# Reliability Function of Quantum Information Decoupling

Quantum information decoupling is an important quantum information processing task, which has found broad applications. In this paper, we characterize the performance of catalytic quantum information decoupling regarding the exponential rate under which perfect decoupling is approached, namely, the reliability function. We have obtained the exact formula when the decoupling cost is not larger than a critical value. In the situation of high cost, we provide upper and lower bounds. This result is then applied to quantum state merging and correlation erasure, exploiting their connection to decoupling. As technical tools, we derive the exponents for smoothing the conditional min-entropy and max-information, and we prove a novel bound for the convex-split lemma. Our results are given in terms of the sandwiched Rényi divergence, providing it with operational meanings in characterizing how fast the performance of quantum information tasks approach the perfect.

• 123 publications
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## I Introduction

Quantum information decoupling HOW2005partial ; HOW2007quantum ; ADHW2009mother is an important quantum information processing task, which has found broad applications, ranging from quantum Shannon theory HOW2005partial ; HOW2007quantum ; ADHW2009mother ; DevetakYard2008exact ; YardDevetak2009optimal ; BDHSW2014quantum ; BCR2011the ; BBMW2018conditional ; BFW2013quantum to quantum thermodynamics BrandaoHorodecki2013area ; Aberg2013truly ; DARDV2011thermodynamic to black-hole physics HP2007black ; BraunsteinPati2007quantum ; BSZ2013better . It has later been studied in the one-shot setting BCR2011the ; DBWR2014one ; MBDRC2017catalytic ; ADJ2017quantum . By introducing an independent catalyst, tight one-shot characterization has been derived in MBDRC2017catalytic ; ADJ2017quantum , which is able to provide the exact second-order asymptotics.

The reliability function in information theory was introduced by Shannon Shannon1959probability . In the context of communication over noisy channels, it is defined as the rate of exponential decay of the error with the increasing of code-length, in the case that the transmitting rate is lower than the capacity. Thus the reliability function provides the desired precise characterization of how rapidly an information processing task approaches the perfect with the increasing of blocklength Gallager1965simple . The study of the reliability function has become a major topic in information theory, now also called error exponent analysis. Some works in this topic have been carried out in the quantum regime Holevo2000reliability ; Dalai2013lower ; DalaiWinter2017constant ; CHT2019quantum . However, the reliability function is not known even for classical-quantum channels. Nevertheless, see References KoenigWehner2009strong ; SharmaWarsi2013fundamental ; WWY2014strong ; GuptaWilde2015multiplicativity ; CMW2016strong ; MosonyiOgawa2017strong ; CHDH2020non for a partial list of the fruitful results for the strong converse exponent in the quantum setting, which characterizes how fast a quantum information task is getting the useless.

In this paper, we investigate the reliability function for the task of quantum information decoupling, when an independent catalyst is available. We have obtained the exact formula when the decoupling cost is not larger than a critical value. In the high-rate situation, we provide upper and lower bounds. This result is then applied to quantum state merging HOW2005partial ; HOW2007quantum ; ADHW2009mother and correlation erasure GPW2005quantum , exploiting their connection to decoupling. As technical tools, we derive the exact exponents for smoothing the conditional min-entropy and the max-information, as well as prove a novel bound for the convex-split lemma ADJ2017quantum , which should be of independent interest.

Our results, along with LiYao2021exponent , are given in terms of the sandwiched Rényi divergence MDSFT2013on ; WWY2014strong , providing it with operational meanings in characterizing how fast the performance of quantum information tasks approach the perfect. This is in stark contrast to those of the previous ones MosonyiOgawa2015quantum ; MosonyiOgawa2015two ; CMW2016strong ; HayashiTomamichel2016correlation ; MosonyiOgawa2017strong ; CHDH2020non , which are concerned with the strong converse exponents.

