Entanglement in channel discrimination with restricted measurements

We study the power of measurements implementable with local quantum operations and classical communication (LOCC) measurements in the setting of quantum channel discrimination. More precisely, we consider discrimination procedures that attempt to identify an unknown channel, chosen uniformly from two known alternatives, that take the following form: (i) the input to the unknown channel is prepared in a possibly entangled state with an ancillary system, (ii) the unknown channel is applied to the input system, and (iii) an LOCC measurement is performed on the output and ancillary systems, resulting in a guess for which of the two channels was given. The restriction of the measurement in such a procedure to be an LOCC measurement is of interest because it isolates the entanglement in the initial input-ancillary systems as a resource in the setting of channel discrimination. We prove that there exist channel discrimination problems for which restricted procedures of this sort can be at either of the two extremes: they may be optimal within the set of all discrimination procedures (and simultaneously outperform all strategies that make no use of entanglement), or they may be no better than unentangled strategies (and simultaneously suboptimal within the set of all discrimination procedures).

Figure 1

Operational scenario for the NE norm. The unknown channel ${\Phi }_i$ goes from Alice to Bob, and it is not in the hands of either of them. Bob performs the final measurement. In this case, Alice’s input to the channel is uncorrelated with any other system in the hands of Alice or Bob.

Figure 2

Operational scenario for the diamond norm. Alice and Bob share an initial—possibly entangled—probe-ancilla state which is optimal for channel discrimination. Alice sends the probe down the channel and Bob proceeds to the measurement of the output probe-ancilla state.

Figure 3

Operational scenario for the LOCC norm. Alice creates an entangled probe-ancilla state, keeps the ancilla, and sends the probe down the channel. Alice and Bob can then proceed to an LOCC measurement of the output probe-ancilla state.

Figure 4

Structure of some exemplary pairs of channels for which the LOCC norm is equal to the diamond norm, and strictly greater than the NE norm. Single lines indicate quantum systems or messages; double lines classical ones. The channel chooses at random a unitary among a discrete set. An incomplete projective measurement in the computational basis with two possible outcomes is then performed. The result of the measurement is further flipped or not, depending on the channel (on the value of $i$ in the figure). The output of the channel is classical, meaning that it is diagonal in a fixed computational basis: a register containing the index of the chosen unitary, and a bit containing the (possibly flipped) result of the measurement. A full description of the action of the channels can be found in the main text.