Entropic singularities give rise to quantum transmission

Abstract

When can noiseless quantum information be sent across noisy quantum devices? And at what maximum rate? These questions lie at the heart of quantum technology, but remain unanswered because of non-additivity— a fundamental synergy which allows quantum devices (aka quantum channels) to send more information than expected. Previously, non-additivity was known to occur in very noisy channels with coherent information much smaller than that of a perfect channel; but, our work shows non-additivity in a simple low-noise channel. Our results extend even further. We prove a general theorem concerning positivity of a channel’s coherent information. A corollary of this theorem gives a simple dimensional test for a channel’s capacity. Applying this corollary solves an open problem by characterizing all qubit channels whose complement has non-zero capacity. Another application shows a wide class of zero quantum capacity qubit channels can assist an incomplete erasure channel in sending quantum information. These results arise from introducing and linking logarithmic singularities in the von-Neumann entropy with quantum transmission: changes in entropy caused by this singularity are a mechanism responsible for both positivity and non-additivity of the coherent information. Analysis of such singularities may be useful in other physics problems.

Generic nonadditivity of quantum capacity in simple channels

“…The theory of quantum capacities is far richer and more complex than the corresponding classical theory [15,16]. This richness includes many synergies and surprises: super-additivity of coherent information [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31], private information [32][33][34], and Holevo information [35], superactivation of quantum capacity [36][37][38][39][40], and private communication at a rate above the quantum capacity [41,42]. Over the past two decades, there have been numerous exciting discoveries about these phenomena, but they remain mysterious.…”

“…The situation for quantum information transmission is quite different. There is a plethora of concrete channels with super-additive coherent information [17][18][19][20][23][24][25][26][27][28][29][30]. The only known class of channels with strongly additive coherent information are the entanglement-breaking channels, but they are somewhat trivial -their quantum capacity is zero.…”

“…where 0 ≤ s ≤ 1/2, and the input H a , output H b and environment H c have dimension 3, 3, and 2, respectively. The channel N s was first introduced in [28], and extensively studied in the companion paper [1]. From [1,28], the channel coherent information is always positive and can be attained on inputs of the form σ(u) := (1 − u)|0 0| + u|2 2|:…”

Bounding quantum capacities via partial orders and complementarity

“…where the optimization in the last line is over classical-quantum states as in Equation (2.1) with mixed ensemble states, and the mutual informations I(U : B) and I(U : E) are evaluated on the states (id U ⊗ N )(ρ U A ) and (id U ⊗ N c )(ρ U A ), respectively. Similar to the classical capacity above, both the regularizations in the quantum capacity formula (2.6) and in the private capacity formula (2.9) are necessary as well because of superadditivity of the underlying information quantities Q (1) (•) and P (1) (•) [42,13,18,48,47,30,5,4,45,44,43]. The coding theorems for the quantum and private capacity state that the single-letter information quantities are achievable lower bounds on the true capacities: For every quantum channel N ,…”

“…In this work we are particularly interested in the first inequality, and whether there is a gap between the quantum and private capacity. Only a few channels with a strict separation between Q and P are known, among them the Horodecki channel [23,24,22,36], the 'half-rocket' channel [32], and the recently introduced 'platypus' channel [43,29,28]. On the other hand, Watanabe [58] gave sufficient criteria implying Q(N ) = P (N ), which was previously known for degradable channels [46] and antidegradable channels (for which both capacities vanish).…”