraded version of C-QMA-WTC in the i.i.d. regime.

Motivation

Some papers are written to construct new decoding approaches, such as simultaneous position-based decoding 1014. In this paper, we want to have a comprehensive study about C-QMA-WTC in i.i.d. and one-shot regimes. We also construct a new method based on introducing dual channels to bypass the multiple quantum hypothesis testing problem and the smoothing bottleneck of the multipartite convex split lemma.

Preliminaries

Let A (Alice), B (Bob) and C (Charlie) be three quantum systems. These quantum systems can be denoted by their corresponding Hilbert spaces as H A , H B and H C . ρ A , ρ B and ρ C are density operators of the above quantum systems, while ρ ABC is the shared state (joint state) between Alice, Bob, and Charlie. A density operator is a positive semi-definite operator with a unit trace. Every quantum state can be defined by a partial trace operator over the shared state (joint state). The partial trace is used to model the lack of access to a quantum system. Thus, ρ A = T r BC {ρ ABC } is Alice's density operator using partial trace. |ψ⟩ A denotes the pure state of system A. Also, ψ A = |ψ⟩ ⟨ψ| A is the corresponding density operator. H(A) ρ = −T r{ρ A log ρ A } is the von Neumann entropy of the state ρ A . The quantum conditional entropy can be defined as H(A|B) σ = H(A, B) σ − H(B) σ for an arbitrary bipartite state σ AB . The quantum mutual information and the conditional quantum mutual information are defined as follows:

Quantum operations can be denoted by completely positive trace-preserving (CPTP) maps N A→B .

CPTP maps take state A as input and produce state B. The distance between two quantum states, such as A and B, is defined by trace distance. The trace distance between two arbitrary states, such as σ and ρ, is:

where |ψ|= ψ † ψ. This quantity is zero for two perfectly distinguishable states.

For two arbitrary density operators such as (ρ, σ), fidelity and purified distance can be defined as

, and P (ρ, σ) := 1 − F (ρ, σ) 2 , respectively. Most of the above definitions are given in 16.

Definition 1 (Hypothesis testing mutual information). Hypothesis testing mutual information is denoted by

, ϵ ∈ (0, 1) and is considered as quantum hypothesis testing divergence 14 where D ϵ H (.∥.) is hypothesis testing relative entropy 14. ϵ is the smoothing variable, ρ H X H Y is the joint classical-quantum state of input and output over their Hilbert spaces (H X , H Y ), and it can be shown as ρ XY :

where p X is the input distribution.

Definition 2 (Quantum relative entropy 18). Consider states ρ X , σ X ∈ D(H X ). The quantum relative entropy is defined as:

where supp (σ X ) refers to the set-theoretic support of σ. supp (σ) is the subspace of H spanned by all eigenvectors of σ with non-zero eigenvalues.

Fact 1.

The following relation exists between the quantum relative entropy and hypothesis testing relative entropy for ϵ ∈ (0, 1) 14:

where

Definition 3 (Max mutual information 19). Consider a bipartite state ρ XY and a parameter ϵ ∈ (0, 1). The max mutual information can be defined as follows:

where ρ refers to the state ρ XY and D max (.∥.) is the max-relative entropy 20 for ρ X , σ X ∈ H X :

Definition 4 (Quantum smooth max relative entropy 20). Consider states ρ X , σ X ∈ D(H X ), and ϵ ∈ (0, 1). The quantum smooth max relative entropy is defined as:

where

Definition 5 (Quantum smooth max mutual information 19). Consider ρ XY := x∈X p X (x) |x⟩ ⟨x| X ⊗ ρ x Y as a classical-quantum state and a parameter ϵ ∈ (0, 1). The smooth max mutual information between the systems X and Y can be defined as follows:

Definition 6 (Conditional smooth hypothesis testing mutual information 21). Consider ρ XY Z := z∈Z p Z (z) |z⟩ ⟨z| Z ⊗ ρ z XY be a tripartite classical-quantum state and ϵ ∈ (0, 1). We define,

be a tripartite classical-quantum state and ϵ ∈ (0, 1).

