# Semidefinite programming strong converse bounds for classical capacity

###### Abstract

We investigate the classical communication over quantum channels when assisted by no-signalling (NS) and PPT-preserving (PPT) codes, for which both the optimal success probability of a given transmission rate and the one-shot -error capacity are formalized as semidefinite programs (SDPs). Based on this, we obtain improved SDP finite blocklength converse bounds of general quantum channels for entanglement-assisted codes and unassisted codes. Furthermore, we derive two SDP strong converse bounds for the classical capacity of general quantum channels: for any code with a rate exceeding either of the two bounds of the channel, the success probability vanishes exponentially fast as the number of channel uses increases. In particular, applying our efficiently computable bounds, we derive an improved upper bound on the classical capacity of the amplitude damping channel. We also establish the strong converse property for the classical and private capacities of a new class of quantum channels. We finally study the zero-error setting and provide efficiently computable upper bounds on the one-shot zero-error capacity of a general quantum channel.

^{†}

^{†}thanks: A preliminary version of this paper was presented at the 20th Annual Conference on Quantum Information Processing and the IEEE International Symposium on Information Theory in 2017 Wang2017b ().

## I Introduction

The reliable transmission of classical information via noisy quantum channels is central to quantum information theory. The classical capacity of a noisy quantum channel is the highest rate at which it can convey classical information reliably over asymptotically many uses of the channel. The Holevo-Schumacher-Westmoreland (HSW) theorem Holevo1973 (); Holevo1998 (); Schumacher1997 () gives a full characterization of the classical capacity of quantum channels:

(1) |

where is the Holevo capacity of the channel given by , is an ensemble of quantum states on and is the von Neumann entropy of a quantum state. Throughout this paper, denotes the binary logarithm.

For certain classes of quantum channels (depolarizing channel King2003 (), erasure channel Bennett1997 (), unital qubit channel King2002 (), etc. Amosov2000 (); Datta2006 (); Fukuda2005 (); Konig2012 ()), the classical capacity of the channel is equal to the Holevo capacity, since their Holevo capacities are all additive. However, for a general quantum channel, our understanding of the classical capacity is still limited. The work of Hastings Hastings2008a () shows that the Holevo capacity is generally not additive, thus the regularization in Eq. (1) is necessary in general. Since the complexity of computing the Holevo capacity is NP-complete Beigi2007 (), the regularized Holevo capacity of a general quantum channel is notoriously difficult to calculate. Even for the qubit amplitude damping channel, the classical capacity remains unknown.

The converse part of the HSW theorem states that if the communication rate exceeds the capacity, then the error probability of any coding scheme cannot approach zero in the limit of many channel uses. This kind of “weak” converse suggests the possibility for one to increase communication rates by allowing an increased error probability. A strong converse property leaves no such room for the trade-off, i.e., the error probability necessarily converges to one in the limit of many channel uses whenever the rate exceeds the capacity of the channel. For classical channels, the strong converse property for the classical capacity is established by Wolfowitz Wolfowitz1978 (). For quantum channels, the strong converse property for the classical capacity is confirmed for several classes of channels Ogawa1999 (); Winter1999 (); Koenig2009 (); Wilde2013a (); Wilde2014a (). Winter Winter1999 () and Ogawa and Nagaoka Ogawa1999 () independently establish the strong converse property for the classical capacity of classical-quantum channels. Koenig and Wehner Koenig2009 () prove the strong converse property for particular covariant quantum channels. Recently, for the entanglement-breaking and Hadamard channels, the strong converse property is proved by Wilde, Winter and Yang Wilde2014a (). Moreover, the strong converse property for the pure-loss bosonic channel is proved by Wilde and Winter Wilde2013a (). Unfortunately, for a general quantum channel, less is known about the strong converse property of the classical capacity and it remains open whether this property holds for all quantum channels. A strong converse bound for the classical capacity is a quantity such that the success probability of transmitting classical messages vanishes exponentially fast as the number of channel uses increases if the rate of communication exceeds this quantity, which forbids the trade-off between rate and error in the limit of many channel uses.

