Quantum coding with finite resources

The quantum capacity of a memoryless channel determines the maximal rate at which we can communicate reliably over asymptotically many uses of the channel. Here we illustrate that this asymptotic characterization is insufficient in practical scenarios where decoherence severely limits our ability to manipulate large quantum systems in the encoder and decoder. In practical settings, we should instead focus on the optimal trade-off between three parameters: the rate of the code, the size of the quantum devices at the encoder and decoder, and the fidelity of the transmission. We find approximate and exact characterizations of this trade-off for various channels of interest, including dephasing, depolarizing and erasure channels. In each case, the trade-off is parameterized by the capacity and a second channel parameter, the quantum channel dispersion. In the process, we develop several bounds that are valid for general quantum channels and can be computed for small instances.

: Second order approximation for a qubit dephasing channel with entanglement assistance. Second order approximation of the achievable region (with and without entanglement assistance) for a qubit dephasing channel with ε = 1% and γ = 0.1. The achievable region is enlarged in the presence of entanglement, as discussed in Supplementary Note 1.

Supplementary Note 1: Entanglement-Assisted Communication
Finally, we would like to note that the recent results about finite resource entanglement assisted classical communication [1] can be transformed to entanglement assisted quantum communication (and this then also gives outer bounds on the achievable rate region for unassisted codes). This is accomplished by using the equivalence results in [16,App. B] which make use of quantum teleportation and superdense coding. In particular, one finds that for covariant channels N (which includes the qubit dephasing channel and the erasure channel) the boundary of the entanglement assisted achievable regionR E (n; ε) satisfieŝ with the mutual information of the channel, I(N ), and its variance, V ε (N ), as defined in [1]. As an example, we mention again the qubit dephasing channel Z γ for whicĥ where h(γ) denotes the binary entropy and v(γ) the corresponding variance as defined in Theorem 1 in the main text. The entanglement-assisted achievable region is compared with the unassisted achievable region in Supplementary Figure 1.

