Experimental observation of coherent-information superadditivity in a dephrasure channel 1, 2 1, 2 3, 1, 2 1, 2 1, 2 1, 2 Shang Yu, Yu Meng, Raj B. Patel, ∗ Yi-Tao Wang, Zhi-Jin Ke, Wei Liu, Zhi-Peng Li, 1, 2 1, 2 1, 2, 1, 2, 1, 2 Yuan-Ze Yang, Wen-Hao Zhang, Jian-Shun Tang, † Chuan-Feng Li, ‡ and Guang-Can Guo 1CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, China 2CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, 230026, P.R. China 3Clarendon Laboratory, Department of Physics, Oxford University, Parks Road OX1 3PU Oxford, United Kingdom We present an experimental approach to construct a dephrasure channel, which contains both dephasing and erasure noises, and can be used as an efficient tool to study the superadditivity of coherent information. By using a three-fold dephrasure channel, the superadditivity of coherent information is observed, and a substantial gap is found between the zero single-letter coherent information and zero quantum capacity. Particularly, we find that when the coherent information of n channel uses is zero, in the case of larger number of channel uses, it will become positive. These phenomena exhibit a more obvious superadditivity of coherent information than previous works, and demonstrate a higher threshold for non-zero quantum capacity. Such novel channels built in our experiment also can provide a useful platform to study the non-additive properties of coherent information and quantum channel capacity. Introduction.—Channel capacity is at the heart of in- capacity [16, 17]. The first example of superadditivity formation theory and is used to quantify the ability of a is given by DiVincenzo et al. [7], and then has been practical communication channel to transmit the infor- extended by Smith et al. [18], in which they provide a mation from a sender to a receiver. Regarding the quan- heuristic for designing codes for general channels. Just tum channel ε, the notion of channel capacity has mul- recently, Cubitt et al. [15] find that for any number of tiple formulations in different settings: classical capacity channel uses, channels exist for which there is positive C(ε) [1, 2], which denotes the ability of the channel to capacity and zero coherent information. deliver classical information; entanglement-assisted clas- These additive-violated results indicate our limited un- sical capacity CE(ε) [3, 4], which shows the same ability derstanding of the superadditivity of coherent informa- when the sender and receiver share some entanglement tion, and the limitations in studying the quantum capac- resources; and private capacity P (ε) [5], which describes ity. These limitations come down to the fact that we do the maximum classical bit that can be transmitted while not have a useful tool to exhibit the superadditive phe- negligible information can be intercepted by third par- nomenon of coherent information, especially in an exper- ties. In quantum communication theory, the ability to imental testing. Therefore, to understand the coherent convey quantum information is denoted as the quantum information and, directly, the quantum capacity more ef- capacity Q(ε) [6–9], which is defined by the maximum ficiently and profoundly, a family of quantum channels of coherent information Ic. Many recent studies focused that can exhibit non-additivity is urgently needed. on exploring various properties of quantum capacity [10]: In this letter, we build a dephrasure channel [9] that for example, the lower bounds of quantum capacity in a can exert both the dephasing and the erasure evolution noisy quantum channel have been detected [8, 11], and on the input state. Assisted by this antidegradable chan- the upper bounds of quantum capacity are derived in nel, the superadditivity of coherent information has been thermal attenuator channels [12]. studied. By coding the information with different num- The amount of coherent information present in a quan- bers of photons and applying the corresponding number arXiv:2003.13000v2 [quant-ph] 15 Apr 2020 tum channel plays a central role in quantum communi- of channel uses, we find that a substantial gap of coher- cation. The quantum capacity of a channel [10] can be ent information exists between the n- and (n + 1)-letter computed using the methods of Lloyd, Shor and Deve- level. This evidence well demonstrates the superadditive tak [6, 13, 14], while coherent information itself can be property of the coherent information and also shows a considered analogous to the role of mutual information channel with zero coherent information can have positive in classical channels. In contrast to the mutual infor- capacity [15]. The results obtained in our study provide mation, which is additive in classical channels, coherent a deeper understanding of the nonadditivity of coherent information exhibits superadditivity. Consequently, the information, and supply a solid basis for inspiring many quantum capacity cannot be computed by a single op- new ideas on coherent information and quantum capac- timization like the classical one [15], which makes the ity. measurements of quantum capacity challenging. For this Coherent information, dephrasure channel, and exper- reason, the superadditive properties of coherent infor- imental setup.