Capacity estimation and verification of quantum channels with arbitrarily correlated errors

The central figure of merit for quantum memories and quantum communication devices is their capacity to store and transmit quantum information. Here, we present a protocol that estimates a lower bound on a channel’s quantum capacity, even when there are arbitrarily correlated errors. One application of these protocols is to test the performance of quantum repeaters for transmitting quantum information. Our protocol is easy to implement and comes in two versions. The first estimates the one-shot quantum capacity by preparing and measuring in two different bases, where all involved qubits are used as test qubits. The second verifies on-the-fly that a channel’s one-shot quantum capacity exceeds a minimal tolerated value while storing or communicating data. We discuss the performance using simple examples, such as the dephasing channel for which our method is asymptotically optimal. Finally, we apply our method to a superconducting qubit in experiment.

Next, we make some general conventions. Conventions : Throughout this document, we make use of the following conventions: • log denotes the binary logarithm (base 2) and ln denotes the natural logarithm (base e).
• Quantum systems are assumed to be finite-dimensional, and the symbol H always denotes a finite-dimensional complex inner product space.
• A single subscript of H refers to the system associated with the space (for example, H A is the space of system A).
• We use multiple subscripts of H to refer to a space of a joint system (for example, H AB = H A ⊗ H B ).
The formal definition of a quantum channel goes as follows. Definition 2 (Quantum channel): Let A and B be quantum systems.
• The identity channel on A, denoted by I A , is the linear map • A quantum channel from A to B is a linear map which is trace-preserving, i.e. tr(Λ(ρ A )) = tr(ρ A ) ∀ ρ A ∈ End(H A ) , and which is completely positive. That is, for any quantum system E of any dimension d E ∈ N + , the map Λ ⊗ I E is a positive map, Such a map is called a trace-preserving completely positive map (abbreviated as TPCPM). (According to this definition, the terms "channel" and "TPCPM" are equivalent. In practice, the term channel is preferred when speaking of the evolution of a system in a physical sense, while the term TPCPM refers to the map as a mathematical object. However, this distinction is often not very strict.) In order to define the the one-shot quantum capacity of a channel and the smooth entropies of quantum channels below, we need to make use of some distance measures. Definition 3 (Distance measures): On the above operator spaces, we define the following distance measures: • The trace norm on End(H) is defined as L 1 = tr √ L † L .
• The generalized fidelity [14] on S ≤ (H) is defined as F (ρ, σ) : where The inner maximization ranges over all channels D : End(H B ) → End(H A ) (decoding channels).
• The one-shot capacity of entanglement transmission Q ε ent (Λ) of the channel is defined as where The maximization over D is as above, and the state is a maximally entangled state on the subsystem H M and a copy H M of it.
Although these two capacities are distinct, we will see below that they are comparable in the sense that they bound each other (see inequality Supplementary Equation 13 below). It is important to note that in Definition 4, we follow the definitions in [2] and [13] by using the fidelity as the figure of merit. Other sources, such as [5], define the one-shot capacities with the trace distance as the figure of merit. This will have consequences in Section 2, when a bound on Q ε ent (Λ) is converted from one definition to another (see inequality Supplementary Equation 20 below). Now we define the (smooth) min-and max-entropy of a bipartite quantum state. Definition 5 (Min-and max-entropy): Let ρ AB ∈ S ≤ (H AB ) be a subnormalized bipartite state.
• The min-entropy of A conditioned on B for the state ρ AB is defined [9] as • The max-entropy of A given B for the state ρ AB is defined [4] as Definition 6 (Smooth min-and max-entropy): Let ρ AB ∈ S ≤ (H AB ) be a bipartite state and let ε ≥ 0.
• The ε-smooth max-entropy of A conditioned on B is defined as • The ε-smooth min-entropy of A conditioned on B is defined as For states that are defined on more systems than labeled in the entropy, the entropy is evaluated for the according reduced state. For example, given a state ρ ABE ∈ H ABE , the smooth min-entropy H ε min (A|E) ρ is evaluated for ρ AE = tr B (ρ ABE ).
