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arXiv:0810.3608 · Physical Review A

Phase boundaries in deterministic dense coding

Michael R. Beran, Scott M. Cohen

Published Mar 6, 2009• 7 citations
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Abstract

We consider dense coding with partially entangled states on bipartite systems of dimension $d\ifmmode\times\else\texttimes\fi{}d$, studying the conditions under which a given number of messages, $N$, can be deterministically transmitted. It is known that the largest Schmidt coefficient, ${\ensuremath{\lambda}}_{0}$, must obey the bound ${\ensuremath{\lambda}}_{0}\ensuremath{\le}d/N$, and considerable empirical evidence points to the conclusion that there exist states satisfying ${\ensuremath{\lambda}}_{0}=d/N$ for every $d$ and $N$ except the special cases $N=d+1$ and $N={d}^{2}\ensuremath{-}1$. We provide additional conditions under which this bound cannot be reached---that is, when it must be that ${\ensuremath{\lambda}}_{0}<d/N$---yielding insight into the shapes of boundaries separating entangled states that allow $N$ messages from those that allow only $N\ensuremath{-}1$. We also show that these conclusions hold no matter what operations are used for the encoding, and in so doing, identify circumstances under which unitary encoding is strictly better than nonunitary.

DOI
10.1103/physreva.79.032307

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A useful method for deriving analytical results applicable to the standard two-party deterministic dense-coding protocol is introduced and illustrated. In this protocol, communication of KKK perfectly distinguishable messages is attainable via KKK selected local unitary operations performed on one qudit from a pair of entangled qudits of equal dimension ddd in a pure state $|\ensuremath{\psi}⟩$ with largest Schmidt coefficient $\sqrt{{\ensuremath{\lambda}}_{0}}$. The method takes advantage of the fact that the KKK message states, together with ${d}^{2}\ensuremath{-}K$ augmenting orthonormal state vectors, yield a unitary matrix, thereby implying properties of the KKK message states which otherwise are not readily recognized. Employing this augmented message matrix, we produce simple proofs of previously established results including (i) ${\ensuremath{\lambda}}_{0}\ensuremath{\le}d/K$, (ii) ${\ensuremath{\lambda}}_{0}<d/K$ when K=d+1K=d+1K=d+1, and (iii) the impossibility of finding a $|\ensuremath{\psi}⟩$ that can enable transmission of $K={d}^{2}\ensuremath{-}1$ messages but not d2{d}^{2}d2. Additional results obtained using the method include proofs that the ${\ensuremath{\lambda}}_{0}\ensuremath{\le}d/K$ bound is reduced to at least (i) ${\ensuremath{\lambda}}_{0}\ensuremath{\le}(1/2)[1+\sqrt{(d\ensuremath{-}2)/(d+2)}]$ when K=d+1K=d+1K=d+1 and (ii) ${\ensuremath{\lambda}}_{0}\ensuremath{\le}(K\ensuremath{-}m)/(2K\ensuremath{-}m\ensuremath{-}d)$ whenever $(d+1)\ensuremath{\le}K\ensuremath{\le}2d$ and the selected local unitaries include the first mmm non-negative integral powers of the shift operator XXX.
We present an analytical study of the standard two-party deterministic dense-coding protocol, under which communication of perfectly distinguishable messages takes place via a qudit from a pair of nonmaximally entangled qudits in a pure state $|\ensuremath{\psi}⟩$. Our results include the following: (i) We prove that it is possible for a state $|\ensuremath{\psi}⟩$ with lower entanglement entropy to support the sending of a greater number of perfectly distinguishable messages than one with higher entanglement entropy, confirming a result suggested via numerical analysis in Mozes et al. [Phys. Rev. A 71, 012311 (2005)]. (ii) By explicit construction of families of local unitary operators, we verify, for dimensions d=3d=3d=3 and d=4d=4d=4, a conjecture of Mozes et al. about the minimum entanglement entropy that supports the sending of d+jd+jd+j messages, $2\ensuremath{\le}j\ensuremath{\le}d\ensuremath{-}1$; moreover, we show that the j=2j=2j=2 and $j=d\ensuremath{-}1$ cases of the conjecture are valid in all dimensions. (iii) Given that $|\ensuremath{\psi}⟩$ allows the sending of KKK messages and has $\sqrt{{\ensuremath{\lambda}}_{0}}$ as its largest Schmidt coefficient, we show that the inequality ${\ensuremath{\lambda}}_{0}\ensuremath{\le}d/K$, established by Wu et al. [Phys. Rev. A 73, 042311 (2006)], must actually take the form ${\ensuremath{\lambda}}_{0}<d/K$ if K=d+1K=d+1K=d+1, while our constructions of local unitaries show that equality can be realized if K=d+2K=d+2K=d+2 or $K=2d\ensuremath{-}1$.
The Schmidt number quantifies the dimensionality of entanglement in bipartite quantum states. We investigate when the spectrum of a state on Cd⊗Cd\mathbb{C}^d \otimes \mathbb{C}^dCd⊗Cd alone guarantees that its Schmidt number is not maximal. By deriving spectral bounds for (d−1)(d-1)(d−1)-block positive operators, we prove that LSp⊆ASNd−1\mathrm{LS}_p\subseteq \mathrm{ASN}_{d-1}LSp​⊆ASNd−1​ for p≤⌊d2/2⌋p\leq \lfloor d^2/2\rfloorp≤⌊d2/2⌋, where LSp\mathrm{LS}_pLSp​ denotes the set of states whose largest eigenvalue does not exceed the sum of their ppp smallest eigenvalues, and ASNd−1\mathrm{ASN}_{d-1}ASNd−1​ denotes the set of states whose Schmidt number remains at most d−1d-1d−1 under all global unitaries. This also yields a simple sufficient condition involving only the largest eigenvalue. As an application, we show that states satisfying the reduction criterion under arbitrary global unitaries belong to ASNd−1\mathrm{ASN}_{d-1}ASNd−1​ in every dimension. This subsumes the earlier result that absolutely positive partial transpose states belong to ASNd−1\mathrm{ASN}_{d-1}ASNd−1​. The global unitary requirement is nevertheless essential: for every d≥3d\geq3d≥3, we construct states of maximal Schmidt number satisfying the reduction criterion on both subsystems.
One of the primary goals of information theory is to provide limits on the amount of information one can send through communication channels, and to understand encoding methods that achieve such limits. The discovery of dense coding showed that quantum channels can transmit twice as much classical information than is possible with a classical channel. This optimal rate requires using quantum systems maximally entangled between sender and receiver, and then the sender can encode her messages using unitary operations. The use of nonmaximally entangled systems has been intensively studied in recent years, and it has appeared as though unitary encoding might well be optimal in all cases. Indeed, this optimality of unitary operations has been found to hold under far more general conditions. Nonetheless, we here present numerical evidence supported by analytical arguments that there exist circumstances where more classical information can be encoded using nonunitary, rather than unitary, operations.