
1 Optimal superdense coding of entangled states Anura Abeyesinghe, Patrick Hayden, Graeme Smith, and Andreas Winter Abstract— We present a one-shot method for preparing pure bipartite pure state can be written in the form ϕAB = entangled states between a sender and a receiver at a minimal | i i √λi ei fi , where ei ej = fi fj = δij and λi 0 [4]. cost of entanglement and quantum communication. In the case | i| i h | i h | i ≥ Since the numbers √λi, known as Schmidt coefficients, are of preparing unentangled states, an earlier paper showed that P l the only local invariants of ϕAB , they entirely determine a 2 -qubit quantum state could be communicated to a receiver | i by physically transmitting only l + o(l) qubits in addition to the nonlocal features of the state. In the case of one-shot consuming l ebits of entanglement and some shared randomness. superdense coding, we find that it is the largest Schmidt When the states to be prepared are entangled, we find that coefficient that plays a crucial role. More specifically, we show there is a reduction in the number of qubits that need to be how Alice can, with fidelity at least 1 κ, share with Bob transmitted, interpolating between no communication at all for − maximally entangled states and the earlier two-for-one result any pure state that has reduction on Bob’s system of dimen- of the unentangled case, all without the use of any shared sion dS and maximum Schmidt coefficient √λmax by trans- 1 1 randomness. We also present two applications of our result: a mitting 2 log dS+ 2 log λmax+O (log(1/κ) log log dS ) qubits direct proof of the achievability of the optimal superdense coding 1 1 and consuming 2 log dS 2 log λmax+O (log(1/κ) log log dS ) protocol for entangled states produced by a memoryless source, ebits. We also show that− these rates are essentially optimal. and a demonstration that the quantum identification capacity of an ebit is two qubits. In the spirit of [5], this new protocol can be viewed as the “father” of the noiseless, visible state communication Index Terms— concentration of measure, entanglement, iden- protocols. Composing it with teleportation generates an op- tification, remote state preparation, superdense coding timal remote state preparation [6], [7] protocol. Applying it to the preparation of states drawn from a memoryless I. INTRODUCTION source generates all the optimal rate points of the triple A sender’s power to communicate with a receiver is fre- cbit-qubit-ebit trade-off studied in [8], when combined with quently enhanced if the two parties share entanglement. The quantum-classical trade-off coding [9], [10]. An inspiration best-known example of this phenomenon is perhaps super- for the present work was Harrow’s alternative construction of dense coding [1], the communication of two classical bits optimal protocols in this memoryless setting that made use of of information by the transmission of one quantum bit and coherent classical communication [11] and pre-existing remote consumption of one ebit. If the sender knows the identity of state preparation protocols [12]. Harrow’s techniques provided the state to be sent, superdense coding of quantum states also strong circumstantial evidence that the protocol we present becomes possible, with the result that, asymptotically, two here should exist. qubits can be communicated by physically transmitting one The rest of the paper is structured as follows. We begin, in qubit and consuming one bit of entanglement [2], [3]. In [2] Section II, by presenting the universal protocol for superdense it was furthermore shown that a sender (Alice) can asymptot- coding of entangled states and then prove its optimality, along ically share a two qubit entangled state with a receiver (Bob) with that of the associated remote state preparation protocol, in Section III. Section IV contains an easy application of typical arXiv:quant-ph/0407061v2 9 Aug 2006 at the same qubit and ebit rate, along with the consumption of some shared randomness. That result, however, failed to subspace techniques to the task of developing an optimal pro- exploit one of the most basic observations about superdense tocol for preparing states generated by a memoryless source. coding: highly entangled states are much easier to prepare Section V provides another application of the protocol, this than non-entangled states. Indeed, maximally entangled states time to the theory of identification [13], [14]. Specifically, we can be prepared with no communication from the sender at show that the quantum identification capacity of an ebit is two