Classical Information Capacity of Superdense Coding

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Classical information capacity of superdense coding

Garry Bowen∗ Department of Physics, Australian National University, Canberra, A.C.T. 0200, Australia. (February 1, 2001) Classical communication through quantum channels may be enhanced by sharing entanglement. Superdense coding allows the encoding, and transmission, of up to two classical bits of information in a single qubit. In this paper, the maximum classical channel capacity for states that are not maximally entangled is derived. Particular schemes are then shown to attain this capacity, ﬁrstly for pairs of qubits, and secondly for pairs of qutrits.

03.67.Hk

Quantum information exhibits many features which do ters, then the maximal amount of classical information not have analogues in classical information theory [1]. that may be transferred is given by the Kholevo bound For this reason, when a quantum channel is used for com- [10], munication, there exist a number of different capacities for the different types of information transmitted through C = max S p (U k I )ρ (U k I )† k k A B AB A B the channel [2–5]. {U ,pk} ⊗ ⊗ A " k ! Superdense coding (referred to in this paper simply as X dense coding), first proposed by Bennett and Wiesner [6], k k † pkS (U IB )ρAB(U IB ) . (1) is where the transmission of classical information through − A ⊗ A ⊗ k # a quantum channel is enhanced by shared entanglement X between sender and receiver. The classical information This bound has been shown to be asymptotically attain- capacity for a channel where sender and receiver share able by using product state block coding [8,11]. k entanglement has been called the entanglement-assisted As the operators U IB are unitary, applying one A ⊗ classical capacity CE [5]. The classical capacity for dense of the operators to ρAB will not change the eigenval- coding, denoted here by C, provides a lower bound on ues. Hence the entropy, which depends only on the CE. eigenvalues, of each summand in the second term of Eq. For Completely General Dense Coding (CGDC) [7], (1) remains unchanged, and the second term reduces to the sender, Alice, and receiver, Bob, share qubits in the S(ρAB). To maximize the capacity we must therefore state ρAB, Alice may encode a message using a set of maximize the first term, unitary transformations U k , with a priori probabilities { A} pk , on her qubit. Alice then sends her qubit to Bob, 0 k k † { } S(ρ ) = S pk(U IB)ρAB (U IB ) . (2) who decodes the message by doing joint measurements AB A ⊗ A ⊗ k ! on both qubits. X For pure states of pairs of D state systems, where A general density matrix of a two qubit bipartite sys- ρAB = ΨAB ΨAB , the channel capacity has been de- tem may be expanded as, rived by| bothihHausladen,| et. al. [8], and, Barenco and Ekert [9], and was shown to be, C = log D +S(ρB). Here i j ρAB = λij σ σ , (3) S is the von Neumann entropy S(ρ) = Trρ log ρ, where A ⊗ B ij the logarithm is base 2. − X Bose, Plenio and Vedral [7] have further proven that if where the σ’s consist of a scaled version of the set of Pauli Alice’s alphabet of operators is restricted to the set of the matrices and the identity, that is, k identity and three Pauli matrices, UA I, σx, σy, σz , then the capacity for a pair of qubits is maximized∈ { by set-} 0 1 1 1 0 σ = I2 = (4) ting p = 1/4. This scheme was labeled by the authors 2 2 0 1 k as Special Dense Coding (SDC). 1 1 1 0 1 σ = σ = (5) In this paper, a bound on the channel capacity for 2 x 2 1 0 dense coding is derived for arbitrary sets of unitary operators on pairs of qubits. It is shown that the scheme 2 1 1 0 i σ = σy = − (6) 2 2 i 0 of SDC attains that bound. Further, the proof for the case of pairs of qutrits is outlined, utilizing the higher 3 1 1 1 0 σ = σ = . (7) dimensional analogue of SDC. 2 z 2 0 1 Suppose Alice and Bob share pairs of qubits in the − state ρAB, and Alice is restricted to using unitary oper- By linearity we can obtain the reduced density matrices 0 ators and sending her message as a product state of let- of ρAB and ρAB, by tracing over the expansions,