To avoid confusion with other sources that define the smooth min-and max-entropies, it is important to note two things.
• Firstly, the max-entropy, as we defined it in Definition 5, coincides with the Rényi entropy of order 1/2, whereas in some older sources, it was defined as the Rényi entropy of order 0 [9].
• Secondly, the smooth entropies, as we defined them in Definition 6, measure the distance in the purified distance, whereas in some older sources, it was defined with respect to the trace distance [9].
There are several reasons for making the definitions as we use them here. One important reason is that this way, the smooth min-and max-entropies satisfy a duality relation [14] that we will exploit in Section 3 (see Lemma 9).
Supplementary Note 2: Background: Proof of the min-entropy bound on the one-shot quantum capacity In supplementary note 2, we explain the details of the min-entropy bound on the one-shot quantum capacity (inequality (17) of the main article) and show how it is derived. This is not a new result, but an application of results that are well-established in quantum information science, which we provide here for convenience of the reader.
As mentioned in the main article, as the first step in the derivation of inequality (17), we note that the one-shot quantum capacity Q ε (Λ) of a quantum channel Λ can be lower-bounded by the one-shot capacity of entanglement transmission Q ε ent (Λ). More precisely, Barnum, Knill and Nielsen [1] have shown that (here we use the form presented in [2]) In particular, In the next step, we will bound Q ε ent (Λ). Before we do that, it is helpful to extend our picture with the Stinespring dilation of the channel and a purification of the input state, as shown in Supplementary Figures 1  and 2. Readers who are already familiar with these concepts may skip this part and continue reading below Supplementary Figure 2.
Recapitulate the situation that we consider: we are given a quantum channel Λ that takes a quantum system on Alice's side as its input and outputs another quantum system on Bob's side. It is helpful to give these input and output systems their own labels. We denote the input system on Alice's side by A and the output system on Bob's side by B (the reason for choosing A instead of A will become clear below). The situation is depicted in Supplementary Figure 1 Isometries have the important property that they map pure input states to pure output states. This will be important below, where we purify the input state ρ A (see Supplementary Figure 2).
For our purposes, it is useful to extend this picture. Mathematically speaking, a quantum channel is a tracepreserving completely positive map that maps density operators ρ A to density operators ρ B , The Stinespring dilation theorem [11] states that for every such completely positive map Λ, there is a system E of dimension d E ≤ d 2 A and a linear isometry The extended picture is shown in Supplementary Figure 1 (right). The map V Λ is an isometric extension or Stinespring dilation of the channel Λ, and is a standard tool in quantum information theory [18]. We extend this picture further using another standard tool in quantum information. The input state ρ A of the channel may not be a pure state but a mixed state. This is inconvenient, as many useful mathematical statements require the involved states to be pure. However, we can work around this by considering a purification of ρ A , that is, a system A of dimension d A ≤ d A and a pure state ρ AA such that tr A (ρ AA ) = ρ A . Every state has such a purification, but it is not unique [6]. Here, for every state ρ A , we consider a purification with d A = d A , which is called the canonical purification ψ ρ AA , and is given by Here, |Φ AA is the maximally entangled basis with respect to some bases {|i A } i and {|i A } i for H A and H A , respectively (their choice is irrelevant for what we consider), By extending system A to system AA in this way, we arrive at the overall picture shown in Supplementary  Figure 2. After the channel Λ acted on system A , we not only consider the output system B but the tripartite system ABE, which is in a state ρ ABE . Since isometries map pure input states to pure output states, is is a pure state. This is the reason for this extension. It will allow us to apply the duality relation in Section 3 (see Lemma 9). Figure 2: The fully purified diagram for a quantum channel. A quantum channel taking input A and mapping it to output B can be purified. By taking the environment E suitably large, the overall state is always pure: Since the input state ψ ρ AA is pure and I A ⊗ V Λ is an isometry, the output state ρ ABE is pure. Now we are ready