On the Probabilistic Quantum Error Correction

Probabilistic quantum error correction is an error-correcting procedure which uses postselection to determine if the encoded information was successfully restored. In this work, we analyze the probabilistic version of the error-correcting procedure for general noise. We generalize the Knill-Laflamme conditions for probabilistically correctable errors. We show that for some noise channels, the initial information has to be encoded into a mixed state to maximize the probability of successful error correction. Finally, the probabilistic error-correcting procedure offers an advantage over the deterministic procedure. Reducing the probability of successful error correction allows for correcting errors generated by a broader class of noise channels. Significantly, if the errors are caused by a unitary interaction with an auxiliary qubit system, we can probabilistically restore a qubit state by using only one additional physical qubit.

In this work, we study a particular QEC procedure called probabilistic quantum error correction (pQEC) [14][15][16].To outline how pQEC procedure works, let us present an example of classical probabilistic error correction.Consider the scenario, when the encoded data is harmed by a single bit error, that is with the probability p ∈ [0, 1] an arbitrary bit will be flipped.To secure a one bit of information, we use two physical bits.If we expect that p ≤ 2  3 , then we can encode 0 → 00 and 1 → 11.If we receive information 00 at the decoding stage, we are certain the encoded message was 0 (and 1 for 11).We dismiss the cases 01 and 10 as they do not give conclusive answers.Otherwise, if p > 2  3 it would be beneficial to use encoding 0 → 00 and 1 → 01 with the accepting states 10 and 11.It is worth mentioning, that to secure a one bit of information perfectly, it is necessary to use three physical bits, for example 0 → 000, 1 → 111.
Let us return to the quantum case.The heart of pQEC procedure is the probabilistic decoding operation [17,18].This operation uses a classical postselection to determine if the encoded information was successfully restored.The clear drawback is that the procedure may fail with some probability.In such case, we should reject the output state and ask for a retransmission [19].In the context of QEC, probabilistic decoding operations have found application in stabilizer codes [20] especially for iterative probabilistic decoding in LDPC codes [11,21,22], error decoding [23,24] or environment-assisted error correction [25].Moreover, it was noted that they have a potential to increase the spectrum of correctable errors [15] and are useful when the number of qubits is limited [14].It is also worth mentioning, they were used with success in other fields of quantum information theory, e.g.probabilistic cloning [26], learning unknown quantum operations [27] or measurement discrimination [28].
Despite the fact that pQEC procedure has been studied in the literature for a while, there is lack of a formal description of its application for a general noise model.In this work, we fill this gap.Inspired by celebrated Knill-Laflamme conditions [29], we provide conditions (Theorem 1) to check, when probabilistic error correction is possible.We discover that optimal error-correcting codes are not always generated with the usage of isometric encoding operations.We give an explicit example of noise channels family (Section V), such that to maximize the probability of successful error correction we need to encode the quantum information into a mixed state.Moreover, we discuss the advantage of pQEC procedure over the deterministic one with a formal statement in Theorem 7. We show in Theorem 13 how to correct noise channels with bounded Choi rank.Also, we observe the advantage of pQEC procedure for random noise channels, which is presented in Theorem 16.Finally, if the errors are caused by a unitary interaction with an auxiliary qubit system, we show that it is possible to restore a qubit logical state by using only two physical qubits.We present a procedure how to achieve this in Algorithm 17.
The rest of the paper is organized as follows.In Section II we introduce the notation and define pQEC protocol.In Section III we present equivalent conditions for probabilistically correctable noise channels.Then, we investigate a realization of pQEC procedure in Section IV.In Section V we present a family of noise channels for which, it is necessary to use mixed state encoding to maximize the probability of successful error correction.Then, we study an advantage of pQEC procedure in Section VI and Section VII.In Section VIII we define a generalization of pQEC protocol.Finally, we place all proofs in Appendix A.