1 ρB = TrA [ρAB] (8) where Eq. (22) follows from Eq. (21) due to the relation- j i j 1 j 1 i 1 i ship σ σ σ = δij σ σ = σ , for i, j 1, 2, 3 , 2 − 4 4 ∈ { } i j and Eq. (23) is obtained by comparing Eq. (22) with Eq. = TrA λij σ σ (9)  A ⊗ B  ij (11). Thus, the capacity for SDC is equal to the bound X given in Eqs. (16)-(19), and SDC has been shown to be  i j  = λij TrA σA σB (10) an optimal method for CGDC. ij A similar result applies for two qutrits, where Alice X j uses the operators, = λ0j σB, (11) j X 0 0 1 where the trace of each of the Pauli matrices is zero. U0 = 1 0 0 (24) Also,  0 1 0 

0 k k †  0 1 0  ρB = TrA pk(UA IB)ρAB (UA IB ) (12) ⊗ ⊗ 1 " k # U = 0 0 1 (25) X   k i k † j 1 0 0 = λij pkTrA UAσA(UA) σ (13) B  1 0  0 ij k 2 X X i 3 π j U2 = 0 e 0 (26) = λ0 σ (14)  4  j B 0 0 ei 3 π j X  1 0 0  = ρ , (15) 4 B i π U3 = 0 e 3 0 (27) using the fact that the trace of a matrix does not change  i 2 π  0 0 e 3 under unitary transformations.  i  Combining the above derivations leads to the main re- U4 = [U0, U2] (28) sult of this paper. The amount of information that may −√3 k be transferred for any U , pk using an arbitrary, two i { A } U5 = [U0, U3] (29) qubit mixed state ρAB, is given by, √3 k k † i C = S( pk(U IB )ρAB(U IB) ) A ⊗ A ⊗ U6 = [U1, U2] (30) k √3 X k k † i pkS((UA IB )ρAB(UA IB ) ) (16) U7 = [U1, U3] (31) − ⊗ ⊗ k −√3 X 0 U8 = I3, (32) = S(ρAB) S(ρAB) (17) 0 − 0 S(ρA) + S(ρB) S(ρAB) (18) ≤ − with a priori probability pj = 1/9, where [Ui, Uj ] denotes log 2 + S(ρB) S(ρAB). (19) ≤ − the commutator. Expanding the density matrix ρAB in Here, Eq. (17) follows from the discussion following Eq. terms of the identity and the traceless Hermitian gener- (1), and the ﬁrst term rewritten as for Eq. (2). Eq. (18) ators λi of SU(3) [12], we ﬁnd, uses the subadditivity of the entropies of a bipartite sys- { } 0 tem, and Eq. (19) from the relations S(ρ ) = S(ρB), by 0 † B ρ = pj Uj ρABU (33) Eq. (15), and the bound S(ρ0 ) log 2 for a qubit. AB j A j This bound is attainable using≤ Special Dense Coding X k k 1 (SDC), where Alice uses the operators UA = 2σA, each = I3 ρ , (34) 3 B occurring with a priori probability pk = 1/4. Using this ⊗ scheme, the state received by Bob is completely disentangled, that is, and the capacity is given by,

0 k i j k † C = log 3 + S(ρB) S(ρAB). (35) ρ = pkU λij σ σ (U ) (20) − AB A  A ⊗ B  A k ij X X Similar constructions for arbitrary N M state sys-   × k i k j tems may easily be considered using analogues of the = λij σ σ σ σ (21) A A A ⊗ B unitary transformations used in SDC. The transforma- ij k ! X X tions consist of the set of cyclic permutations of the DA 0 j = λ0j σ σ (22) basis states of A, where DA is the dimension of the A ⊗ B H j Hilbert space A of Alice’s state, the set of unitary ma- X H 1 trices derived from the cyclic group generated by the ma- = IA ρB, (23) 2 ⊗ trix consisting of the DA roots of unity on the diagonal

2 (up to overall phase), and the normalized commutators APPENDIX A between elements of these two sets of transformations. The connection between sets of unitary depolarizers, the existence of orthonormal bases of maximally entangled Proof of Eq. (39). Suppose ρ is disentangled, states, and dense coding has previously been noted by AB then the density matrix may be written in the form Werner [13]. i i ρAB = i pi ωA ωB, with i pi = 1 and pi > 0, where We thus make the conjecture that for an N M state ⊗ i × the reduced density matrices ω are all pure states. By system ρAB, the dense coding capacity is given by, P P the convexity of the expression S(ρB) S(ρAB) [18], we have, − C = log DA + S(ρB ) S(ρAB), (36) − with DA = N. i i i S(ρB) S(ρAB) piS(ω ) piS(ω ω ) − ≤ B − A ⊗ B The result obtained in this paper agrees with the pre- i i X X viously obtained results in the case of pure states. The = 0 capacity may also be rewritten in the form,