The reminder of the paper is organized as follows. In Section II necessary notation, definitions and properties are presented. In Section III we describe the problem of quantum information decoupling and give the main result. In Section IV we discuss the relevant quantum information processing tasks including correlation erasure and state merging. At last, Section V is devoted to the proofs.

## Ii Notation and preliminaries

We define  () as a Hilbert space (the Hilbert space associated with system ) and denote by

the tensor product of

and . We use the notation for the dimension of . Let  () and  () denote the set of quantum states and subnormalized states on  () respectively. The set of unitary operators on  () is denoted by  (). The set of the positive semidefinite operators on  () is denoted by  (). The norm of an operator on is defined as .

A quantum channel is a linear, completely positive, trace-preserving (CPTP) map. We denote by a quantum channel from system to system . If is a self-adjoint operator with spectral projections , then the associated pinching map is given by

 EA:X→n∑i=1PiXPi.

The following concepts or quantities will be used. We refer Wilde2013quantum ; Tomamichel2015quantum for these definitions.

###### Definition 1

We define

1. Fidelity: For ,

 F(ρA,σA)\emphdef=∥√ρA√σA∥1+√(1−TrρA)(1−TrσA).
2. Purified distance: For ,

 P(ρA,σA)\emphdef=√1−F2(ρA,σA).
3. Relative entropy: For ,

 D(ρA∥σA)\emphdef={Tr(ρA(logρA−logσA)) if supp(ρA)⊆supp(σA),+∞ otherwise.
4. Sandwiched Rényi divergence MDSFT2013on ; WWY2014strong : For ,

 Dα(ρA∥σA)\emphdef=⎧⎨⎩1α−1logTr(σ1−α2αAρAσ1−α2αA)α if α>1∧supp(ρA)⊆supp(σA) or 0≤α<1,+∞ otherwise.
5. Sandwiched Rényi mutual information GuptaWilde2015multiplicativity : For , ,

 Iα(A:B)ρ\emphdef=minσB∈S(HB)Dα(ρAB∥ρA⊗σB).
6. Max relative entropy Datta2009min : For ,

 D\emphmax(ρA∥σA)\emphdef=inf{λ | ρA≤2λσA}.
7. symmetric states: Let denote the permutation group. Then,

 Ssym(An)\emphdef={ρAn|ρAn∈S(An), WπρAnW∗π=ρAn, ∀π∈Sn},Ssym≤(An)\emphdef={ρAn|ρAn∈S≤(An), WπρAnW∗π=ρAn, ∀π∈Sn},

where is the permutation operator on associated with .

The following properties will be used.

###### Proposition 2

The following facts hold.

1. Pinching inequality Hayashi2002optimal : If is positive semidefinite and A is self-adjoint, we have

 X≤v(A)EA(X),

where

denotes the number of different eigenvalues of

.

2. Data processing inequalities Lindblad1975completely ; FrankLieb2013monotonicity ; Beigi2013sandwiched ; MDSFT2013on ; WWY2014strong : Let be a CPTP map, and . Then

 D(ρA∥σA)≥D(Φ(ρA)∥Φ(σA)),Dα(ρA∥σA)≥Dα(Φ(ρA)∥Φ(σA)),P(ρA,σA)≥P(Φ(ρA),Φ(σA));
3. Universal symmetric state MosonyiOgawa2017strong ; HayashiTomamichel2016correlation : For every Hilbert space , there exists a universal symmetric state such that

 σAn≤gn,|A|ω(n)An, ∀σAn∈Ssym(An),v(ω(n)An)≤gn,|A|,

where and denotes the number of different eigenvalues of .

4. Duality relation of Sandwiched Rényi mutual information HayashiTomamichel2016correlation : For , let be a purification of . Then, we have

 Iα(A:B)ρ=−minσC∈S(C)Dβ(ρAC∥ρ−1A⊗σC),α,β∈[12,+∞), 1α+1β=2.