The following relation holds,

(Alternate smooth max-mutual information). Consider a bipartite state ρ XY and a parameter ϵ ∈ (0, 1). The alternate definition of the smooth max-mutual information between the systems X and Y can be defined as follows:

Fact 3 (Relation between two definitions of the smooth max mutual information 22). Let ϵ ∈ (0, 1) and γ ∈ (0, ϵ) For a bipartite state ρ XY , it holds that:

Definition 8 (Conditional smooth max mutual information 21). Consider ρ XY Z := z∈Z p Z (z) |z⟩ ⟨z| Z ⊗ ρ z XY be a tripartite classicalquantum state and ϵ ∈ (0, 1). We define,

be a tripartite classical-quantum state and ϵ ∈ (0, 1).

The following relation holds,

Definition 9 (Quantum Rényi relative entropy of order α 14). The quantum Rényi relative entropy of order α for a state ρ ∈ D(H), a positive semidefinite operator σ, and α ∈ [0, 1) ∪ (1, +∞) can be defined as follows:

Also, Rényi entropy of order α can be defined as follows:

Definition 10 (One-shot inner bound of a classical-quantum multiple access channel 11).

A two-user C-QMAC under the one-shot setting is a triple (

, where X 1 and X 2 are the alphabet sets of two classical inputs, and Y is the output system. ρ Y x1x2 is a quantum state. The CPTP of the channel is N X1X2→Y . A one-shot inner bound of a C-QMAC is as follows:

+ log ϵ with decoding error at most 49 √ ϵ, where I ϵ H (.) is the hypothesis testing mutual information defined in Definition 1 with respect to the controlling state:

and Q is a time-sharing variable.

Note that I ϵ H (:) is the difference between a Rényi entropy of order two and a conditional quantum entropy.

Lemma 1. [16] Given the control state in Equation 4

(without time-sharing variable), δ ′ > 0 and 0 < ϵ ′ < δ ′ , let {x 1 , . . . , x K1 } and {y 1 , . . . , y K2 } be i.i.d. samples from the distributions p X and p Y , respectively. Then, if

Proof: see 13. Lemma 2 (Convex split lemma). 1618 Let p XY be an arbitrary state and suppose that τ X1,...,X k B be the following state:

Proof: See 18. Lemma 3 (Hayashi-Nagaoka inequality).

[27] Suppose that S, T ∈ P(H X ) such that I −S ∈ P(H X ) are operators such that T ≥ 0 and 0 ≤ S ≤ I, then for all positive constant c, the following relation holds:

, where X i , i ∈ {1, 2} denote the input alphabet sets and Y, Z denote the output systems (Y denotes the channel output at the legitimate receiver (Charlie), and Z is the channel output at the eavesdropper). ρ Y Z x1x2 is the system output's quantum state. Both users want to transmit their messages as securely as possible over a CQ-MA-WTC to the receiver.

The main channel is illustrated in Figure 1.

Consider the main channel illustrated in Figure 1. Each user chooses its message m i ; i ∈ {1, 2} from its message set M i = [1 : |M i |= 2 Ri ]; i ∈ {1, 2} (R 1 and R 2 are the transmitting rates corresponding to the first and the second messages, respectively) and send it over a CQ-MA-WTC. The users also use two junk variables k i ; i ∈ {1, 2} from two amplification sets

for randomizing Eve's knowledge. We have two doubly indexed codebooks x 1 (m 1 , k 1 ), and x 2 (m 2 , k 2 ), for user-1 and user-2, respectively.

The above channel model is the same as what is described in 7810, but here, we have considered the main channel with randomness-assisted codes.

Main Results

In this section, we present the main results.