C(ρAB) = log DA S(ρA) and hence S(ρAB) S(ρB). Similarly for S(ρAB) − ≥ ≥ +S(ρA) + S(ρB) S(ρAB) (37) S(ρA). − = C(ρA) + S(A, B), (38) for C(ρA) the capacity of sending qubit A without access to qubit B, and S(A, B) = S(ρA) + S(ρB) S(ρAB) the − von Neumann mutual entropy of ρAB. In this way it is shown that the capacity due to the joint measurement ∗ of both qubits is enhanced over the use of a single qubit Current address: Centre for Quantum Computation, by a factor equal to the von Neumann mutual entropy of Clarendon Lab, Parks Road, Oxford OX1 3PU, United the combined state. Kingdom. [1] C. H. Bennett and D. P. DiVincenzo, Nature 404, 247 The capacity also gives an exact bound on the mixed- (2000). ness of a state for when dense coding with that state may [2] H. Barnum, M. A. Nielsen, and B. Schumacher, Phys. be said to fail [14]. For an arbitrary bipartite state the Rev. A 57, 4153 (1998). capacity will not exceed log N, for an N M state sys- × [3] C. Adami and N. J. Cerf, Phys. Rev. A 56, 3470 (1997). tem, with N, M 2, 3 , whenever S(ρB) S(ρAB) 0. ∈ { } − ≤ [4] S. Lloyd, e-print, quant-ph/9604015 (1996). Disentangled states [15] satisfy the inequality, [5] C. H. Bennett, P. W. Shor, J. A. Smolin, and A. V. Thapliyal, Phys. Rev. Lett. 83, 3081 (1999). S(ρAB) max S(ρA), S(ρB ) , (39) ≥ { } [6] C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. 69, 2881 (1992). and therefore cannot be used to transmit more than log N [7] S. Bose, M. Plenio and V. Vedral, J. Mod. Opt. 47, 291 bits per state. The proof of Eq. (39) is given in Appendix (2000). A. [8] P. Hausladen, R. Jozsa, B. Schumacher, M. Westmore- It may also be noted [16], that the result for the ca- land, and W. K. Wootters, Phys. Rev. A 54, 1869 (1996). pacity of a two qubit system also proves the conjecture [9] A. Barenco and A. K. Ekert, J. Mod. Opt. 42, 1253 that the capacity for dense coding is bounded by [7], (1995). [10] A. S. Kholevo, Probl. Peredachi Inf. 9, 3 (1973) [Probl. C 1 + ED, (40) ≤ Inf. Transm. (USSR) 9, 177 (1973)]. [11] B. Schumacher and M. D. Westmoreland, Phys. Rev A where E is the (one way) distillable entanglement D 56, 131 (1997). of ρAB, provided the Hashing Inequality [17], ED ≥ [12] C. M. Caves and G. J. Milburn, Opt. Commun. 179, 439 S(ρB) S(ρAB), is true. − (2000). In summary, the classical information capacity of dense [13] R. F. Werner, e-print, quant-ph/0003070 (2000). coding, through a noiseless channel, using arbitrary [14] S. Bose and V. Vedral, Phys. Rev. A. 61, 040101(R) mixed states of two qubits or two qutrits has been de- (2000). rived. A method of generalizing to N M state systems 57 × [15] V. Vedral and M. B. Plenio, Phys. Rev. A 1619 (1998). has been outlined, and a conjecture made about the clas- [16] S. Bose (private communication). sical capacity of dense coding using such systems. [17] M. Horodecki, P. Horodecki, and R. Horodecki, Phys. The author would like to thank Craig Savage and Tim Rev. Lett. 85, 433 (2000). Ralph for discussions. The author is supported by the [18] E. H. Lieb and M. B. Ruskai, J. Math. Phys. 14, 1938 DETYA. (1973).