On quantum illumination, quantum reading, and the capacity of quantum computation
aa r X i v : . [ qu a n t - ph ] D ec On quantum illumination, quantum reading,and the capacity of quantum computation
Stefano Pirandola
Department of Computer Science, University of York, York YO10 5GH, UK andResearch Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge MA 02139, USA
In this brief note, I clarify the crucial differences between three different protocols of quantumchannel discrimination, after some confusion has appeared in recent literature.
Reason for this note
In some recent literature, there has been confusion be-tween the protocols of quantum illumination [1, 2], quan-tum reading [3], and a scheme of communication withina discrete-variable quantum computer [4]. All these pro-tocols are based on the idea of quantum channel dis-crimination (QCD), but they have completely differentapplications and features, which is the reason why theyhave different names and should not be confused. Let meprovide some clarifications below.
Quantum Illumination
Quantum illumination [1, 2] (see also [5–21]) is the useof input quantum resources (such as entanglement) andoutput quantum measurements to enhance the detectionof a remote low-reflectivity object in a bright thermal-noise environment. It can be represented as a QCD prob-lem where the bit of information associated with the pres-ence or absence of the target is associated with the binarydiscrimination of two channels, one including a partialreflection from the target and the other one being a com-pletely thermalizing channel (replacing the input withthe state of the environment). Here one can show that,despite initial entanglement is lost in the sender-receiverpath, the benefits of quantum illumination still survivein the forms of output correlations. These allow one toenhance the sensitivity of detecting the presence of thetarget-object with respect to the use of classical sourcesof light (in particular separable states in the DV versionof the protocol [1], and mixtures of coherent states inthe CV version [2]). It is called “quantum” illuminationbecause because it proves a quantum advantage with re-spect to classical strategies under the same conditions(e.g., the same mean number of input photons).
Quantum Reading
Quantum reading [3] (see also [22–37]) is the use of in-put quantum resources (such as entanglement) and out-put quantum measurements to enhance the retrieval ofclassical information stored in the cells of an optical mem-ory. It can be represented as a QCD problem where the bit of information is encoded in two different reflectivi-ties of the memory cell. These are two different bosonicGaussian channels that are generally characterized by dif-ferent losses and thermal noises. Contrary to quantumillumination, the scheme is in the very near range, workswith high reflectivities and allows one to use codewordsto encode information in blocks of many cells (so thatquantum reading capacities can be defined). Here onecan show that the use of quantum resources (e.g. en-tanglement) allows one to enhance the data readout interms of bits per cell with respect to the use of classi-cal strategies (in particular the use of coherent states ortheir mixtures). It is called “quantum” reading becauseit proves a quantum advantage with respect to classicalstrategies using the same amount of energy (mean num-ber of photons).The two schemes of quantum illumination and quan-tum reading have a specific peculiarity (quantum en-hancement) that gives them the “quantum name”. Atthe same time it is clear that they are both schemes ofQCD, where classical information is retrieved from a box(target object or memory cell). For instance, see thediscussion in Section V.H “Gaussian channel discrimina-tion and applications” of the Gaussian information re-view [38]. For more details on these protocols, see alsothe recent review on photonic quantum sensing [39].
Capacity of quantum computation
The scheme of Ref. [4] is about the communication ca-pacity of quantum computation. Clearly, it is not abouttarget detection or optical storage, but rather commu-nication between registers of a dicrete-variable quantumcomputer. In this scheme, there is a “memory” register( M ) where the sender encodes a classical variable i in N pure quantum states | i i M h i | with some probability p i .Then, the receiver has a computation register ( C ) pre-pared in some initial state ρ C . The initial state of thetwo registers is therefore the tensor-product X i p i | i i M h i | ⊗ ρ C . (1)The two registers are then fed into a quantum computer,which applies the unitary ˆ U i onto register C conditionallyon the value i of register M . Here ˆ U i represents a series ofquantum gates which describes some quantum algorithm.For instance, i may be an integer, and the computationaloutput ρ iC = ˆ U i ρ C ˆ U † i may be its factorization accordingto Shor’s algorithm.In general, for the input state as in Eq. (1), the quan-tum computer provides the output X i p i | i i M h i | ⊗ ρ iC . (2)The receiver measures register C so as to discriminatebetween the possible output states ρ iC , or equivalentlybetween the possible unitary operations ˆ U i . The optimalinformation accessible to the receiver is the Holevo bound I ( C : i ) = S ( C ) − S ( C | i ) , (3)where S ( C ) is the von Neumann entropy of the reducedstate of C , and S ( C | i ) is the corresponding conditionalvon Neumann entropy. This is clearly maximized when p i is uniform and the states ρ iC are pure and orthogonal,so that it takes the maximum value I ( C : i ) = log N .By construction, it is clear that I ( C : i ) representsthe capacity of the quantum computation { ˆ U i } becauseit tells you how good the quantum computer is in provid-ing distinguishable output states (solutions) for differentinputs. When the maximum log N is achieved, it meansthat the quantum computation is perfect over the entireinput alphabet of N letters. Clarifications
Apart from being interpreted as protocol of QCD, thescheme of Ref. [4] is clearly different from both quantumillumination and quantum reading. • First of all, Ref. [4] is a communication scheme,where sender’s input alphabet is decoded by a re-ceiver. More specifically, it is spatial communica-tion between two registers which is mediated by aquantum computation. It is a two-register descrip-tion where the unitaries are ˆ U i are not stored inthe computational register M but rather appliedin the dynamical process of the quantum computer(they are in fact control-unitaries). In this regardit is clearly different from the static scenario wherea classical variable is physically and stably storedinto a black box by an ensemble of channels (to de-scribe presence/absence of a target, or the differentreflectivities of a memory cell). This means thatRef. [4] is not about readout from storage . • The input-output process is based on a single sig-nal system processed by a unitary. Today, we knowthat unitary discrimination can be perfectly solvedin a finite number of uses [40]. It is thereforedifferent from what happens in the more general discrimination of quantum channels, where perfectdiscrimination is not guaranteed (at finite energies)and the optimal states may require the use of idlersystems, which are not sent through the box butdirectly to the output measument in order to assistthe entire process. • The signal-idler structure, which is missing inRef. [4], is one of the main features for both quan-tum illumination and quantum reading. The useof input entanglement and, more generally, quan-tum correlations, is the main working mechanism ofthese two protocols under completely general condi-tions of decoherence. As a matter of fact, as alreadysaid above, the “quantum” name in “illumination”and “reading” exactly comes from the comparisonof a quantum resource at the input (entanglement)with respect to the use of classical input states (sep-arable states, mixtures of coherent states). • The communication scheme of Ref. [4] is for adiscrete-variable Hilbert space. The main settingfor both quantum illumination and quantum read-ing is bosonic . Quantum illumination provides thepossible working mechanism of a lidar (in the op-tical case) and a radar (in the microwave case).Quantum reading is also working at the optical fre-quencies, which is the physics in optical storage.In summary, all the protocols of quantum illumination,quantum reading, and communication of quantum com-putation can be represented as schemes of QCD. How-ever, they are protocols with different aims and features,which is the reason why they should not be confusedone with the other. In particular, the scheme of Ref. [4]is about communication between registers of a quantumcomputer, clearly not “quantum reading” of a classicalmemory nor “quantum illumination” of a remote target.With these two protocols, it only shares the basic struc-ture, that of QCD, which is ultimately about the retrievalof classical information from some type of black box. [1] Lloyd, S. Enhanced sensitivity of photodetection viaquantum illumination.
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