A. Mathematical framework
In this section, we will introduce the notation and recall necessary basic facts of quantum information theory.We will denote complex Euclidean spaces by symbols X , Y, . ... The set of linear operators M : X → Y will be written as M(X , Y) and M(X ) := M(X , X ).The identity operators will be denoted by 1l X ∈ M(X ).For any operator M ∈ M(X , Y) we will consider its vectorization |M ∈ Y ⊗ X , which is defined as where |i are elements of computational basis.In the space M(X ), we distinguish the set of positive semi-definite operators P(X ), the space of Hermitian operators H(X ) and the set of unitary operators U(X ).We use the convention that for non-invertible operator M , by M −1 , we denote its Moore-Penrose pseudo-inverse [30].We consider the set of quantum states D(X ), that is, the set of positive semi-definite operators with unit trace.We say that a quantum state ρ is a pure state if rank(ρ) = 1, otherwise, if rank(ρ) > 1, we say that ρ is a mixed state.The maximally mixed state will be denoted by ρ * X := 1 dim(X ) 1l X .We also consider transformations between linear operators.We denote by I X : M(X ) → M(X ) the identity map.Let us define the set of quantum subchannels sC(X , Y) [31].A quantum subchannel Φ ∈ sC(X , Y) is a linear map Φ : M(X ) → M(Y), which is completely positive [30,Theorem 2.22], i.e.
and trace non-increasing In particular, the subchannel Φ which is trace preserving, i.e.
will be called a quantum channel.We denote by C(X , Y) the set of quantum channels Φ : M(X ) → M(Y).We will also use the following notation, sC(X ) := sC(X , X ) and C(X ) := C(X , X ).
In this work, we will consider the following representations of subchannels: • Kraus representation: Each subchannel Φ ∈ sC(X , Y) can be defined by a collection of Kraus operators We say that the subchannel Φ is given in a canonical Kraus representation To represent the subchannel Φ by its Kraus representation (K i ) r i=1 , we introduce the notation K : M(X , Y) ×r → sC(X , Y) given by Φ = K ((K i ) r i=1 ).• Choi-Jamiołkowski representation: Each subchannel Φ ∈ sC(X , Y) can be uniquely described by its Choi-Jamiołkowski operator J(Φ) ∈ M(Y ⊗ X ), which is defied as The rank of J(Φ) is called the Choi rank and it determines the minimal number r of Kraus operators K i needed to describe Φ in the Kraus form Φ = K ((K i ) r i=1 ).Therefore, if the Kraus representation (K i ) r i=1 is canonical, then r = rank(J(Φ)).
• Stinespring representation: By the Stinespring Dilatation Theorem any subchannel Φ ∈ sC(X , Y) can be defined as Φ(X) = tr 2 AXA † for X ∈ M(X ), where A ∈ M(X , Y ⊗ C r ) and tr 2 is the partial trace over the second subsystem C r .The minimal dimension r of the auxiliary system is equal to the Choi rank.In particular, for Φ ∈ C(X ), the Stinespring representation of Φ can be written in the form Φ(X) = tr 2 U (X ⊗ |ψ ψ|)U † , where |ψ ψ| ∈ D(C r ) and U ∈ U(X ⊗ C r ).

B. Problem formulation
In this work, we consider the following procedure of probabilistic quantum error correction.We are given a noise channel E ∈ C(Y) and a Euclidean space X .The goal of pQEC is to choose an appropriate encoding operation S ∈ sC(X , Y) and decoding operation R ∈ sC(Y, X ), such that for any state ρ ∈ D(X ) we have RES(ρ) ∝ ρ.In this protocol, the pair (S, R) represents the error-correcting scheme and the quantity tr (RES(ρ)) represents the probability of successful error correction.This protocol may fail with the probability 1 − tr (RES(ρ)).In such a case, the output state is rejected.To exclude a trivial, null strategy, we add the constrain that a valid error-correcting scheme must satisfy tr(RES(ρ)) > 0 for any ρ ∈ D(X ).
In this set-up, the probability of successful error correction does not depend on the input state ρ (see Lemma 19 in Appendix A 1).We use this fact to standardize the definition of pQEC.From now, we say that E ∈ C(Y) is probabilistically correctable for X , if there exists an error-correcting scheme (S, R) such that 0 = RES ∝ I X . ( We say that E is correctable perfectly if RES = I X .In this work, we will be particularly interested in error-correcting schemes (S, R), which maximize the probability of success for given E and X .