to proceed with the next step in the derivation of inequality (17). It turns out that the one-shot capacity of entanglement transmission Q ε ent (Λ) of the channel Λ can be bounded by functions of the state ρ ABE that we described above. Buscemi and Datta [2] have shown that a lower bound on the one-shot capacity of entanglement transmission can be formulated in terms of a maximization of an entropic quantity. Subsequently, Morgan and Winter tightened this bound and translated it to an optimization of the smooth minentropy of the state ρ AE = tr B (ρ ABE ) [5]. Here we use this bound in the following form [13]: The square root in the smoothing parameter is a consequence of the fact that the bound Supplementary Equation 20 was derived through conversion from a bound where the figure of merit for entanglement transmission was the purified distance [5] instead of the fidelity [13]. Next, we drop the maximization over the state ρ A ∈ S(H A ) by choosing the maximally mixed state ρ A = I A /d A . This way, we arrive at another lower bound: This corresponds to the case where the input state ψ ρ AA in Supplementary Figure 2 Supplementary Note 3: Proof of the bound on the min-entropy In Section 2, we have seen that the one-shot quantum capacity Q ε (Λ) can be bounded in terms of the smooth min-entropy H ε min (A|E) of an appropriately defined state ρ AE . In this supplementary note, we prove that this min-entropy is bounded by the smooth max-entropies H ε max (X|B) and H ε max (Z|B) of measurement X and Z on A. More precisely, we will show: (see Theorem 13 below). Using this inequality, we will prove our bound on the one-shot quantum capacity in terms of the protocol parameters in Section 4.
In the following, we will cite some lemmas that we will need for the proof of the bound Supplementary Equation 23. The most important ones are: • an uncertainty relation for the smooth min-and max-entropies [16], • a chain rule theorem for the smooth max-entropy [17] and • a duality relation for the smooth min-and max-entropies [4,14].
We start with the uncertainty relation for the smooth min-and max-entropies.
Consider the states ρ XBE and ρ ZBE that arise from measuring all of the N qubits of system A with respect to X and Z, respectively, and storing the outcomes in a classical register X and Z, respectively, where and analogously for ρ ZBE . Then for ε ≥ 0, where The parameter q is the preparation quality. If X and Z are measurements with respect to mutually unbiased bases, then q = 1.
The chain rule that we will use is actually just one out of a series of chain rule inequalities proved in [17]. The particular form that we use here can be found in [12]. Lemma 8 (Chain rule for smooth max-entropy): Let ρ ABC ∈ S ≤ (H ABC ) be a tripartite state, let ε > 0, ε ≥ 0, ε ≥ 0. Then The duality relation between the smooth min-and max-entropy, or min-max duality, for short, relates the smooth min-entropy of a state to the max-entropy of a purification of the state. It was first proved for the unsmoothed min-and max-entropy König, Renner and Schaffner in [4]. The min-max duality for the smooth entropies is due to Tomamichel, Colbeck and Renner [14]. Lemma 9 (Min-max duality): Let ρ ABE ∈ S(H ABE ) be a pure tripartite state, let ε ≥ 0. Then Apart from these three main ingredients, we will also make use of three smaller lemmas. The first one states that the smooth min-and max-entropies are invariant under isometries [12]. Lemma 10 (Invariance under isometries): Let ρ AB ∈ S ≤ (H AB ) be a bipartite state, let ε ≥ 0. Then for all isometries V : In simple terms, the following lemma states that "forgetting" side information cannot decrease one's uncertainty. It is a special case of a more general theorem, called the data processing inequality [12]. We only state the more special case that we are interested in. Lemma 11 : Let ρ ABC ∈ S ≤ (H ABC ) be a tripartite state. Then Finally, the last lemma that we add to our list of tools shows how the (unsmoothed) max-entropy simplifies in the case where classical side information is given. Lemma 12 : Let ρ ACX ∈ S ≤ (H ACX ) be a state of the form Then [12] H max (A|CX) ρ = log Now we are ready to state the theorem formally and prove it. Theorem 13 : Let ρ ABE ∈ S(H ABE ) be a pure tripartite state where A and B are each an N -qubit system, let X = {X 0 , X 1 } and Z = {Z 0 , Z 1 } be non-trivial projective measurements on a qubit (that is, both elements are one-dimensional projectors). Consider the states ρ XBE and ρ ZBE that arise from measuring all of the N qubits of system A with respect to X and Z (as in Lemma 7). Then, for ε > 0 and ε , ε ≥ 0, it holds that where q is the preparation quality (as in Lemma 7).