## Iii Quantum information decoupling

For a bipartite quantum state , a catalytic decoupling scheme consists of a catalytic system in the state and a unitary operation , where is the system to be removed and is the remaining system. Without loss of generality, we require

. The cost of the decoupling is given by the number of qubits that is discarded,

, and the performance is measured by the purified distance between the remaining state and the nearest product state of the form . We denote the optimal performance given that the cost is bounded by as , defined as follows.

###### Definition 3

Let be a bipartite quantum state. For given , we define the optimal performance of decoupling as

 PdecR:A(ρRA,r):=minP(TrA2U(ρRA⊗σA′)U∗,ρR⊗ωA1), (1)

where the minimization is over all system dimensions , , such that and , all unitary operations , and all states , .

For copies of the state , the optimal decoupling performance is expected to decrease exponentially with . The reliability function of quantum information decoupling is defined as the rate of such exponential decreasing.

###### Definition 4

Let be a bipartite quantum state, and . The reliability function of quantum information decoupling for the state is defined as

 EdecR:A(r):=limsupn→∞−1nlogPdecRn:An(ρ⊗nRA,nr).
###### Theorem 5

Let be a bipartite quantum state, and consider the problem of decoupling quantum information from the Reference system in the quantum state . We have

 EdecR:A(r) ≥max0≤s≤1{s(r−12I1+s(R;A)ρ)}, (2) EdecR:A(r) ≤  sups≥0 {s(r−12I1+s(R;A)ρ)}. (3)

In particular, when ,

 EdecR:A(r)=max0≤s≤1{s(r−12I1+s(R;A)ρ)}. (4)

## Iv Related quantum information processing tasks

In this section, we discuss several quantum information processing tasks related to decoupling. We define the respective optimal performance and reliability function for these tasks, and relate them to their counterparts for decoupling. Thus results similar to Theorem 5, as well as Proposition 20 and Proposition 21, are established for these tasks.

### iv.1 catalytic quantum correlation erasure

For a bipartite quantum state , we can use a random unitary channel on system to erasure the correlation between and . In order to do so, a catalytic system can be added. Specifically, we can choose a catalytic state on catalytic system and a random unitary channel on system to make the purified distance between the resulting state and a certain product state of the form as small as possible. If the number of the unitaries in is , then the cost of this process is . The optimal performance given that the cost is bounded by is denoted by , defined as follows.

###### Definition 6

Let be a bipartite quantum state. For given , we define the optimal performance of catalytic quantum correlation erasure as

 P\empherasR:A(ρRA,r):=minP(ΛAA′(ρRA⊗σA′),ρR⊗ωAA′), (5)

where the minimization is over all system dimension , all states , and all random unitary channels in the following form:

 ΛAA′(⋅)=1mm∑i=1Ui(⋅)U∗i, (6)

where and .

We define the reliability function of catalytic quantum correlation erasure.

###### Definition 7

Let be a bipartite quantum state, and . The reliability function of catalytic quantum correlation erasure for is defined as

 E\empherasR:A(r):=limsupn→∞−1nlogP\empherasRn:An(ρ⊗nRA,nr). (7)

There is a close relation between , and , which is stated as follows.

###### Proposition 8

For and , we have

 P\empherasR:A(ρRA,2r)=P\emphdecR:A(ρRA,r), (8) E\empherasR:A(ρRA,2r)=E\emphdecR:A(ρRA,r). (9)

Proof. We first prove . We let , and be the states and the unitary operator which satisfy the conditions in Definition 3. Then we choose as the catalytic state for quantum correlation erasure of and construct a random unitary channel as

 ΛAA′(ωAA′)=1|A2|2|A2|2∑i=1UiA2UAA′ωAA′U∗AA′Ui∗A2, (10)

where are all the Heisenberg-Weyl operators on and . Hence, by the definition of , it can be evaluated as