Another fundamental problem, of both theoretical and practical interest, is the trade-off between the channel uses, communication rate and error probability in the non-asymptotic (or finite blocklength) regime. In a realistic setting, the number of channel uses is necessarily limited in quantum information processing. Therefore one has to make a trade-off between the transmission rate and error tolerance. Note that one only needs to study one-shot communication over the channel since it can correspond to a finite blocklength and one can also study the asymptotic capacity via the finite blocklength approach. The study of finite blocklength regime has recently garnered great interest in classical information theory (e.g., Polyanskiy2010 (); Hayashi2009 (); Matthews2012 ()) as well as in quantum information theory (e.g., Matthews2014 (); Wang2012 (); Renes2011 (); Tomamichel2013a (); Berta2011a (); Leung2015c (); Tomamichel2015 (); Beigi2015 (); Tomamichel2015b (); Tomamichel2016 (); Fang2017 (); Cheng2017b (); Chubb2017 ()). For classical channels, Polyanskiy, Poor, and Verdú Polyanskiy2010 () derive the finite blocklength converse bound via hypothesis testing and Matthews Matthews2012 () provides an alternative proof of this converse bound via classical no-signalling codes. For classical-quantum channels, the one-shot converse and achievability bounds are given in Mosonyi2009a (); Wang2012 (); Renes2011 (). Recently, the one-shot converse bounds for entanglement-assisted and unassisted codes were given in Matthews2014 (), which generalizes the hypothesis testing approach in Polyanskiy2010 () to quantum channels.

To gain insights into the generally intractable problem of evaluating the capacities of quantum channels, a natural approach is to study the performance of extra free resources in the coding scheme. This scheme, called a code, is equivalently a bipartite operation performed jointly by the sender Alice and the receiver Bob to assist the communication Leung2015c (). The PPT-preserving codes, i.e. the PPT-preserving bipartite operations, include all operations that can be implemented by local operations and classical communication (LOCC) and were introduced to study entanglement distillation in an early paper by Rains Rains2001 (). The no-signalling (NS) codes refer to the bipartite quantum operations with the no-signalling constraints, which arise in the research of the relativistic causality of quantum operations Beckman2001 (); Eggeling2002a (); Piani2006 (); Oreshkov2012 (). Recently these general codes have been used to study the zero-error classical communication Duan2016 () and quantum communication Leung2015c () over quantum channels. Our work follows this approach and focuses on classical communication via quantum channels assisted by NS and NSPPT codes.

## Ii Summary of results

In this paper, we focus on the reliable classical communication over quantum channels assisted by no-signalling and PPT-preserving codes under both non-asymptotic (or finite blocklength) and asymptotic settings. The summary of our results is as follows.

In Section IV, we formalize the optimal average success probability of transmitting classical messages over a quantum channel assisted by NS or NSPPT codes as SDPs. Using these SDPs, we establish the one-shot NS-assisted (or NSPPT-assisted) -error capacity, i.e., the maximum rate of classical communication with a fixed error threshold. We further compare these one-shot -error capacities with the previous SDP-computable entanglement-assisted (or unassisted) converse bound derived by the technique of quantum hypothesis testing in Matthews2014 (). Our one-shot -error capacities, which consider potentially stronger assistances, are always no larger than the previous SDP bounds, and the inequalities can be strict even for qubit channels or classical-quantum channels. This means that our one-shot -error capacities can provide tighter finite blocklength converse bounds for the entanglement-assisted and unassisted classical capacity. Moreover, our one-shot -error capacities also reduce to the Polyanskiy-Poor-Verdú (PPV) converse bound Polyanskiy2010 () for classical channels. Furthermore, in common with the quantum hypothesis testing converse bound Matthews2014 () and the bound of Datta and Hsieh Datta2013c (), the large block length behaviour of our one-shot NS-assisted -error capacity also recovers the converse part of the formula for entanglement-assisted capacity Bennett1999 () and implies that no-signalling-assisted classical capacity coincides with the entanglement-assisted classical capacity.

In Section V, we derive two SDP strong converse bounds for the NSPPT-assisted classical capacity of a general quantum channel based on the one-shot characterization of the optimal success probability. These bounds also provide efficiently computable strong converse bounds for the classical capacity. As a special case, we show that is a strong converse bound for the classical capacity of the amplitude damping channel with parameter , and this improves the best previously known upper bound in Brandao2011c (). Furthermore, applying our strong converse bounds, we also prove the strong converse property for the classical and private capacities of a new class of quantum channels.

In Section VI, we consider the zero-error communication problem Shannon1956 (), which requires that the communication is with zero probability of error. To be specific, based on our SDPs of optimal success probability, we derive the one-shot NS-assisted (or NSPPT-assisted) zero-error capacity of general quantum channels. Our result of the NS-assisted capacity provides an alternative proof of the NS-assisted zero-error capacity in Duan2016 (). Moreover, our one-shot NSPPT-assisted zero-error capacity gives an SDP-computable upper bound on the one-shot unassisted zero-error capacity, and it can be strictly smaller than the previous upper bound in Duan2013 ().