Proof. Starting from ρ ABE , we construct a purification ρ AXX BE of ρ XBE . Further below, we will expand the smooth max-entropy of this state using the chain rule (Lemma 8). Reformulating the terms in that expansion will lead us to the desired result. Consider the product POVM elements We construct ρ AXX BE from ρ ABE by performing a coherent measurement on the A system with respect to the POVM formed by the elements Supplementary Equation 39. The outcome of this measurement is stored in two copies X and X of a classical register. For x ∈ {0, 1} N , let V x be the map We define the state ρ AXX BE := V (ρ ABE ), where The map V is an isometry that maps the pure state ρ ABE to the pure state ρ AXX BE . Thus, by virtue of Lemma 9, it holds that We will use Supplementary Equation 42 further below. Now we expand the max-entropy of ρ AXX BE using the chain rule, Lemma 8: The states ρ AB and ρ AXX B only differ by an isometry, so by Lemma 10, we have Now we expand the term H ε+2ε max (AX|B) using the chain rule:

Combining Supplementary Equation 46 with Supplementary Equation 47 allows us to infer
Reordering terms and using Lemma 11 and the uncertainty relation for the smooth min-and max-entropy (Lemma 7), we get Applying the duality relation (Lemma 9) to the left hand side of Supplementary Equation 51, we get We are left to show that H max (A|X) ρ is upper bounded by 0. We show, more precisely, that H max (A|X) ρ = 0. This goes as follows. where Now we can apply Lemma 12 to Supplementary Equation 57: By setting the system C in the lemma to a trivial system (H C C), we can deduce that where H max (A) ρ x A reduces to the unconditional form of the max-entropy, Since the Π X (x) are one-dimensional projectors, we have that and therefore H max (A|X) ρ = 0, as claimed. Thus, we have proved that which is what we wanted to show.

Supplementary Note 4: Comparison to min-entropy estimation in QKD
In Section 2, we have seen that the one-shot quantum capacity of a channel is bounded by the min-entropy. In the last section, we have seen how the smooth min-entropy H 3ε+ε +4ε min (A|E) can be bounded in terms of the max-entropy H ε max (X|B) and H ε max (Z|B) of the classical measurement outcomes X and Z on A. This puts us in a very good position, because we already know from quantum key distribution how to bound these max-entropies: a modern approach to quantum key distribution based on smooth entropies proves security by bounding exactly such a quantity.
In that approach, a QKD protocol is devised in which after sifting, Alice and Bob have n systems where they both measured in the X-basis and k systems where they both measured in the Z-basis. Then they exchange their outcomes in the Z-basis and determine the error rate λ. If the error rate λ does not exceed a specified error tolerance e z , then they conclude that [15,8,7] H ε max (Z|B) ρ ≤ nh(e z + µ(ε)) , where h denotes the binary entropy function and Here p pass is the probability that the correlation test (which checks whether λ ≤ e z ) is passed, where p = 1 − p pass is the parameter given in the theorem. The state ρ in inequality Supplementary Equation 64 is the state of the n qubits that have actually been measured in X. This means that from the error rate λ in one part of the qubits, one can infer a bound on H ε max (Z|B) for the other part of the qubits. This is illustrated in Supplementary Figure 3. In the QKD scenario we just described, the goal was to infer a bound on H ε max (Z|B) on only a part of the total system from the error rate λ on its complement. In our one-shot quantum capacity estimation and verification protocols, the situation is a bit different. It is easier to discuss the verification protocol first, because it is conceptually closer to the QKD protocol from which we adopt the estimation techniques.

