Dense coding using entangled input states

Dense coding using entangled input states

2 July 2001 Physics Letters A 285 (2001) 115–118 www.elsevier.nl/locate/pla Dense coding using entangled input states Garry Bowen Centre for Quantum...

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2 July 2001

Physics Letters A 285 (2001) 115–118 www.elsevier.nl/locate/pla

Dense coding using entangled input states Garry Bowen Centre for Quantum Computation, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, UK Received 30 January 2001; received in revised form 2 May 2001; accepted 4 May 2001 Communicated by P.R. Holland

Abstract Quantum entanglement may be used to increase the classical information capacity of a quantum channel. The capacity of dense coding using entangled input states though a noiseless channel is shown to be no better than the capacity for product input states, when using a pure entanglement resource.  2001 Elsevier Science B.V. All rights reserved. PACS: 03.67.Hk; 89.70.+c Keywords: Quantum information; Dense coding; Entanglement

1. Introduction

entangled resource ρAB was shown to be CX→Y = log dim X + S(ρY ) − S(ρAB ),

Entanglement of quantum systems leads to many phenomena with no classical counterpart. Superdense coding [1], or quantum dense coding, uses the sharing of quantum entanglement to assist in transmitting a higher rate of classical information through a noiseless quantum channel than is possible without shared entanglement. The initial implementations of dense coding involve product state coding, where individual letters of the message are encoded onto single qubits. The message is decoded using combined measurements of the whole message. Special dense coding (SDC) [2] has been shown to be the optimal method of dense coding using unitary operators and product state coding, even when the entangled resource is not maximally entangled [3,4]. The capacity for dense coding using SDC and an

E-mail address: [email protected] (G. Bowen).

(1)

where X and Y are determined by which state, A or B, is used for the encoding, dim X is the dimension of this Hilbert space, and S(ω) = − Tr ω log ω the von Neumann entropy of the state ω. By convention the logarithm is taken to be base 2. In this Letter, the capacity of dense coding is determined for pure entangled states, for the general case where Alice is no longer restricted to using product state encoding, but may use entangled coding of all N copies of the qubits available to her. The case where Alice uses superoperators to encode a product state message follows as a corollary. A conjecture is then made about the bounds of the combined von Neumann entropy of a bipartite system when a superoperator acts on one part of the system. Provided this conjecture can be shown to be true, the capacity for dense coding through a noiseless channel, using any quantum resource, is bounded above by the maximum of either Eq. (1) or the classical capacity of one bit per qubit.

0375-9601/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 3 2 8 - 0

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2. Unitary encoding The encoding of the classical information onto Alice’s state may be written as a unitary transformation on an extended system containing an encoding state ρ C . In the case of special dense coding, we have ρC = i pi |ii|C , for pi = 1/4 and {|iC } a set of orthogonal states, and  ECi ⊗ σAi , UCA = (2) i Ei

where = |ii|C is a projection operator onto the state |iC , and {σ i } the set of Pauli operations and the identity. The quantum correlations between Alice and Bob following the encoding and transmission of the qubit may be expressed using the von Neumann mutual information, S(X : Y ) = S(ρX ) + S(ρY ) − S(ρXY ). By applying a memory to Alice’s coding state,  i pi ρM ⊗ ρCi , ρMC = (3) i

and, using the ρC state to encode the qubit A, the von Neumann mutual information between the memory state and the shared entangled state is given by     − S ρMAB , S(M : AB) = S(ρM ) + S ρAB (4)  † i i with ρMAB = TrC { i pi ρM ⊗ UCA (ρC ⊗ ρAB )UCA } = Tr {ρ and ρAB M MAB }. From the disentangled nature of the state ρMC and convexity of the term S(ρM ) − S(ρMAB ) [5], we obtain the Holevo bound [6–8], H (M : AB)  S(M : AB)  i     − , pi S ρAB  S ρAB

(5)

i

S(M : AB)  S(M : CAB)   i  , pi S ρM  S(ρM ) −

   †  = TrC UCA ρCi ⊗ ρAB UCA = ΛiA ρAB ,

where the final inequality follows from the fact that ρMCAB is  disentangled. Similarly, S(M : AB)  S(ρC ) − i pi S(ρCi ). As the memory states only restrict the capacity in terms of Eq. (9), they can be choi = |ii| , so sen as a set of orthogonal pure states ρM M that the bound is limited by the values of the a priori probabilities pi , as in the case of a classical message coding.

3. Entangled coding Given N copies of an entangled quantum state, instead of restricting the encoding operations to single copies of A and C, Alice may use an encoding which entangles all the copies together. The message state and encoding state are written as a probability distribution over all possible messages,  i pi ρM ⊗ ρCi . MC = (10) j

(6)

for a superoperator acting on state A, and H (M : AB) is the Shannon mutual information following measurement. The Shannon mutual information gives the shared classical information between Alice’s message, stored in M, and Bob’s decoded message AB. The capacity can thus be bounded above by  i     − pi S ρAB C  S ρAB (7)

For N copies of the entanglement resource ρAB , the total density operator then becomes

ΛiA

i

 log dim A + S(ρB ) −

(9)

i

i

where i ρAB

In the case that Alice is restricted to unitary ΛiA , using SDC and appropriate memory states, the bound given by Eq. (8) may be attained. This then reduces to the capacity given in Eq. (1) with X = A and Y = B. The requirement for memory states of an appropriate nature is due to the fact that there are bounds on the capacity determined by the choices of memory and encoding states. These are given by

 i



 pi S ΛiA ρAB .