all. qubits. In this paper, we construct a family of protocols that take Notation: We use the following conventions throughout the advantage of this effect, finding that even partial entanglement paper. log and exp are always taken base 2. Unless otherwise in the state to be shared translates directly into a reduction stated, a “state” can be pure or mixed. The density operator in the amount of communication required. Recall that every ϕ ϕ of the pure state ϕ will frequently be written simply | ih | | i as ϕ. If ϕAB is a state on A B, we refer to the reduced A. Abeyesinghe, P. Hayden and G. Smith are with the Institute for ⊗ Quantum Information, Caltech 107–81, Pasadena, CA 91125, USA (email: state on A as ϕA. Sometimes we omit subscripts labelling [email protected], [email protected], [email protected]) subsystems, in which case the largest subsystem on which the A. Winter is with the Department of Mathematics, University of Bris- state has been defined should be assumed: ϕ = ϕAB in the tol,Bristol BS8 1UB, United Kingdom (email: [email protected]) bipartite system A B, for example. A system we call A AA, PH, and GS acknowledge the support of the US National Science ⊗ Foundation under grant no. EIA-0086038. PH acknowledges the support of will have a Hilbert space also called A with a dimension dA. the Sherman Fairchild Foundation. AW is grateful for the hospitality of the U(d) denotes the unitary group on Cd, and (Cd) the set of Caltech’s Institute for Quantum Information, during a visit to which part of Cd B the present work was done; he furthermore acknowledges support by the EC linear transformations from to itself. We write the fidelity project RESQ under contract no. IST-2001-37559. between two states ρ and σ as F (ρ, σ) = √ρ√σ 2 and the k k1 2 2 1 1 dB von Neumann entropy of a state ρ as S(ρ)= Tr ρ log ρ. where we may choose C = (8π ln 2)− , and β = . − ln 2 rdA It generalizes the following lemma for rank-one ϕ, which II. THE UNIVERSAL PROTOCOL was proved in [3]. To begin, suppose that Alice would like to share a maxi- mally entangled state with Bob. Clearly, this can be accom- Lemma II.2 Let ϕ be chosen according to the Haar mea- | i plished without any communication – Alice need only perform sure on A B. Then, if 3 dB dA, operations on her half of a fixed maximally entangled state ⊗ ≤ ≤ shared between them. In particular, if ψ is an arbitrary maxi- Pr S(ϕB) < log dB α β | i d − − mally entangled state and we denote by Φ = 1 i i α2C d √d i=1 (5) | i | i| i exp (dBdA 1) 2 , a fixed maximally entangled state, then ψ can be expressed ≤ − − (log dB) | i P as 2 1 1 dB where C = (8π ln 2)− as before and β = ln 2 d . ψ = Vψ IB Φd , (1) A ⊓⊔ | i ⊗ | i Proof (of Lemma II.1) If we let R be a space of dimension r where Vψ is a unitary transformation of Alice’s system which and τABR be a uniformly distributed state on A B R, then depends on ψ. This identity is equivalent to the following 1 | i ⊗ ⊗ Π is equal in distribution to U ϕU † , where Π circuit diagram, in which time runs from left to right: r τAB AB AB τAB is the projector onto the support of τAB. Let σ denote the r 1 I A unitary transformation − e(j+1 mod r) ej + ΠτAB that j=0 | ih | − Vψ implements a cyclic permutation on the eigenvectors ej of τ corresponding to non-zeroP eigenvalues. (There are{| r i}such Φd ψ AB | i B | i eigenvalues except on a set of measure zero, which we will (2) ignore.) We then have r 1 Of course, in general, we would like to prepare an arbitrary 1 1 − k k ΠτAB = σ τAB σ− . (6) state ψAB that may not be maximally entangled, and to do r r k=0 so by| usingi as few resources as possible. Our general method X is as follows. Alice and Bob initially share a fixed maximally Eq. (6), together with the concavity of entropy, implies r entangled state Φd , to which Alice applies an isometry Vψ. | B i 1 1 k k She then sends a subsystem of dimension to Bob, who S TrA ΠτAB S(TrA σ τAB σ− ), (7) A2 dA2 r ≥ r applies a fixed unitary † . Alice’s goal is to make as k=1 UA2B dA2 X small as possible while still reliably preparing ψAB . The which in turn gives procedure can again be summarized with a circuit| diagram,i Pr S(TrA UABϕU † ) < log dB α β although this time it is much less clear whether there exist AB − − choices of the operations Vψ and UA2B that will do the job: r 1 k k Pr S(TrA σ τABσ− ) < log dB α β A ≤ r − − 1 k ! X=1 Vψ k k r Pr S(TrA σ τABσ− ) < log dB α β A2 ≤ − − ψ = r Pr S(τ ) < log d α β , (8) B B ΦdB | i − − | i ❯ B where the final step is a result of the unitary invariance of UA† 2B (3) τABR.
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