Corollary 1 gives a one-shot achievable secrecy rate region for sending classical messages over a CQ-MA-WTC based on Sen's quantum joint typicality lemma 11. The second theorem presents a novel approach to decode both messages over a CQ-MA-WTC reliably and confidentially (simultaneous position-based decoder). It should be noted that Corollary 1 and Theorem 1 use the same method to prove the security requirements. Also, we present a theorem that tries to overcome the bottlenecks connected to Theorem 1.

Corollary 1 (One-shot achievable rate region for CQ-MA-WTC). Consider a two-user CQ-MA-WTC that accepts X 1 and X 2 as inputs and Y, Z as outputs. ρ Y Z x1x2 is the channel density operator. For any fixed ϵ ∈ (0, 1), ϵ ′ ∈ (0, δ ′ ) and δ, δ ′ such that δ, δ ′ > 0, the rate pair (R 1 , R 2 , 49 √ ϵ + 20δ ′ 1 8 ) is achievable to satisfy the following inequalities:

where η = δ ′ − ϵ ′ and the union is taken over input distribution p Q (q)p X1|Q (x 1 |q)p X2|Q (x 1 |q). Q is the time-sharing random variable, and all of the mutual information quantities are taken with respect to the following state:

Proof: See Appendix 1.

Sketch of proof : The proof has two steps: 1-Reliable decoding based on Sen's quantum one-shot joint typicality (Definition 10). 2-Secure decoding based on Lemma 1. Theorem 1 (One-shot lower bound for CQ-MA-WTC). For any fixed ϵ ∈ (0, 1), ϵ ′ ∈ (0, 1), and δ, δ ′ such that δ ∈ (0, ϵ), δ ′ ∈ (0, ϵ ′ ), there exists a one-shot code for the channel N X1X2→Y Z , if rate pair (R 1 , R 2 , ϵ + 2δ + 20δ ′ 1 8 ) satisfies the following bounds:

where η = δ ′ −ϵ ′ and the union is taken over input distribution p Q (q)p X1|Q (x 1 |q)p X2|Q (x 2 |q). Q is the timesharing random variable, and all mutual information quantities are taken with respect to the state Equation 5.

Proof: In Appendix 2.

Sketch of proof:

The proof has two steps: 1-Reliable decoding based on the simultaneous positionbased technique: for simplicity of analysis, we merge reliability and security criteria into a single criterion 18. 2-Secure decoding based on the Lemma 1. Remark 1. It should be noted that both of the above theorems tend to the same result if and only if δ = ϵ.

As mentioned before, the simultaneous positionbased decoder tends to a multiple hypothesis testing problem, which is unsolvable in the general case. Also, the convex split lemma (Lemma 2) does not make sense in the simultaneous decoding, because it runs to the famous smoothing bottleneck of quantum information theory. Now, consider the channel illustrated in Figure 2. This channel accepts two or more messages from one user. We call this channel Figure 2. The PP-QWTC model a point-to-point quantum wiretap channel with multiple messages (PP-QWTC). Consider PP-QWTC with classical messages. This channel is studied in 23 under a different scenario wherein a sender wants to send classical and quantum messages simultaneously to a legitimate receiver.

Information processing task: Two classical messages, (m 1 , m 2 ) ∈ M 1 ×M 2 are possessed by a sender (Alice) and are transmitted to a receiver (Bob) in the presence of a passive wiretapper over a point-to-point quantum channel under the one-shot scenario. Both of the messages should be kept as securely as possible from the wiretapper. The PP-QWTC is a triple

, where X denotes the input alphabet sets, and Y, Z denote the output systems (Y denotes the channel output at the legitimate receiver (Bob), and Z is the channel output at the eavesdropper).

is the system output's quantum state.

Alice chooses its message m i ; i ∈ {1, 2} from its message set M i = [1 : |M i |= 2 Ri ], i ∈ {1, 2} and sends it over a PP-QWTC. Alice also uses two junk variables k i ; i ∈ {1, 2} from two amplification sets

Ri ], i ∈ {1, 2} for randomizing Eve's knowledge. We have two doubly indexed codebooks, u 1 (m 1 , k 1 ), and u 2 (m 2 , k 2 ).

