Perfect and Quasi-Perfect Codes for the Bosonic Classical-Quantum Channel

In this article, we explore perfect and quasi-perfect codes for the Bosonic channel, where information is generated by a laser and conveyed in the form of coherent states. In particular, we consider the phase-modulation codebook for coherent states in a Bosonic channel. We show that these phase-modulation codes are quasi-perfect as long as the cardinality of the code is the same as the dimension of the coherent states. These codes feature the smallest error probability among all codes of the same cardinality and the same dimension of the channel Hilbert space. We study the performance of these codes in terms of error probability, incorporating the degradation caused by a depolarizing or an erasure quantum channel.


I. INTRODUCTION
We study the limits of communication systems in the finiteblocklength regime. In these cases, nonasymptotic bounds on the error probability provide a description of the system performance. In order to derive these bounds, it is common to use hypothesis testing concepts. Hypothesis testing was used by Shannon et al. in [1] in order to derive the sphere-packing exponent, and also by Blahut in [2]. More recently, hypothesis testing has been used by Nagaoka in order to derive strong converse bounds in classical-quantum channels [3], and by Hayashi [4,Sec. 4.6] in order to derive the converse part of the channel-coding theorem. Polyanskiy et al. [5,Th. 27] introduced a hypothesis testing finite-blocklength bound for classical channels. Matthews and Wehner [6] obtained finitelength bounds for general quantum channels. In classical channels, perfect and quasi-perfect codes were generalized beyond binary alphabets in [7]. These codes attain the metaconverse bound and so they are optimum. In quantum channels, quasi-perfects codes were introduced in [8] and the Bell codebook was provided as an example.
This work follows as a continuation of [8] and focuses on quasi-perfect codes for coherent states. Information is conveyed by the electromagnetic field generated by a laser. The received signal is represented by a coherent state that describes the photon statistics of the field. The coherent state was first introduced by Klauder (see [9]) and later on formally defined by Glauber (see [10]). The channel used to transmit coherent states is the Bosonic channel. This channel can be modeled using an infinite dimensional Hilbert space, but for practical purposes, it is necessary to consider an equivalent channel with a reduced dimension. We consider a truncation of the Bosonic channel as in [11], where the coherent states has instead a finite dimension (with the corresponding normalization).
The line of work in this article and the one in [8] are focused on solving problems related to state discrimination. Helstrom [23] and [24] developed the theory of quantum detection and quantum hypothesis testing. Our work studies the optimality of a codebook used to transmit classical information over a quantum channel rather than the optimality of the detector. The classical information is recovered directly by means of an optimum measurement operation. A different line of work in the literature considers error correction codes, which are codes that use redundancy in order to correct errors that may be caused by the channel or other effects. In quantum systems, Shor showed in [12] that errors can be corrected by encoding the state of the system in a quantum code and perform measurements on the redundant parts of the code to detect errors. Detected errors can be corrected by simply applying unitary transformations to the state. Later quantum stabilizer codes were introduced by Calderbank [13], [14] and Gottesman [15]. Error correction codes include, for example, surface codes (see [16], [17]) and LDPC quantum codes (see [18], [19], [20]). The results obtained in this work could also be applied in an error correction setting by considering a classical-quantum channel where the error correction procedure is included in the channel model. Here, we only focus on the state discrimination problem for the Bosonic channel.
The rest of this article is organized as follows. In Section II, we formalize the problems of binary and multiple hypothesis testing and establish a connection between them. We also define quasi-perfect codes and show that they attain the meta-converse bound. In Section III, we define a quasi-perfect code for the Bosonic channel based on phasemodulation and study its performance in terms of error probability. We also consider the inclusion of a depolarizing and an erasure channel, since they are symmetric and it is possible to show results for both of them. Finally, Section IV concludes this article.

A. NOTATION
Let D(H) denote the space of density operators acting on a Hilbert space H. In the general case, a quantum state is described by a density operator ρ ∈ D(H). Density operators are self-adjoint, positive semidefinite, and have unit trace. A measurement on a quantum system is a mapping from the state of the system ρ to a classical outcome m ∈ {1, . . . , M}. For self-adjoint operators A, B, the notation A ≥ B means that A − B is positive semidefinite. Similarly A ≤ B, A > B, and A < B means that A − B is negative semidefinite, positive definite, and negative definite, respectively.
For a self-adjoint operator A with spectral decomposition This corresponds to the projector associated to the positive eigenspace of A. We shall also use {A ≥ 0} i:λ i ≥0 E i and {A = 0} i:λ i =0 E i .