(8)

MCAB =





 i pi ρM

⊗ ρCi

i



N

j ρAB

(11)

j =1

= MC ⊗ AB .

(12)

Using an encoding which entangles all N copies, † MCAB = UCA MCAB UCA 

† i = pi ρM ⊗ UCA ρCi ⊗ AB UCA , i

(13) (14)

G. Bowen / Physics Letters A 285 (2001) 115–118

the reduced density matrix may then be written in the form  †  = TrC UCA MCAB UCA MAB (15)  

 † i = pi ρM ⊗ TrC UCA ρCi ⊗ AB UCA i

=



(16) i ⊗ AB ,

i pi ρM

(17)

i where AB may be an entangled state of all N copies j of ρAB . The mutual information between the memory state and the output state is then given by       S(M : AB) = S M (18) + S AB − S MAB     i   S AB + pi S ρM





pi S



i i MAB



(19)

i

 i     − , pi S AB = S AB

(20)

i where the last equality follows from the fact that MAB is a disentangled state. By using the subadditivity of the entropy of a bipar )  log(dim A)N and tite system and the fact that S(A  j S(B ) = S( N j =1 ρB ) = NS(ρB ), we have     i  − C  S AB (21) pi S AB i

 N log dim A + NS(ρB ) −



 i  . pi S AB

(22)

i

The capacity per state is obtained by dividing through by N , which will give the bound for dense coding with product states in Eq. (8) provided     i   j  (23) N pi S AB pk S ΛkA ρAB , i

k

where {ΛkA } is the set of superoperators generated by the product state encoding. For pure states ρAB , the capacity for an entangled state encoding is no larger than that for product state encodings. It can easily be seen from Eq. (23), that the right-hand side is equal to zero for pure states if all Λk are chosen to be unitary. Thus, the classical capacity

117

for pure states through a noiseless channel is C = H (M : AB) = log dim A + S(ρB ),

(24)

which is attained using SDC, or higher-dimensional analogues. The relationship in Eq. (24) has been shown to be the capacity for dense coding, with product state encoding, using pure states [9,10]. To determine the relationship between entangled and product state encodings for mixed states, much more must be known about bounds for both sides of Eq. (23). In the case of restrictions to unitary alphabets, the right-hand side becomes NS(ρAB ), which in some cases may not be optimal.

4. A conjectured bound using superoperators When a superoperator acts on one part of a bipartite system, it may be conjectured that the entropy of the combined system obeys the relationship   S(ΛA ωAB )  min S(ωB ), S(ωAB ) , (25) which in the entangled coding case gives the inequality   j   i   N min S(ρB ), S ρAB . S AB (26) Eq. (25) is known to hold whenever ΛA is unitary or unital [11]. 1 For non-unital operators it is not known whether the relationship still holds. If Eq. (26) holds, combining this with Eq. (23) would mean the generalization of SDC is optimal for dense coding, though a noiseless channel whenever S(ρAB ) < S(ρB ). In the case where S(ρAB )  S(ρB ), Alice may project her qubits onto a pure state and encode her message “classically” using orthogonal pure states, always giving a capacity of C = log(dim A). If Eq. (26) does not hold, then we may look for entangled coding schemes which, for mixed states, exceed the capacity provided by SDC. The capacity for SDC may be rewritten in the form CA→B = log dim A − S(ρA ) + S(ρA ) + S(ρB ) − S(ρAB ) = C(ρA ) + S(A : B),

(27) (28)

1 A unital operator is one which preserves the identity Λ 1 = 1. A

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G. Bowen / Physics Letters A 285 (2001) 115–118

where C(ρA ) is the capacity of sending and decoding Alice’s qubit only. If we are to improve on this capacity, then the entangled coding of Alice’s qubits must involve some form of a “unmixing” process, where the entropy of ρA is reduced without reducing the mutual entropy S(A : B) of the combined system by a larger amount. Note that the mutual entropy of the combined system cannot increase under the action of a superoperator on one part of the system. In the case where SDC gives CA→B < log dim A, j then by purifying each ρA , C(ρA ) → log dim A and S(A : B) → 0, giving the classical capacity of one bit per qubit.

5. Conclusion In this Letter, it has been shown that the capacity for dense coding through a noiseless channel cannot be improved for pure entangled states by using entangled input states. A conjecture has also been made about the inability to improve the capacity when using entangled input states for a shared mixed entangled state, based on an entropy inequality for superoperators acting on part of a bipartite mixed state.

Acknowledgement The author would like to thank Sougato Bose for helpful discussions. The work is supported by the Harmsworth Trust, the Oxford–Australia Fund, and the CVCP.

References [1] [2] [3] [4] [5] [6] [7] [8]

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