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Zero-entropy encoders and simultaneous decoders in identification via quantum channels

Pau Colomer, Christian Deppe, Holger Boche, Andreas Winter

TL;DR

This work revisits the problem of identification via quantum channels with the additional restriction that the message encoding must use pure quantum states, rather than general mixed states, and suggests a two-dimensional spectrum of different identification capacities.

Abstract

Motivated by deterministic identification via classical channels, where the encoder is not allowed to use randomization, we revisit the problem of identification via quantum channels but now with the additional restriction that the message encoding must use pure quantum states, rather than general mixed states. Together with the previously considered distinction between simultaneous and general decoders, this suggests a two-dimensional spectrum of different identification capacities, whose behaviour could a priori be very different. We demonstrate two new results as our main findings: first, we show that all four combinations (pure/mixed encoder, simultaneous/general decoder) have a double-exponentially growing code size, and that indeed the corresponding identification capacities are lower bounded by the classical transmission capacity for a general quantum channel, which is given by the Holevo-Schumacher-Westmoreland Theorem. Secondly, we show that the simultaneous identification capacity of a quantum channel equals the simultaneous identification capacity with pure state encodings, thus leaving three linearly ordered identification capacities. By considering some simple examples, we finally show that these three are all different: general identification capacity can be larger than pure-state-encoded identification capacity, which in turn can be larger than pure-state-encoded simultaneous identification capacity.

Zero-entropy encoders and simultaneous decoders in identification via quantum channels

TL;DR

This work revisits the problem of identification via quantum channels with the additional restriction that the message encoding must use pure quantum states, rather than general mixed states, and suggests a two-dimensional spectrum of different identification capacities.

Abstract

Motivated by deterministic identification via classical channels, where the encoder is not allowed to use randomization, we revisit the problem of identification via quantum channels but now with the additional restriction that the message encoding must use pure quantum states, rather than general mixed states. Together with the previously considered distinction between simultaneous and general decoders, this suggests a two-dimensional spectrum of different identification capacities, whose behaviour could a priori be very different. We demonstrate two new results as our main findings: first, we show that all four combinations (pure/mixed encoder, simultaneous/general decoder) have a double-exponentially growing code size, and that indeed the corresponding identification capacities are lower bounded by the classical transmission capacity for a general quantum channel, which is given by the Holevo-Schumacher-Westmoreland Theorem. Secondly, we show that the simultaneous identification capacity of a quantum channel equals the simultaneous identification capacity with pure state encodings, thus leaving three linearly ordered identification capacities. By considering some simple examples, we finally show that these three are all different: general identification capacity can be larger than pure-state-encoded identification capacity, which in turn can be larger than pure-state-encoded simultaneous identification capacity.
Paper Structure (9 sections, 9 theorems, 59 equations, 1 figure)

This paper contains 9 sections, 9 theorems, 59 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Transmission and identification via quantum channels
  3. Zero-entropy encoders (with simultaneous decoders) achieve double exponential identification codes
  4. Simultaneous identification capacity is attained with pure state encodings
  5. Separations between capacities and further analysis
  6. Conclusions
  7. Acknowledgments.
  8. Disclosure of Interests.
  9. Fixed superposition phases fail to prove Theorem \ref{['th:PureStateIDcode']}

Key Result

Theorem 2

For all $\lambda\in (0;1)$, we have the capacity of communication and the strong converse given by

Figures (1)

  • Figure 1: Let Alice encode a message $m$ from a set $\mathcal{M}=\{1,\dots,M\}$ into a code word of length $n$ and send it through a discrete memoryless channel (DMC) described by the stochastic matrix $W^n$. In the usual transmission scheme (above), when Bob receives $y^n$ he can decode the message, aiming to recover some $\hat{m}\approx m$. In an identification scheme (below), he instead chooses any message $m'\in\mathcal{M}$ and checks whether it is equal to $m$ with a particular hypothesis testing decoder, obtaining a binary answer.

Theorems & Definitions (22)

  • Definition 1
  • Theorem 2: Shannon:TheoryCommunicationWolfowitz:converse
  • Definition 3
  • Proposition 4: AD:ID_ViaChannels
  • Theorem 5: AD:ID_ViaChannelsHanVerdu:ID
  • Definition 6
  • Theorem 7: holevo:capacitySW:capacityON:Strong_converseWinter:Strong_converse
  • Definition 8
  • Definition 9
  • Definition 10
  • ...and 12 more