Supplementary Note 2: Analysis of General Channels
In this note we detail the derivations of Theorems 4 and 5 in the main document.
Outer Bounds on the Achievable Rate Region Let us first give a self-contained proof of our outer bound for one use of the channel, using the notation introduced in the main document. Let us restate [2,Lemma 2]: For every σ AB ∈ PPT * (A : B), we have for all maximally entangled states φ AB of local dimension |M |. Lemma 1. Let N A→B be a quantum channel. Then, for any fixed ε ∈ (0, 1), the achievable region with cpp-assistance satisfiesR Proof of Lemma 1. First, observe that the encoding operation E M →AQ can be chosen to be an isometry without loss of generality, because we may include any extension systems needed for the Stinespring dilation into Q. Then we may express the entanglement fidelity as follows Here † denotes the adjoint map with regards to the Hilbert-Schmidt inner product.
Since E is an isometry, the state ρ AĀQQ = E M →AQ ⊗ E M →ĀQ (φ M M ) is pure, and therefore there exists an isometry W A →ĀQQ such that |ρ AĀQQ = W A →ĀQQ |ψ ρ AA . Thus, Now consider the entanglement fidelity of any σ A B ∈ PPT * (A : B) instead of N A→B (ψ ρ AA ). By (3) we have as the operations on σ A B are all PPT-preserving. We may write this bound in terms of the hypothesis-testing relative entropy, because is a feasible test to discriminate between N A→B (ψ ρ A A ) and σ A B . That is, the former since D is completely-positive and trace-preserving and E and W are isometries, the latter by assumption that F ≥ 1 − ε. From (8) we then obtainR Since the bound holds for all σ A B ∈ PPT * (A : B), we may take the minimum over this set. The resulting bound depends on the precise channel input ρ A ∈ S(A) used by the code, but we can remove the dependence by taking the maximum over all possible inputs. The result is (4).
Relaxation. We may relax the bound from Lemma 1 by restricting the form of the possible states σ AB in the definition of the hypothesis testing Rains relative entropy I ε R (N ). For this purpose, a quantum channel N A→B is called PPT preserving if a PPT state input necessarily results in a PPT state output. It turns out that PPT-preserving channels output PPT states for any input, since they have PPT Choi states [3] (see the discussion after Eq. 4.13). Channels with PPT Choi states were also called PPT-binding in [4]. For our purposes here, we may consider trace non-increasing PPT-preserving channels, and write M A→B ∈ PPT when the Choi state M AB of M A→B satisfies T A (M AB ) ≥ 0 and tr B M AB ≤ 1 A . Corollary 2. Let N A→B be a quantum channel. We define the function Proof. Suppose that σ AB = (I A ⊗ M A →B )(ψ ρ AA ) for some trace non-increasing PPT-preserving (PPT-binding) channel M A→B . Any such σ AB is in PPT * (A : B), therefore we may use this choice in the right-hand side of (4), which yieldŝ Now we may define Λ AB = (T A (ρ A )) 1/2 Λ AB (T A (ρ A )) 1/2 and find Finally for fixed channel input ρ A , we can reverse the order of the inner optimizations in (14) by von Neumann's minimax theorem, since the objective function is linear and the sets are both convex and compact. This concludes the proof of (12).
Furthermore, f (N , ε) can be expressed as a semidefinite optimization program that satisfies strong duality. This is discussed in Supplementary Note 4.
Symmetries. Applied to the channel N ⊗n we immediately get for any fixed ε ∈ (0, 1), This bound is generally hard to evaluate even for moderately large n. In the following we show that symmetries of the channel can further simplify the outer bounds. Suppose G is a group represented by unitary operators U g on A and V g on B. A quantum channel N A→B is covariant with respect to G when Alternatively we can also write this as an invariance of the channel Now the main workhorse to simplify our outer bounds for channels with symmetries is [5,Prop. 2], which states that we may restrict the optimization in Lemma 1 to covariant input states. Due to the form of the hypothesis testing Rains relative entropy, we may then also choose group invariant states σ and test operators Λ to obtain the tightest bound. Note that the semidefinite optimization outer bound in Corollary 2 inherits these symmetry simplifications. For general tensor product channels, which are invariant to permutation of the inputs and outputs, this allows us to restrict attention to pure states that are permutation invariant. Moreover, if the channel is covariant, then the channel input state can be chosen to be maximally mixed.
Asymptotics. Now let N A→B be a covariant quantum channel and φ AA a maximally entangled state. Then, we bound where we voluntarily restricted the minimization to product states σ ⊗n AB in PPT * (A : B). Moreover, since these states have tensor product structure, the outer bound can be expanded using [6,7] This leads to the following proposition.
Proposition 3. Let N ≡ N A→B be a quantum channel and let φ AA be maximally entangled. We define the channel Rains information of N as where we let Π ⊂ PPT * (A : B) be the set of states that achieve the minimum. The variance of the channel Rains information is If N is covariant, then for any fixed ε ∈ (0, 1), the achievable region with cpp-assistance satisfieŝ Since we are here interested in outer bounds, we are also free to chose a potentially suboptimal σ AB ∈ PPT * (A : B) to further relax this bound. As we see in the discussion of the qubit dephasing channel and the erasure channel with classical post-processing assistance, the bound from Proposition 3 is tight up to the second order asymptotically.

Inner Bounds on the Achievable Rate Region
We start with the following result [8,Prop. 20]: Lemma 4. Let N A→B be a quantum channel with complementary channel N c A→E . Then {R, 1, ε} is achievable if, for any η ∈ (0, ε] and any state ρ A ∈ S(A), we have where . Note that the authors of [8] use the purified distance as their figure of merit whereas we use the fidelity criterion. This accounts for the square root in the smoothing parameter of the conditional min-entropy. They also state their result for the special case n = 1, but this can be generalized to arbitrary n ∈ N if we simply consider N ⊗n A→B as a single channel. This leads immediately to the following inner bound on the achievable region.
Corollary 5. Using the notation of The problem with this bound is that it is generally hard to evaluate, even for moderately large values of n. Hence we are interested to further simplify the expression on the right-hand side in this regime. To do so, we choose η = 1/ √ n and use input states of the form ρ ⊗n A . This yields the following relaxation, which holds if n > 1 ε : Here we introduced ε n = √ ε− 1 √ n and ω AE as in Lemma 4. Using standard second order expansion methods [7], we can give an asymptotic expansion ofR inner (n; ε) in (25) as follows. Proposition 6. Let N ≡ N A→B be a quantum channel. We define its coherent information as and let Π ⊂ S(A) be the set of states that achieve the maximum. Define Then, for any fixed ε ∈ (0, 1), the achievable region satisfieŝ Proof. We analyze the expression in (25) using the following asymptotic expansion of the smooth conditional min-entropy [7], This yields that for any ρ A ∈ S(A), we havê and then by duality of the conditional entropy we find −I(A E) ω = I(A B) ω . Furthermore, it is easy to verify that V (A E) ω = V (A B) ω (see, e.g., [9]). We conclude the proof by choosing an optimal state ρ A ∈ Π depending on the sign of Φ −1 (ε).