A. BINARY HYPOTHESIS TESTING
Consider a binary hypothesis test discriminating among two quantum states ρ 0 and ρ 1 acting on H. We define a test measurement {T,T } such that T andT are positive semidefinite, self-adjoint operators, and T +T = 1 1. We define the probability of false alarm 1|0 and the probability of miss-detection 0|1 as follows: We define α β (ρ 0 ρ 1 ) as the minimum probability of false alarm 1|0 among all tests with 0|1 ≤ β

B. MULTIPLE HYPOTHESIS TESTING
Consider a multiple hypothesis testing problem where we want to discriminate between M quantum states acting on H.
In particular, we consider that the quantum states τ 1 , . . . , τ M are associated with classical probabilities p 1 , . . . , p M , respectively. We define an M-ary test as a POVM P { 1 , 2 , . . . , M } satisfying i = 1 1 and m ≥ 0. This test will decide τ j when the true hypothesis is τ i with a probability of Tr τ i j . The average error probability is We are interested in the minimum average error probability among all test P min P (P ).
The following lemma states that a test is optimum under certain conditions.
Proof: The proof can be found in [8].

C. META-CONVERSE BOUND
We consider the channel coding problem of transmitting M equiprobable messages over a one-shot classical-quantum channel x → W x , with x ∈ X and W x ∈ D(H). A channel code is defined as a mapping from the message set For a source message m, the decoder receives the associated density operator W x m and must decide on the transmitted message. With some abuse of notation, for a fixed code, sometimes we shall write W m W x m . The minimum error probability for a code C is then given by

Lemma 3 (Classical-Quantum Meta-Converse Bound):
Let C be any codebook of cardinality M for a channel W x ∈ D(H). Define the following: Then where the maximization is over auxiliary states μ ∈ D(H), and the minimization is over (classical) input distributions P. Proof: The proof can be found in [8].

D. PERFECT AND QUASI-PERFECT CODES
For any density operator μ in D(H), and t ∈ R we define Definition 1: We say that a classical-quantum channel for every x ∈ X , whereW ∈ D(H) does not depend on x and U x is a unitary linear operator acting on H and parametrized by x. Equivalently, define For any symmetric channel and μ ∈ U W , F x (t, μ), and G x (t, μ) do not depend on x as shown in [8].
We denote byt the smallest value of t such that E • x (t, μ) x∈C are orthogonal to each other for a certain code C. DefineĒ and alsoĒ We define the orthogonal basis {Ē(i)} associated to the eigenspace of W −tμ ≥ 0 such that where I • denotes the set of basis indexes associated to the strictly positive eigenvalues. We also writē where I • denote the set of basis indexes associated to the zero eigenvalues. Definition 2 (See [8]): A code C is perfect for a classicalquantum channel x → W x , if there exist a scalar t and a state μ ∈ D(H) such that the projectors E x (t, μ) x∈C are orthogonal to each other and x∈C E x (t, μ) = 1 1. More generally, a code is quasi-perfect if there exist t and μ ∈ D(H) such that the projectors E • x (t, μ) x∈C are orthogonal to each other, and for where ρ A is the input quantum state and the Isometric channel I A→B (ρ A ) = I A→B ρ A I † A→B is defined using the isometry  μ) is the eigenspace of I A→B (ρ A ) −tμ = 0 and |e e| B does not depend on x (i.e., all codewords share the same eigenvector |e e| B ). The input state has no effect on the term |e e| B , so for this case, we introduce the following generalized definition of quasi-perfect codes, which can accommodate the different input and output dimensions of the erasure channel.
Definition 3: A code C is generalized quasi-perfect if there exists t and μ ∈ D(H) such that the projectors E • x (t, μ) x∈C are orthogonal to each other, fulfilling μ) − |e e| B and c ∈ R, c > 0 is a normalizing constant that depends on the code C.
The following lemma shows that generalized quasi-perfect codes are optimum among all codes of the same dimension of the channel Hilbert space and cardinality.