Finally, in Section VII, we make a conclusion and leave some interesting open questions.

## Iii Preliminaries

In the following, we will frequently use symbols such as (or ) and (or ) to denote (finite-dimensional) Hilbert spaces associated with Alice and Bob, respectively. We use to denote the dimension of system . The set of linear operators over is denoted by . We usually write an operator with subscript indicating the system that the operator acts on, such as , and write . Note that for a linear operator , we define , where is the adjoint operator of , and the trace norm of is given by . The operator norm is defined as the maximum eigenvalue of . A deterministic quantum operation (quantum channel) () is simply a completely positive (CP) and trace-preserving (TP) linear map from to . The Choi-Jamiołkowski matrix Jamiokowski1972 (); Choi1975 () of is given by , where and are orthonormal bases on isomorphic Hilbert spaces and , respectively. A positive semidefinite operator is said to be a positive partial transpose operator (or simply PPT) if , where means the partial transpose with respect to the party , i.e., . As shown in Rains2001 (), a bipartite operation is PPT-preserving if and only if its Choi-Jamiołkowski matrix is PPT. We sometimes omit the identity operator or operation , for example, .

The constraints of PPT and NS can be mathematically characterized as follows. A bipartite operation is no-signalling and PPT-preserving if and only if its Choi-Jamiołkowski matrix satisfies Leung2015c ():

(2) |

where the five lines correspond to characterize that is completely positive, trace-preserving, PPT-preserving, no-signalling from A to B, no-signalling from B to A, respectively. The structure of no-signalling codes is also studied in Duan2016 ().

Semidefinite programming Vandenberghe1996 () is a subfield of convex optimization and is a powerful tool in quantum information theory with many applications (e.g., Matthews2014 (); Leung2015c (); Duan2016 (); Rains2001 (); Wang2016 (); Harrow2015 (); Wang2016c (); Li2017 (); Berta2015 (); Xie2017 ()). There are known polynomial-time algorithms for semidefinite programming Khachiyan1980 (). In this work, we use the CVX software (a Matlab-based convex modeling framework) Grant2008 () and QETLAB (A Matlab Toolbox for Quantum Entanglement) NathanielJohnston2016 () to solve the SDPs. Details about semidefinite programming can be found in Watrous2011b ().

## Iv Classical communication assisted by NS and PPT codes

### iv.1 Semidefinite programs for optimal success probability

Suppose Alice wants to send the classical message labeled by to Bob using the composite channel , where is a bipartite operation that generalizes the usual encoding scheme and decoding scheme , see Fig. 1 for details. In this paper, we consider as the bipartite operation implementing the or assistance. After the action of and , the message results in quantum state at Bob’s side. Bob then performs a POVM with outcomes on the resulting quantum state. The POVM is a component of the operation . Since the results of the POVM and the input messages are both classical, it is natural to assume that is with classical registers throughout this paper, that is, for some completely dephasing channel . If the outcome happens, he concludes that the message with label was sent. Let be some class of bipartite operations. The average success probability of the general code and the -class code is defined as follows.

###### Definition 1

The average success probability of to transmit messages assisted with the code is defined by

(3) |

where and is the computational basis in system .

Furthermore, the optimal average success probability of to transmit messages assisted with -class code is defined by

(4) |

where the maximum is over the codes in class .

We now define the -assisted classical capacity of a quantum channel as follows.

###### Definition 2

(5) |

As described above, one can simulate a channel with the channel and code , where is a bipartite CPTP operation from to which is no-signalling (NS) and PPT-preserving (PPT). In this work we shall also consider other classes of codes, such as entanglement-assisted (EA) code, unassisted (UA) code. The class of entanglement-assisted codes corresponds to bipartite operations of the form , where are encoding and decoding operations respectively, and can be any shared entangled state of arbitrary systems and . we use to denote specific class of codes such as in the following.

Let denote the resulting composition channel of and , written . As both and are quantum channels, there exist quantum channels and , where is an isometry operation and is a quantum register, such that Chiribella2008 ()

(6) |

Based on this, the Choi-Jamiołkowski matrix of is given by Leung2015c ()

(7) |

The operations and can be considered as generalized encoding and decoding operations respectively, except that the register may be not possessed by Alice or Bob. If the Hilbert space with is trivial, and become the unassisted local encoding/decoding operations. Moreover, the coding schemes with register can be designed to be forward-assisted codes Leung2015c ().

We are now able to derive the one-shot characterization of classical communication assisted by NS (or NSPPT) codes.