— The quantum capacity Q(ε), which de- mation becomes important in computing the quantum scribes the capability of the channel to convey quantum 2 (iii) Additional identical channel Coincidence (n + 1 ) ε = I ... ... channel Photon n+1 complementary (n ) ε p,q |0 |1 |1 |0 |e |e Photon n HWP@45° ... ... -fold ... ... ... ... n ... ... |0 |1 |1 |0 Photon 1 dephrasure channel |e HWP@45° |e (1 ) ε p,q (i) State preparation (ii) Dephrasure channel (iv) Tomography dephasing ： box EOM ... BD HWP QWP PBS Mirror Lens Attenuator Fiber PP SPAD FIG. 1: Experimental setup consisting of four parts: (i) state preparation [the β-barium borate (BBO) crystals and the pump laser are not presented in the figure]; (ii) a three-fold dephrasure channel, which will induce a dephasing evolution between 0 and 1 bases in the gray dephasing box via the PPs-EOM sets, and cause an erasure evolution by adding a flag basis e via| i adjustment| i of an attenuator; (iii) additional identical channel platform, which receives the (n + 1)th photon and can prepare| i the repetition code remotely and, meanwhile, also acts as an additional path for building the complementary channel εc; (iv) detection devices. Keywords include: HWP, half-wave plate; QWP, quarter-wave plate; PBS, polarizing beam splitter; PP, phase plate; EOM, electro-optic modulator; BD, beam displacer; SPAD, single-photon avalanche diode. information, can be quantified by the maximum of co- where p and q (p, q [0, 0.5]) denote the dephasing and herent information. The quantity of coherent informa- erasure noise strength∈ in the channel, Π = n σˆ (ˆσ are · tion Ic, depends on the properties of the channel, and the Pauli operators), and e denotes the erasure flag ba- is also strongly related to the form of error-correcting sis which is orthogonal to| thei input space. This channel code used in the communication [7, 17]. Thus, the ad- can be efficiently realized with an optical setup by con- ditive property of Ic will also vary accordingly [9]. The trollable qutrit evolution, and the error-correcting codes quantum capacity of a noisy channel ε can be expressed can be prepared by a multiphoton entangled source. as [5, 6, 19] Fig. 1 presents our experimental setup, which can be recognized as four parts: (i) the state preparation plat- Q(ε) := lim Qn(ε), n (1) form, which contains a multiphoton-entangled source [25] →∞ (not presented in the figure); (ii) three-fold dephrasure 1 where Qn(ε) = n maxρ Ic(ρ, εn) , and Ic(ρ, εn) = channel; (iii) an additional identical channel, which acts c { } S(εn(ρ)) S[εn(ρ))] expresses the coherent information as a complementary channel as well as remote state of the channel− [20, 21]. Here, S( ) is the von Neumann preparation; and (iv) tomography device for the qutrit entropy, and εc denotes the complementary· channel of ε. state. In our experiment, we use spontaneous paramet- Clearly, to study the quantum capacity, the properties ric down conversion process to create a four-photon GHZ of coherent information are an important target to con- state as Φ+ = (1/√2)( HHHH + VVVV ) [25], which | i | i | i duct research and exploration, especially superadditivity. can be used to construct the repetition codes applied in However, studies of the superadditivity of coherent infor- the communication, and the corresponding purification mation are very limited [9], which causes difficulties in codes for coherent information measurements (the de- understanding quantum capacity. tails of the photon source can be found in Supplemental To address this challenge, we construct a dephrasure Material [26]). The dephrasure channel [εp,q( )] is con- · channel with a different number of uses (Fig. 1, mid- stituted by two types of sub-channels: dephasing and dle part). This channel contains both dephasing [22] erasure channels. In the dephasing channel, the input and erasure [23, 24] noises, and can be used to study qubit state, 1/√2( 0 + 1 )( 0 and 1 are the two or- | i | i | i | i the superadditivity of coherent information since its an- thogonal bases in the path degree of freedom), will suffer tidegradability. According to Ref. [9], this dephrasure a dephasing evolution. It is realized by a pair of BD, two channel can be expressed as: sets of waveplates [28, 29], and a dephasing box that con- tains a PPs-EOM set. This structure converts the path ε (ρ) = (1 q)[(1 p)ρ + pΠρΠ†] + qTr(ρ) e e , (2) into the polarization degree of freedom and can achieve p,q − − | ih | 3 (a) (b) ×10- 3 0.5 3 0.4 p = 0.00 ) (a) q = 0.30 q 0.8 , 2 p p 0.4 ( = 0.02 1 0.3 ∆ 0.6 ) p = 0.04 0 ε 0.3 λ 0.0933 0.1092 0.1290 0.1489 0.1687 ρ , ( 0.2 p = 0.06 p 0.4 c p I 0.2 3 0.03 ) (b) ) 0.2 q = 0.35 ε 0.1 q λ 0.02 , 2 ρ , 0.1 ( p c 0.01 ( I p=0.118 p=0.120 p=0.124 p=0.128 p=0.130 1 0.00 0.0 0.0 0.2 0.4 0.6 0.8 1.0 ∆ 0.0 λ 0.0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.0666 0.0799 0.0965 0.1132 0.1298 λ q p 3 ) (c) q = 0.40 q , 2 FIG.