Supplementary Note 3: Discussion of Channel Examples
In this note we detail the derivations of Theorems 1, 2 and 3 in the main document. This specialization builds on the results that are derived in Supplementary Note 2 for general channels.

Qubit Dephasing Channel
We determine the third order asymptotic performance of the qubit dephasing channel. We do this by directly obtaining the finite block length behavior of the qubit dephasing channel from that of the classical binary symmetric channel (BSC). First, consider the converse, particularly that of (18), applied to the channel Z ⊗n γ . Using the Bell states φ + Now, in (18) we are free to pick any PPT * state to obtain a bound. Pick σ AB = 1 (The choice of σ AB is equivalent to using the convex relaxation of the bound, Corollary 2, and choosing M = Z 1/2 in (11).) To connect to the finite block length bounds of the BSC, consider measuring both A and B in the Pauli x basis, and let X and Y be the output random variables for A and B, respectively. For the state ω AB , this results in the distribution P XY in which P X is uniformly-distributed and P [Y = X] = 1 − γ. For σ AB , the distribution is of product form P X Q Y with Q Y also uniform. Moreover, the original quantum states can be reconstructed from the classical random variables X and Y by the map which outputs φ + AB when X = Y and φ − AB otherwise. Therefore, the bound becomeŝ which is precisely the bound obtained for the BSC (see [10,Thm. 26]) which is equivalent to the classical sphere-packing bound [11,Eq. 5.8.19]. This establishes the desired upper bound.
For the achievability, we may directly employ linear codes for the classical BSC to the qubit dephasing channel. Specifically, any linear {R, n, ε} code for the BSC (which recovers the input with probability at least 1 − ε, averaged over a uniform choice of inputs), can be converted into an {R, n, ε} Calderbank-Shor-Steane (CSS) code for entanglement transmission over the dephasing channel. This is possible since, for a linear code, the action of the channel is a mapping among the orthogonal Bell states, which is essentially a classical action.
To formalize the connection, we begin with the description of the classical linear code by its (n − log |M |) × n parity check matrix H. Each row r j ∈ {0, 1} n defines a parity function and the codewords c k of the code must satisfy c k ·r j = 0 for all j. The associated CSS code can be defined as the simultaneous +1 eigenspace of the "stabilizer" operators X r j , where X r j = X r j,1 ⊗ · · · ⊗ X r j,n . (Generically, a CSS code has stabilizers of both X-type, as here, and of Z-type, i.e. composed of products of Pauli Z operators.) Crucially, the action of the channel is to apply an operator of the form Z u , with u ∈ {0, 1} n , according to the distribution P U . At the output, the receiver can simultaneously determine the eigenvalues of all the of the stabilizer operators. This information is precisely equivalent to determining the value of the parity checks of the classical linear code, called the syndrome s. Given the syndrome, the decoder of the classical code determines a guess as to the input codeword, which is equivalent to a guess u (s) of the actual channel error.
We may also utilize this algorithm (whatever its precise details) in the quantum case, and attempt to correct the error by applying Z u (s) . When u (s) is the true error pattern, the quantum state is properly recovered, and the entanglement fidelity is unity. On the other hand, if u (s) is incorrect, then in the worst case the action Z u (s)+u is a logical operation on the code subspace, which results in a state orthogonal to the desired entangled state. Therefore, the error probability of the classical code translates directly into the entanglement fidelity of the quantum code. Thus, we may apply finite-block length bounds for linear codes, particularly the bound in [12] (see also [10,Eq. 65]). This establishes the lower bound.