 PerasR:A(ρRA,2r)≤P(ΛAA′(ρRA⊗σA′),ρR⊗ωA1⊗πA2)=P(TrA2UAA′(ρRA⊗σA′)U∗AA′⊗πA2,ρR⊗ωA1⊗πA2)=P(TrA2UAA′(ρRA⊗σA′)U∗AA′,ρR⊗ωA1), (11)

where denotes the maximally mixed state on . Because Eq. (11) holds for arbitrary , and which satisfy the conditions in Definition 3, we have

 PerasR:A(ρRA,2r)≤PdecR:A(ρRA,r). (12)

Now we deal with the other direction. Let , , be the states and the random unitary channel which satisfy the conditions in Definition 6. can be written as

 ΛAA′(⋅)=1mm∑i=1UiAA′(⋅)Ui∗AA′, (13)

where . We introduce systems and such that and denote by the maximally entangled state on and . The catalytic state for decoupling can be chosen as

 σA′CC′=1mm∑i=1σA′⊗UiCΨmUi∗C,

where denote all the Heisenberg-Weyl operators on and we define the unitary operator for decoupling as

 UAA′CC′=m∑i=1UiAA′⊗UiCΨmUi∗C.

Set be the remaining system, and be the discarded system. We can bound as follows.

 PdecR:A(ρRA,r)≤P(TrC′UAA′CC′(ρRA⊗σA′CC′)U∗AA′CC′,ρR⊗ωAA′⊗πC)=P(ΛAA′(ρRA⊗σA′)⊗πC,ρR⊗ωAA′⊗πC)=P(ΛAA′(ρRA⊗σA′),ρR⊗ωAA′), (14)

where denotes the maximally mixed state on system . Noticing that Eq. (14) holds for any , , which satisfy the conditions in Definition 6, we have

 PdecR:A(ρRA,r)≤PerasR:A(ρRA,2r). (15)

Eq. (12) and Eq. (15) imply Eq. (8) and Eq. (9) follows from Eq. (8).

### iv.2 quantum state merging

Let be a tripartite pure state. Alice, Bob and a referee hold system , and respectively. A quantum state merging protocol with quantum communication, , consists of using a shared bipartite entangled pure state , Alice applying local unitary and sending the system to Bob, Bob applying local unitary and they discarding systems and . A CPTP map is a quantum state merging protocol with classical communication if it consists of using a shared bipartite entangled pure state , applying local operation at Alice’s side, sending classical bits from Alice to Bob and applying local operation to reproduce and at Bob’s side. The performances of both protocols are given by the purified distance between and the final state on the referee’s system and Bob’s system and . The cost of state merging that we are concerned with, is the number of qubits or classical bits that Alice sends to Bob. We define the optimal performances and , given that the cost is bounded by r, as follows.

###### Definition 9

Let be a tripartite pure quantum state. For given , we define the optimal performance of quantum state merging with quantum communication and classical communication respectively as

 P\emphmergA⇒B(ρRAB,r):=minP(M1(ρABR),ρABR), (16)
 P\emphmergA→B(ρRAB,r):=minP(M2(ρABR),ρABR), (17)

where denotes the protocol with quantum communication, denotes the protocol with classical communication and the minimization is over all possible , whose cost are bound by .

The reliability function of quantum state merging can be defined similar to Definition 4.

###### Definition 10

Let be a tripartite pure state and . The reliability functions and of quantum state merging with quantum communication and classical communication respectively, are defined as

 E\emphmergA⇒B(ρRAB,r):=limsupn→∞−1nlogP\emphmergAn⇒Bn(ρ⊗nRAB,nr), (18)
 E\emphmergA→B(ρRAB,r):=limsupn→∞−1nlogP\emphmergAn→Bn(ρ⊗nRAB,nr). (19)

Catalytic decoupling plays an important role in quantum state merging. We have the following Proposition 11 and Proposition 12 on their optimal performances and reliability functions.