###### Theorem 3

For a given quantum channel , the optimal success probability of to transmit messages assisted by NSPPT codes is given by

(8) |

Similarly, when assisted by NS codes, one can remove the PPT constraint to obtain the optimal success probability as follows:

(9) |

###### Proof.

In this proof, we first use the Choi-Jamiołkowski representations of quantum channels to refine the average success probability and then exploit symmetry to simplify the optimization over all possible codes. Finally, we impose the no-signalling and PPT-preserving constraints to obtain the semidefinite program of the optimal average success probability.

Without loss of generality, we assume that and are classical registers with size , i.e., the inputs and outputs are and , respectively. For some NSPPT code , the Choi-Jamiołkowski matrix of is given by , where is isometric to . Then, we can simplify to

(10) |

Then, denoting , we have

where and is any feasible NSPPT bipartite operation . (See FIG. 1 for the implementation of .) Noting that , we can further simplify as

(11) |

The next step is to simplify by exploiting symmetry. For any permutation , where is the symmetric group of degree , if is feasible (satisfying the constraints in Eq. (2)), then it is not difficult to check that

(12) |

is also feasible. And any convex combination of two operators satisfying Eq. (2) can also checked to be feasible. Therefore, if is feasible, so is

(13) |

where is a twirling operation on .

Noticing that , we have

(14) |

Thus, it is easy to see that the optimal success probability equals to

It is worth noting that can be rewritten as Duan2016 ()

for some operators and . Thus, the objective function can be simplified to . Also, the CP and PPT constraints are equivalent to

(15) |

Furthermore, the constraint is equivalent to , i.e.

(16) |

and the TP constraint holds if and only if , i.e.,

(17) |

which is equivalent to

(18) |

As is no-signalling from A to B, we have , i.e.,

(19) |

Since and are orthogonal positive operators, we have

(20) |

Remark: The dual SDP for is given by

(22) |

To remove the PPT constraint, set . It is worth noting that the strong duality holds here since the Slater’s condition can be easily checked. Indeed, choosing , and in SDP (22), we have is in the relative interior of the feasible region.

It is worthing noting that can be obtained by removing the PPT constraint and it corresponds with the optimal NS-assisted channel fidelity in Leung2015c ().

### iv.2 Improved SDP converse bounds in finite blocklength

For given , the one-shot -error classical capacity assisted by -class codes is defined as

(23) |

We now derive the one-shot -error classical capacity assisted by NS or NSPPT codes as follows.

###### Theorem 4

For given channel and error threshold , the one-shot -error NSPPT-assisted and NS-assisted capacities are given by

(24) |

and

(25) |

respectively.

###### Proof.

When assisted by NSPPT codes, by Eq. (23), we have that

(26) |

To simplify Eq. (26), we suppose that

(27) |

On one hand, for given , suppose that the optimal solution to the SDP (27) of is . Then, it is clear that is a feasible solution of the SDP (8) of , which means that . Therefore,

(28) |

On the other hand, for given , suppose that the value of is and the optimal solution of is . It is easy to check that satisfies the constrains in SDP (27) of . Therefore,

(29) |

Noticing that no-signalling-assisted codes are potentially stronger than the entanglement-assisted codes, and provide converse bounds of classical communication for entanglement-assisted and unassisted codes, respectively.

###### Corollary 5

For a given channel and error threshold ,

We further compare our one-shot -error capacities with the previous SDP converse bounds derived by the quantum hypothesis testing technique in Matthews2014 (). To be specific, for a given channel and error thresold , Matthews and Wehner Matthews2014 () establish that

(31) |

and

(32) |

where is a purification of and . Moreover,

(33) |

is the hypothesis testing relative entropy Wang2012 (); Matthews2014 () and is the similar quantity with a PPT constraint on the POVM.

Interestingly, our one-shot -error capacities are similar to these quantum hypothesis testing relative entropy converse bounds. However, there is a crucial difference that our quantities require that a stricter condition, i.e., . This makes one-shot -error capacities ( and ) always smaller than or equal to the SDP converse bounds in Matthews2014 (), and the inequalities can be strict.

###### Proposition 6

For a given channel and error threshold ,

In particular, both inequalities can be strict for some quantum channels such as the amplitude damping channels and the simplest classical-quantum channels.

###### Proof.

This can be proved by the fact that any feasible solution of the SDP (25) of (or ) is also feasible to the SDP (31) of (or ).

We further show that the inequality can be strict by the example of qubit amplitude damping channel , with and . We compare the above bounds in FIG. 3 and FIG. 3. It is clear that our bounds can be strictly better than the quantum hypothesis testing bounds in Matthews2014 () in this case.