Qubit Erasure Channel
For the qubit erasure channel it is not too difficult to directly derive an outer bound and an explicit coding scheme leading to an inner bound, which precisely match for all n. Let us begin with the outer bound. Again we may relate the finite block length performance to a classical coding problem, namely the classical binary erasure channel (BEC). The argument for the outer bound proceeds very similarly to the dephasing example. The optimal channel input state corresponds to the maximally entangled state φ AA , and the state produced by the channel is now where π A denotes the maximally-mixed state. Measurement of A in the Pauli x basis and B in the basis {|+ , |− , |e } produces the distribution P XY with P X uniform and Y = X with probability 1 − β and Y = e with probability β. The original state can be reconstructed using the map which sends (X, Y ) to φ + AB when X = Y , φ − AB when X = Y = e, and to π A ⊗ |e e| B when Y = e otherwise. As before, we make a specific choice of PPT * state in (18), but this time not a product state across channel uses. Instead, consider the classical distribution P ×n X × Q Y n given in [10,Eq. 168]. The Q Y n distribution has the property that any two y n with the same number of erasure symbols e have the same probability, i.e. there is no dependence on the number of 0s versus 1s. The aforementioned map takes the distribution to a quantum state which is diagonal in the standard bases {|0 , |1 } for A and {|0 , |1 , |e } for B, and is therefore a PPT state. This can be seen as follows. Consider a fixed position j in a given a pair (x n , y n ). If y j = e, the state of the jth pair of systems AB is manifestly diagonal in the standard basis. On the other hand, if y j = e, then the state is mapped to either φ + AB or φ − AB depending on the value of x j . But the sequence in which y j takes the other value has identical probability, meaning the two Bell states occur with equal probability, making the AB state diagonal. Since we may map ω ⊗n AB and σ A n B n to the associated classical distributions and back, the following converse holds for the qubit erasure channel, By design in the choice of σ AB , this is precisely the bound for the BEC reported in [10,Thm. 38], as discussed in more detail in [13]. (This also corresponds to using Corollary 2 with M the channel which ignores its input and prepares σ B n at the output.) Next, we construct an explicit coding scheme, involving classical post-processing including communication from the receiver to the sender, which matches the outer bound exactly. The strategy of the coding scheme is to generate maximally entangled qubit states using the quantum channels and then use the successfully transmitted (i.e. not erased) maximally entangled qubit states to distill a an entangled state of local dimension |M |, as required. Note that the number |M | is fixed at the outset of the code, i.e. the entanglement transmission scheme must deliver a maximally entangled state with local dimension |M |, possibly at the expense of low fidelity, rather than outputting a variable number of certifiably high fidelity entangled pairs.
The encoder prepares n maximally entangled qubit states |φ and sends one half of each over the channel. The other halves, together with the untouched system M , are stored in the memory register Q. The decoder now works as follows. The receiver determines which qubits have not been erased and informs the sender of their locations. Let L be the random variable indicating the total number of erasures and note that L follows a binomial distribution with parameters n and β. Let us also fix k = log |M | and consider the following two cases: 1. If L = l ≤ n − k the decoder can extract a maximally entangled state with unit fidelity. To do so, it selects k perfectly transmitted entangled qubits at the sender and receiver. Let us assume (without loss of generality) that these are in a state |φ + ⊗k = 1 |ii . The receiver then prepares a maximally entangled state of local dimension |M | by measuring the k qubits with the projective measure The outcome, a subset S of cardinality |M |, is transmitted to the sender so that both sender and receiver now share a maximally entangled state on the subspace determined by S.