Supplementary Note 5: Proof for the verification protocol
In the verification protocol, the qubits are divided into three subsets: one subset of test qubits that are measured in the X-basis, one subset of test qubits that are measured in the Z-basis, and the data qubits that are not measured at all (see the top part of Supplementary Figure 4). In the main article, we stated the protocol such that each of theses three subets has the same size N . Here, we consider the more general case where each of theses subsets might have a different size n, k, N ∈ N + , respectively, and later specialize the result to n = k = N . This also helps X-qubits Z-qubits H ε max (Z|B) λ n k Supplementary Figure 3: Bounding the max-entropy from an error rate on a different part. In the QKD protocol that we consider [15,8,7], the test qubits are measured in Z and the key qubits are measured in X. For the security of the protocol, H ε max (Z|B) needs to be bounded for the key qubits. This bound can be inferred from the error rate λ on the test qubits.
X-qubits data qubits Z-qubits trace out Z-qubits trace out X-qubits data qubits Z-qubits X-qubits data qubits Figure 4: Inference of the max-entropies in the verification protocol. Our verification protocol can be seen as the parallel execution of two QKD estimation protocols (see Supplementary Figure 3). Thus, if both γ ≤ e x and λ ≤ e z , we get two bounds of the form Supplementary Equation 64. us in the proof to keep track of which number we mean. Here, we denote the smoothing parameter by δ instead of ε (it will become clear below why it is useful to do so).
This situation may look more complicated than in the QKD scenario above. However, it turns out that our verification protocol can be seen as running the above QKD estimation two times in parallel (see the bottom part of Supplementary Figure 4). When we trace out the X-qubits, the remainder is in the same situation as in the QKD case as shown in Supplementary Figure 3, with the N data qubits taking the role of the qubits for which we bound the max-entropy H δ max (Z|B). Therefore, if we find that the error rate λ is below a tolerated error rate e z , we conclude that where and where p z pass is the probability that λ ≤ e z . Likewise, when we trace out the Z-qubits, the remainder looks like in the QKD case, with the X-basis taking the role of the Z-basis and with the data qubits taking the role of the qubits for which we bound the max-entropy H δ max (X|B). If we find that the error rate γ is below a tolerated error rate e x , then where δ = δ p x pass , µ x (δ) = N + n N n n + 1 n ln 1 δ (69) and where p x pass is the probability that γ ≤ e x . According to our verification protocol (see Protocol 2 in the main article), we are interested in the case where both γ ≤ e x and λ ≤ e z . In that case, we can conclude that At this point, we can connect this bound with the bound that we derived in Section 3 (see Theorem 13), which says that Combining inequalities Supplementary Equation 70 and Supplementary Equation 71, we get that This, in turn, can be connected with the min-entropy bound on the one-shot quantum capacity that we recapitulated in Section 2, which reads To connect inequalities Supplementary Equation 72 and Supplementary Equation 73, we make a variable transformation such that where 3δ + δ + 4δ = δ 3 + 1 p z Hence, we get and with µ z and µ x as in Supplementary Equation 67 and Supplementary Equation 69, respectively. This is the general version of our bound for the verification protocol.