Lemma 5 (Generalized Quasi-Perfect Codes Attain the Meta-Converse Bound):
Let the channel x → W x be symmetric and let C be generalized quasi-perfect with parameters t and μ ∈ U W . Then, for M = |C| where |I • | is the cardinality of the set I • of the input state and d •A is the dimension of I •A . Following similar steps as in [8, the proof of Theorem 3], it is possible to show that the optimality conditions from Lemma 1 are satisfied. We have and so where we used that ( (T ) − 1 M W m I A→B I † A→B ) m = 0 as shown in [8].

III. QUASI-PERFECT AND GENERALIZED QUASI-PERFECT CODES FOR THE BOSONIC CLASSICAL-QUANTUM CHANNEL
This section introduces quasi-perfect codes for the Bosonic channel, where coherent states are transmitted. A coherent state is represented by the following expression: where α is a complex amplitude, |α| 2 is the average number of photons associated with state |α and |n is the Fock, or photon number, state. The Bosonic classical-quantum channel is the mapping of a classical variable x to the quantum state W x defined as W x = |α x α x |, x → W x , with α x ae iθ x , θ x ∈ [0, 2π ) and a = |α x |.
Here, we consider a finite-dimensional quantum receiver implementing collective measurements in the form of a POVM defined in a Hilbert space of dimension N, H N , i.e., restricting the coherent state measurements to Fock states |n for n ∈ {0, . . . , N − 1}.

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Consider the Nth-order approximation |α N ∈ H N to coherent state |α Similar to the Bosonic classical-quantum channel, the truncated Bosonic classical-quantum channel is the mapping of a classical variable x to the quantum state W x defined as , and a = |α x |. In order to assess the closeness of the approximation to the original state we will use the concept of pure-state fidelity, which can be easily computed as a function of the order of approximation N n! . Note that lim N→∞ N = 0 so that if N is big enough, the Fidelity between the original and the approximated state is close to one. The fidelity and the trace distance |||α α| − |α α| N || 1 for pure states can be related as follows: Now, since lim N→∞ N = 0, for sufficiently big values of N, we know that any measurement using an arbitrary operator on the approximated state |α N succeeds with high probability, it also does so if applied to the original state since We consider the channel coding problem of transmitting M equiprobable messages. Messages are modeled by the classical random variable x, over a one-shot approximated coherent quantum channel We consider the properties of this channel codes for the particular case of defining the dimension of the collective measurement's Hilbert space N equal to the cardinality of the message set M, i.e., N = M.
Let ρ A be the density matrix as observed by the Mdimensional decoder. For M ≥ 2 it follows that where Notice that |n is the photon number state, so the density matrix in (51) is in the photon basis. Let us define the state density when message x m is transmitted as W m , i.e., W m |α x m α x m | A . Also, in order to simplify notation, let α m α x m and δ m δ x m . We consider the decoder which proves that (P ) − 1 M W m ≥ 0. We check the symmetry of the one-shot coherent channel x → |α x α x | A , with |α x | = a. In our channel we have α y = α x e i(θ y −θ x ) , i.e., |α y = |α x , where is a diagonal matrix which elements incorporate the corresponding phase shifts. Note that H = I. Since (26) holds, we conclude that the channel is symmetric.
Next we show that C is quasi-perfect for μ = μ 0 and t = t 0 , which will be defined as follows. Recall that a code is quasi-perfect with respect to μ 0 and t 0 if it satisfies that {E • x (t 0 , μ 0 )} for x ∈ C are orthogonal to each other and also that x∈C E • x (t, μ) = cI • . From the optimality condition of the decoder (56), we can see that x (t 0 , μ 0 ) is the null eigenspace of 1 M W x − (P ). This also implies that E • x (t, μ 0 ) = 0, hence E • x (t 0 , μ) for x ∈ C are orthogonal to each other. Also, d • = n = M because d • = 0, which implies that c = 1.
We obtain the eigenvector associated to the largest eigenvalue of |α x α x | − tμ 0 . To this end, we consider an arbitrary unitnorm vector |v . The largest eigenvalue of |α m α m | − tμ 0 is given by We can observe that t = t 0 corresponds to the case for which the maximum eigenvalue of |α m α m | − tμ 0 is equal to zero, which implies |α x α x |v = t 0 μ 0 |v .
Note that (60) implies Multiplying by α x |μ −1 0 at both sides of (60) we obtain So, we conclude that the code is quasi-perfect. We can also easily find the error probability of this code, which is the minimum error probability among all possible codes of cardinality M for this channel. Using the optimal decoder P, we obtain that the probability of error is