2.
On the other hand, if L = l > n − k sender and receiver simply select the successfully transmitted qubits and embed them in a space of local dimension |M |. The fidelity with the target state |φ = 1 √ |M | |M | i=1 |ii is given by To complete the decoding operation, the sender and receiver perform quantum teleportation to teleport M to the receiver, using the maximally entangled state prepared above as a resource. The fidelity of the state prepared above with the target state φ M M is then just the expected fidelity over L, which evaluates to This is exactly the expression reported in the aforementioned outer bound in [10,Thm. 38], meaning the inner bound coincides with the outer bound when we allow classical post-processing and communication from the receiver to the sender.

Qubit Depolarizing Channel
The qubit depolarizing channel is covariant since it is a qubit Pauli channel. Using the Bell Now choosing σ AB = 1 2 φ + AB + 1 6 (φ − AB + ψ + AB + ψ − AB ) in (18) gives the outer bound As in the case of the qubit dephasing channel, we can convert the hypothesis test between ω AB and σ AB into a test between classical distributions, in fact precisely those distributions which were used in the dephasing example. This follows by considering the map which generates φ + AB when X = Y and otherwise randomly generates one of the other Bell states when X = Y . Therefore, we obtain the same outer bound for the qubit depolarization channel as for the qubit dephasing channel.

Supplementary Note 4: Semidefinite Optimization
Here we describe how to formulate the outer bound from Theorem 4 in the main text as a semidefinite optimization program that satisfies strong duality.
A semidefinite program (SDP) is simply an optimization of a linear function of a matrix or operator over a feasible set of inputs defined by positive semidefinite constraints. We give only the bare essentials here, for more detail see [14,15]. The maximization form of an SDP is defined by a Hermiticity-preserving superoperator E A→B taking L(A) to L(B), a constraint operator C ∈ L(B), and an operator K ∈ L(A) which defines the objective function. Here L(A) denotes the set of linear operators on A. The SDP is the following optimization, which we will also refer to as the primal form, α = supremum tr [KX] subject to E(X) ≤ C X ≥ 0 .
When the feasible set is empty, i.e. no X satisfy the constraints, we set α = −∞. The dual form arises as the optimal upper bound to the primal form, and takes the form Again, when the set of feasible Y is empty, β = ∞. Weak duality is the statement that α ≤ β, that indeed the dual form gives upper bounds to the primal (or that the primal lower bounds the dual). Strong duality is the statement that the optimal upper bound equals the value of the primal problem, α = β. This state of affairs often holds in problems of interest, and can be established by either of the following Slater conditions. In the first, called strict primal feasibility, strong duality holds if β is finite and there exists an X > 0 such that E(X) < C. Contrariwise, under strict dual feasibility strong duality holds when α is finite and there exists a Y > 0 such that E † (Y ) > K. For strongly dual SDPs we also have the so-called complementary slackness conditions E † (Y )X = KX and E(X)Y = CY that relate the primal and dual optimizers.
Proposition 7. With the notation from Corollary 2, the outer bound f (N , ε) can be written as or, equivalently, Here PPT(A : B) denotes the subset of S(AB) consisting of positive operators with positive partial transpose.
Proof. The proof is straightforward: we simply use the dual of the inner optimization in (11) to obtain the minimization problem (44). Then we use Slater's condition to show that strong duality holds and obtain (45).
Consider the function Combining this with the outer optimization in (11) gives the minimization program (44). The equality statement is precisely strong duality of the primal and dual forms of the inner optimization. To construct the maximization program, we simply dualize the minimization program. In particular, f (N , ε) is a dual-form semidefinite program in the variable Y = (φ A , Λ AB , Γ AB , ξ A ) with C = (0, 0, 0, 1 A ), K = (1 − ε, −1, 0, 0), and Choosing primal variables X = (m, n, R AB , M AB ) leads to the maximization in (45). Equality again follows from Slater's condition: f is finite (in particular the bound on f 0 used above), while a feasible choice of dual variables is given by M AB = R AB = 1 2|B| 1 AB , n = 1, and m = 1 2|A||B| . The choice of m ensures the first constraint holds strictly, since any Choi operator of a trace-preserving map satisfies N AB ∞ = |A| (largest singular value).
No discussion of strong duality of semidefinite programs is complete until the complementary slackness conditions have been formulated. Often, these give considerable insight into the form and properties of the optimizing variables.
Supplementary References