Encoding: An encoding operation by Alice E :

where for each message ρ M1M2Z and ρ M1M2 are appropriate marginal of the state

Also, ˜ ρ Z can be any arbitrary state.

Decoding: Decoding operation by Bob

A rate pair (R 1 , R 2 ) is (ϵ 1 , ϵ 2 )-achievable if, for such encoding and decoding maps (E, D), the conditions stated in Equation 6 and Equation 7 are satisfied.

As it can be understood from criterion (6), the reliability and security conditions are merged into a single criterion. This idea is used in 24 and 18 for the first time. Theorem 2 (An inner bound on the one-shot capacity region of PP-QWTC). For any fixed ϵ 1 ∈ (0, 1), ϵ 2 ∈ (0, 1) and δ 1 , δ 2 such that δ 1 ∈ (0, ϵ 1 ) and δ 2 ∈ (0, ϵ 2 ), there exists a one-shot code for the channel N X→Y Z , if rate pair (R 1 , R 2 , 3ϵ 1 + 2(

satisfies the following bounds:

with respect to state

Remark 2. The proof of Theorem 2 has two advantages over the proof of Theorem 1: The first is that the proof of Theorem 2 is based on solving a binary hypothesis testing problem against the proof of Theorem 1, which is based on solving a multiple hypothesis testing problem. The second is that in the privacy proof of Theorem 1, Lemma 1 is used. But, in the proof of Theorem 2, the convex split lemma (Lemma 2) can be used.

Remark 3.

From comparing the results of Theorem 1 and Theorem 2, it can be understood that the proof of Theorem 2 does not give the sum rate (R 1 +R 2 ). This is because of using the successive decoding technique. This issue should not cause doubts about whether PP-QWTC is a dual for CQ-MA-WTC. To solve this doubt, we propose the issue of quantum broadcast channels.

Quantum Broadcast Channels (QBCs)

The quantum broadcast channel accepts one user and two or more receivers. In the basic case, the sender (Alice) wishes to transmit three separate messages: m 1 is the personal message for the first receiver Y 1 , m 2 is the personal message for the second receiver Y 2 , and m c is the common message for both of the receivers.

The basic QBC is illustrated in Figure 3. It should be noted that, for ease of analysis, we removed the security constraint from the problem.

The problem of QBC is widely studied in the i.i.d. case in 2526 and in the one-shot case in 27. In the following, we want to achieve a one-shot inner bound is the system output's quantum state. Theorem 3. Let U be an auxiliary random variable, p = p X|U (x|u)p U (u) be the code probability function. The one-shot achievable rate consists of all rate pairs (R 1 , R c ) such that:

H (X; Y 1 ) ρ − 2 + log ϵ is achievable, and all information quantities are taken with respect to the following state:

Proof: In Appendix 4.

Now, consider the extended version of the above theorem:

Corollary 2 (one-shot inner bound for QBC with three personal messages for the first receiver). Let U be an auxiliary random variable, p = p U (u)p X1|U (x 1 |u)p X2|U X1 (x 2 |ux 1 ) be the code probability function. The one-shot achievable rate region consists of all rate tuples (R 1 , R c , R 2 ) in order to sending (m 1 , m 2 , m c ) such that:

log ϵ is achievable, and all information quantities are taken with respect to the following state:

Proof: The proof follows the extended version of Theorem 3's proof.