###### Proposition 11

For a tripartite pure state and , we have

 PmergA⇒B(ρABR,r)=PdecR:A(ρRA,r), (20) EmergA⇒B(ρABR,r)=EdecR:A(ρRA,r). (21)

Proof. Let us prove at first. We set , , to be a pair of states and a unitary operator which satisfy the conditions in Definition 3. Let , be the purification of , respectively. By Uhlmann’s theorem, there exists an unitary such that

 P(TrA2UAA′(ρAR⊗σA′)U∗AA′,ωA1⊗ρR)=P(VA2BB′→AB1BUAA′(ρABR⊗σA′B′)U∗AA′V∗A2BB′→AB1B,ωA1B1⊗ρABR). (22)

Let , the cost of quantum communication of this protocol is . Hence, by the definition of and Eq. (22), we have

 PmergA⇒B(ρRAB,r)≤PdecR:A(ρRA,r). (23)

Now, we turn to the proof of the other direction. Let be a merging protocol with quantum communication whose cost does not exceed . Then we can write as

 M1(⋅)=TrA1B1VA2BB′→ABB1UAA′→A1A2((⋅)⊗ϕA′B′)U∗AA′→A1A2V∗A2BB′→ABB1,

where , are unitary operators and . By using Uhlmann’s theorem again, we know there exists a pure state such that

 P(M1(ρABR),ρABR)=P(VA2BB′→ABB1UAA′→A1A2(ρABR⊗ϕA′B′)U∗AA′→A1A2V∗A2BB′→ABB1,ρABR⊗φA1B1). (24)

Because the purified distance decreases under the action of partial trace, we get

 (25)

We choose as the catalytic state for decoupling and discard the system after the action of the unitary operation . The cost of the decoupling is . Hence, by the definition of and Eq. (25), we have

 PdecR:A(ρRA,r)≤PmergA⇒B(ρRAB,r). (26)

Eq. (23), Eq. (26) give Eq. (20) and Eq. (21) comes from Eq. (20) directly.

For a protocol with quantum communication , we can use the teleportation protocol and a merging protocol with classical communication to simulate . Conversely, for a protocol with classical communication , we can use the dense coding protocol and a merging protocol with quantum communication to simulate . So, we obtain the simple relations , .

###### Proposition 12

For a tripartite pure state and , we have

 P\emphmergA→B(ρRAB,2r)=P\emphdecR:A(ρRA,r),E\emphmergA→B(ρRAB,2r)=E\emphdecR:A(ρRA,r). (27)

### iv.3 entanglement-assisted quantum source coding

Here Alice and a referee share a pure state and Alice hopes to send the system to Bob with the assistance of unlimited entanglement and using noiseless quantum communication or classical communication as resource. It is obvious that this task is a special case of quantum state merging where Bob holds no side information. So, we can easily get the bound of its reliability function from Proposition 11 and it has a simpler formula involving only the Rényi entropy of one single system due to the following lemma.

###### Lemma 13

Let be a bipartite pure state and . Then we have

 I1+s(R:A)ρ=2s+1slogTrρ12s+1A=2H12s+1(ρA).

Proof. By the duality relation of Sandwiched Rényi mutual information (see Proposition 2 (5)),

 I1+s(R:A)ρ=−D1+s2s+1(ρR∥ρ−1R)=2s+1slogTrρ1+s2s+1Rρ(1+s2s+1−1)R=2s+1slogTrρ12s+1A. (28)

## V Proof of main result

### v.1 covex-split lemma

We prove a version of the covex-split lemma ADJ2017quantum , with a new bound in terms of the sandwiched Rényi divergence.

###### Lemma 14

Let and be quantum states such that . Consider the following state

 τRA1A2⋯Am:=1mm∑i=1ρRAi⊗[σ⊗(m−1)]Am/Ai,

where denotes the composite system consisting of and is the product state on these systems. Let denote the number of distinct eigenvalues of . Then for any ,

 D(τRA1A2⋯Am∥∥ρR⊗σ⊗m)≤vs(ln2)sexp{−(ln2) s(logm−D1+s(ρRA∥ρR⊗σA))}.

Proof. We use the shorthand