To derive the form of the bound that we presented in the main article, we make two simplifications. Firstly, we consider the probability that both γ ≤ e x and λ ≤ e z , and denothe this joint probability by p pass . It bounds both probabilities from below, i.e. p pass ≤ p z pass and p pass ≤ p x pass . We set p = 1−p pass . Thus, the bound Supplementary Equation 76 also holds with Secondly, we set k = n = N , and get µ z (δ) = µ x (δ) = µ(δ) = 2(N + 1) Supplementary Note 6: Proof for the estimation protocol Our estimation protocol has one essential difference to the verification protocol. In the verification protocol, the two error rates γ and λ that are measured do not enter the bound directly. Instead, they are compared with some maximally tolerated error rates e x and e z , and the bound is a function of these values. In the estimation protocol, there are no preset maximally tolerated error rates. Alice and Bob simply measure two error rates e x and e z , and the bound that they use is a function of these measured error rates. This may seem different to the verification protocol, but using a simple argument, we can see that the situation in the estimation protocol is analogous to the situation in the verification protocol. (This will also explain why we use the same notation as for the preset values e x and e z ). Suppose that Alice and Bob run the estimation protocol (see Protocol 1 of the main article) up to the point where they determine the error rates. For now, let us denote these error rates by γ and λ. Imagine that at that point, Alice and Bob decide that they actually wanted to make a test in which they check whether both γ ≤ e x and λ ≤ e z holds. However, in contrast to the verification protocol, where e x and e z are preset values, Alice and Bob say that they simply want to make the test for values of e x and e z that are exactly equal to the the error rates that they have just measured, e x = γ and e z = λ. Obviously, if Alice and Bob design the test in this way, they will always pass the test. Moreover, the interpretation of the passing probability changes: it is no longer the probability that the measured error rates are below some preset values. Instead, it becomes the probability that in any run, the measured error rates stay below the rates that have been measured in this run (more precisely, it is a lower bound on it). This probability can be seen as a measure for the typicality of the protocol run, so we may denote it by p typical . In Protocol 1 in the main article, we use the complementary probability p = 1 − p typical . Using this argument, we can consider the state conditioned on passing a correlation test, just as in the case of the verification protocol.
For this reason, the same general form of the bound Supplementary Equation 76 with the same function δ as in Supplementary Equation 77 holds as for the verification protocol. However, in the estimation protocol, we use the measured error rates to infer the max-entropies H ε max (Z|B) and H ε max (X|B) for all qubits, rather than just on a part that has not been measured. Therefore, the functions µ x and µ z differ from the functions for the verification protocol. In order to derive the form of these functions, we again consider a slight generalization of the protocol that we considered in the main article. In the main article, we assumed that the N qubits that go through the channel are divided into N/2 qubits that are prepared and measured in the X-basis and N/2 qubits that are measured in the Z-basis. Here we assume that n qubits are measured in X and k qubits are measured in Z, with n + k = N . We denote the measured error rate in X by γ and the measured error rate in Z by λ. This is shown in Supplementary Figure 5. In order to bound H ε max (Z|B) from λ, we follow the original derivation of Supplementary Equation 65 as in reference [15], adjusted to the situation shown in Supplementary Figure 5. For a detailed derivation, see also [8,7]. In the estimation protocol, the measured error rates are used to bound the max-entropies on the total system of all qubits. This is in contrast to the verification protocol, where the measured error rates were used to bound the max-entropies on only a part of the total system. This is why we cannot simply use the function µ as in Supplementary Equation 65 but need to derive them for this particular situation.
We consider the Gedankenexperiment in which all the bits have been measured in the Z-basis. We denote random variable of the error rate λ in Z in the Z-bits by Λ = Λ z , the error rate in Z in the X-bits by Λ x and the total error rate in Z by Λ tot . Then it holds that nΛ x + kΛ z = (n + k)Λ tot .
The division of the qubits into X-qubits and Z-qubits is fully random. Therefore, the error number probabilities follow a hypergeometric distribution. This means that Serfling's bound [10] applies. Here, we use the particular