The channel described in Corollary 2 will be converted to the channel described in Theorem 2 (PP-QWTC) without secrecy constraint by choosing m c = ∅. Set R c = 0 in Corollary 2:

+ log ϵ where the above last two rates are redundant. Then, we have the following region:

Consider the results of Theorem 2 without the leaked information terms. By choosing δ 2 = ϵ 2 , we have:

Comparing Equation 9, Equation 10, and Equation 46 the argument stated in Remark 3 is proved. Also, the region (9) is a near-optimal achievable rate region compared to Corollary 1. As it can be understood from a comparison between the results of Corollary 1, Theorem 2, and Corollary 2 by considering Equation 46, this idea can be proved that converting the CQ-MA-WTC to PP-QWTC can be a helpful approach to bypass the bottlenecks connected to the multiple hypothesis testing problem (Theorem 1) and the smoothing bottlenecks of quantum information theory (Corollary 1 and Theorem 1).

Asymptotic Analysis

In this subsection, we want to evaluate secrecy rate region presented in Theorem 2 in the asymptotic i.i.d. case (asymptotic limit of many uses of a memoryless channel). It should be noted that, all of the process can be repeated for asymptotic analysis of Theorem 3 and Corollary 1.

. The capacity region of the channel can be expressed as follows:

where C ϵ1,ϵ2 (N ⊗n ) ≡ max p(u1,u2) R ϵ1,ϵ2 (N ⊗n ). Let R(N ) be the set of the maximum rate pairs (R ′ 1 , R ′ 2 ),

Then, the capacity region C ∞ (N ) is the union over n uses of the channel N :

we aim to prove the expression above. Consider both single rates. Applying Fact 2 (and its conditional version), we have,

To prove the achievability, consider the one-shot lower bounds presented in Theorem 2 and apply quantum AEP 28 for the conditional smooth hypothesis testing and max-mutual information. From Theorem 2, for r uses of the channel N , the following lower bound C ϵ1,ϵ2 (N ⊗n ) can be obtained:

where R(N ⊗n ) is the set of all rate pairs (R ′ 1 , R ′ 2 ) satisfying:

We can assume that the sequences of the random variables are i.i.d. according to their distributions. This is due to the fact that the region above is a lower bound on the capacity region. This enable us to use of quantum AEP, as described below. From Fact 3, we have,

Also, using Fact 4, we have the following:

Putting

:

Given the argument above, using Equation 11 and Equation 13 completes the proof.

Comparison

In the following, we have compared the advantages of our method in comparison to the previously introduced methods. As mentioned before, the main novelty of this paper is introducing a new method that tries to investigate C-QMA-WTC using other channels (PP-CQ-WTC and Classical-Quantum Broadcast Wiretap Channel (CQ-B-WTC)). This method led us to achieve sub-optimal (Theorem 2) and nearoptimal (Corollary 2) achievable rate regions. We have shown that we can bypass some of the quantum information theory bottlenecks using the simulation of a quantum channel by another quantum channel.

It should be mentioned that the previously investigated techniques suffer from quantum information theory bottlenecks. For example, the introduced method in the paper 10 suffers from the multiple quantum hypothesis testing limits, which tend to a lower bound on the secrecy capacity region of entanglement-assisted CQ-MA-WTC under the oneshot setting (in a special case in which output states are commutative). Also, the introduced method in the paper 7 suffers from the smoothing bottleneck of the tripartite convex split lemma.

Using the successive position-based decoder helps us to bypass the intractability of asymmetric multiple quantum hypothesis testing problem. Also, the successive position-based decoder enables us to use the convex split lemma, while the simultaneous positionbased decoder does not have the capability to use it.

Conclusion

In this paper, we studied the problem of secure communication over a CQ-MA-WTC using three tech-niques: 1-Sen's joint typicality lemma. 2-simultaneous position-based decoding and 3-successive positionbased decoding. The first and the second decoding techniques use a newly introduced smooth technique 13 to analyze the privacy, while the third technique uses convex splitting 16. We realized that the simultaneous position-based decoder tends to a multiple hypothesis testing problem, which is unsolvable in the general case. We introduced a new channel (PP-QWTC), which can be considered as a dual for CQ-MA-WTC. Also, this channel can be derived from the quantum broadcast channel. The results show that Theorem 2 has a sub-optimal achievable rate region to CQ-MA-WTC. Also, Corollary 2 validates our claim by providing a near-optimal achievable rate region.

Appendix

Appendix 1 (Proof of Corollary 1). Proof. As mentioned, the proof has two steps: Reliable decoding and secure decoding. To these ends, consider two junk variables k i ; i ∈ {1, 2} for each user m i ; i ∈ {1, 2}. These junk variables are used to make two doubly indexed codebooks

. Bob should be able to detect the pair messages (m 1 , m 2 ), and the junk variables k 1 and k 2 Using Definition 10 (Sen's inner bound for QMAC), we have the following relation:

with decoding error at most 49 √ ϵ, and privacy leakage at most 20δ ′ 1 8 (Lemma 1). Also, R Sen refers to Sen's inner bound for QMAC (Definition 10), and R leaked refers to the leaked information from senders to Eve.

From Lemma 1, we have the following:

This completes the proof. Appendix 2 (Proof of Theorem 1). Proof. Both of the messages are uniformly distributed on their sets. The receiver has to be able to decode both messages with negligible error probability. Before communication begins, Alice (A) and Bob (B) share randomness with Charlie (C) and the wiretapper (Z).

be shared-randomness between (A,C,Z) and shared-randomness between (B,C,Z), respectively:

Alice has X ′ 1 system, Bob has X ′ 2 system, and Charlie has (X 1 , X 2 ) system, and wiretapper has (X

Then, the overall controlling state of the channel is as follows:

Sketch of the coding scheme: For each of the messages m i ; i ∈ {1, 2}, there exist local keys k i ∈ [1 : |K i |], i ∈ {1, 2} as uniform randomness for randomizing Eve's knowledge about the sent messages. These local keys are not accessible to Charlie or Eve. Before the communication begins, assume that Alice, Charlie, and Eve share |M 1 ||K 1 | copies of the state 19 and Bob, Charlie, and Eve share |M 2 ||K 2 | copies of the state (23):

To send the pair messages m 1 and m 2 , Alice and Bob pick k 1 ∈ [1 : |K 1 |] and k 2 ∈ [1 : |K 2 |], respectively, and uniformly at random. They send (m 1 , k 1 )-th system X ′ 1 and (m 2 , k 2 )-th system X ′ 2 through the channel

There exists a simultaneous decoder for communication over a CQ-MA-WTC with the upper bound on the average error probability as follows:

where,

As it can be understood from Equation 27, the security criterion is merged into the reliability criterion 14. The simultaneous position-based decoder can be constructed as follows:

where,

Multiple quantum hypothesis testing: As mentioned several times, there is no a general simultaneous decoder for QMACs (more than two users) in the i.i.d.

case. There is a helpful discussion about the multiple quantum hypothesis testing problem in 14. In summary, the multiple quantum hypothesis testing problem is remained an open problem yet. There are two kinds of hypothesis testing: Symmetric and asymmetric. Chernoff distance from symmetric hypothesis testing gives a lower bound on the randomnessassisted error exponent 29; on the opposite point, the asymmetric hypothesis testing tends to a lower bound on the one-shot randomness-assisted capacity (for QMAC with or without secrecy constraint) and in turn on the second-order coding rate for randomnessassisted communication.

In other words, from 4, we know that there exists a general simultaneous decoder if the output states be commutative, and from 14, we know that the multiple hypothesis testing problem can be solvable if the alternative composite hypothesis forms a commutative set of operators. This means that, for a test operator T , a finite set of positive semi-definite operators θ ≡ {θ i : 1 ≤ i ≤ r}, for which supp(ρ) ⊆ supp(θ i ) and min i D(ρ∥θ i ) > 0, there are two hypotheses, and we have:

where δ is a positive integer. The last inequality holds when the set θ forms a commutative set of operators. More information can be found in 14. With these explanations, we use asymmetric hypothesis testing for our problem. Note that we want to decode two messages simultaneously. Consider the upper bound on error probability in Equation 35. Then, we rewrite that as follows:

that if the first decoder fails, the second decoder fails, too. This decoding order can be shown as m 1 → m 2 .

Constructing the first position-based decoder is the same as that presented in 18. To decode m 2 , Bob performs his second position-based decoder conditioned on U 1 , which works for all u 1 ∈ U 1 . It should be noted that the feeding state of the second decoder differs from the main state of the channel.

Alice, Bob, and Eve are allowed to pre-share some quantum state as randomness. Also, Alice has access to two sources of uniform junk randomness k i ; i ∈ {1, 2}. The pre-shared randomness is as follows:

(44)

• The probability of error for decoding m 1 : and τ m1,k1

U1Y is a test operator to discriminate between two hypotheses ρ U1Y , and ρ U1 ⊗ ρ Y . Also, it is obvious that to decode m 1 , it does not matter for the second position-based decoder, which copy is selected by Alice among |M 1 ||K 1 | copies.

• We face a hypothesis testing problem. The null hypothesis is ρ U1Y , and the alternative hypothesis is ρ U1 ⊗ρ Y . Therefore, the probability of success in guessing null and alternative hypotheses are T r{τ U1Y ρ U1Y } and T r{(I U1Y −τ U1Y )(ρ U1 ⊗ ρ Y )}.

The rest of the decoding process for m 1 is analogous to 18. Therefore, we have:

Now, we turn our attention to decoding the second message. As mentioned before, the channel state changes after the first measurement. There is a detailed discussion in 30.

Let σ (m1,k1),(m2,k2) U1U Also, Bob's second POVM is as follows:

inner bound for QBC), the above error probability exponents can be calculated as follows:

The CQ-MA-WTC model the following holds,

The QBC model for QBC with classical messages. Suppose that Alice has not a personal message for the second receiverY 2 (m 2 = ∅ → R 2 = 0).The QBC under the one-shot setting is a triple(X , N X →Y1Y2 ≡ ρ Y1Y2 x , H Y1 ⊗ H Y2 ),where X denotes the input alphabet set, and Y 1 , Y 2 denote the output systems. ρ Y1Y2 x

system. Let ρ X1X ′′ 1 Y Z and σ X2X ′′ 2 Y Z denote the state resulting from sending X

m 1 : The arguments connected to the decoding process for m 1 are listed as follows:D m1 Y U1→ˆM1U1→ˆ U1→ˆM1 ρ (m1,k1),(m2,k2) U ⊗|M 1 ||K 1 | 1 Y := |K1| k1=1 |M1| m1=1 T r Λ m1,k1 U |M 1 ||K 1 | 1 Y ρ (m1,k1),(m2,k2) X ⊗|M 1 ||K 1 | 1 Y ⊗ Λ m1,k1 U |M 1 ||K 1 | 1 Y ρ (m1,k1),(m2,k2) U ⊗|M 1 ||K 1 | 1 Y Λ m1,k1 U |M 1 ||K 1 | 1 Y T r Λ m1,k1 U |M 1 ||K 1 | 1 Y ρ (m1,k1),(m2,k2) U ⊗|M 1 ||K 1 | 1 Y • Λ m1,k1 U |M 1 ||K 1 | 1 Yis a pretty good measurement (POVM) for m 1 ∈ [1 :|M 1 |]:

U1 (u 1 ) |u 1 ⟩ ⟨u 1 | U1 |u 1 ⟩ ⟨u 1 | U ′ 1 ⊗ σ u1(m1,k1) U2U ′ 2 ⊗ . . . ⊗ σ u1(m1,k1),(m2,k2) U2U ′ 2 Y Z ⊗ . . . ⊗ σ u1(m1,k1),(|M2|,|K2|) U2U ′ 2

X;Y1)ρ−2+log ϵ + O(ϵ)

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