III. PROBABILISTIC QUANTUM ERROR CORRECTION
To inspect pQEC procedure, first, we should state conditions which determine when given noise channel is probabilistically correctable.For deterministic QEC, such conditions have been known for a long time and in the literature as the Knill-Laflamme conditions [29].Let E = K ((E i ) i ) ∈ C(Y) be a given noise channel.Then, according to the Knill-Laflamme Theorem, E is perfectly correctable for X if and only if for all i, j and some isometry operator S ∈ M(X , Y).In the following theorem we generalize the above, to cover probabilistically correctable noise channels.
The following conditions are equivalent: (A) There exist error-correcting scheme (S, R) ∈ sC(X , Y) × sC(Y, X ) and p > 0 such that (B) There exist (D) There exist S * ∈ M(X , Y) and R * ∈ M(Y, X ) such that and there exists i 0 , for which it holds R * E i0 S * = 0.
Moreover, if point (A) holds for S = K ((S k ) k ) and R = K ((R l ) l ), then R ∈ P(Y) from points (B) and (C) can be chosen to satisfy R = l R † l R l .It also holds that R l E i S k ∝ 1l X for any i, k, l.The proof of Theorem 1 is presented in Appendix A 2. Let us discuss the meaning of the conditions stated in Theorem 1.The condition (B) presents a general form of probabilistically correctable noise channels E. Such channels, after applying post-processing √ R behave as mixed isometry operations.They hide parts of an initial quantum information on orthogonal subspaces.The condition (C) may be used to calculate the maximum value of the probability p of successful error correction.For r = rank(J(E)) and d = dim(X ), s = dim(Y) we can introduce the optimization procedure: maximize: tr(M ) Moreover, one may get the form of a recovery subchannel R based on R, S = K ((S k ) k ) and M obtained from this optimization in the following way (see Appendix A 2): 1. Let M = U † DU be the spectral decomposition of M .

For each
4. The recovery subchannel is given as .
Finally, the condition (D) gives us a simple method to check if E = K ((E i ) r i=1 ) is probabilistically correctable for X .Let us compare the point (D) with Knill-Laflamme conditions.The latter, is a constraint satisfaction problem with r 2 quadratic constrains S † E † j E i S ∝ 1l X for the variable S ∈ M(X , Y), which satisfies S = 0.The parameters In comparison, the conditions in the point (D) represent a constraint satisfaction problem with r bilinear constrains RE i S ∝ 1l X for the variables S ∈ M(X , Y) and R ∈ M(Y, X ).Additionally, it must hold RE i0 S = 0 for some i 0 ∈ {1, . . ., r}.In this problem, the parameters E i are arbitrary operators from M(Y), which satisfy span im(E † i ) : i = 1, . . ., r = Y (although a stronger condition holds i E † i E i = 1l Y , we will see in Section VI, it is more convenient to use the weaker version).

IV. REALIZATION OF PQEC PROCEDURE
In this section, we will investigate the form of error-correcting scheme (S, R) which provides the maximal probability of successful error correction.For perfectly correctable noise channels, the encoding S can be realized by the isometry channel.This observation meaningfully reduces the complexity of finding error-correcting schemes -it is enough to consider a vector representation of pure states.Inspired by that, we ask if a similar behavior occurs in the probabilistic quantum error correction.The following proposition gives us some insight in the form of encoding and decoding.
Proposition 2. For a given channel E ∈ C(Y), let us fix an error-correcting scheme (S, R) ∈ sC(X , Y) × sC(Y, X ) such that RES = pI X , for some p > 0.Then, the following holds: The proof of Proposition 2 is presented in Appendix A 3. We may use Proposition 2 (A) to state a realization of pQEC procedure (see Figure 1).For a given noise channel E ∈ C(Y) let (S, R) ∈ C(X , Y) × sC(Y, X ) be an error-correcting scheme for which RES = pI X , where p > 0. The encoding channel S can be realized using the Stinespring representation given in the form S(X) = tr 2 U S XU † S .The state is then sent through E. The decoding subchannel R ∈ sC(Y, X ) can be realized by implementing the channel In summary, the output of the whole procedure consists of a quantum state σ ∈ D(X ) and a classical label i ∈ {0, 1}.If the label i = 0 is obtained, we know that σ ∝ RES(ρ) = pρ, and hence, the output state can be accepted.Otherwise, if i = 1, the output state σ ∝ ΨES(ρ) should be rejected, as in general it may differ from ρ.
In Proposition 2 (C), we observed that using non-isometric channels S or formal subchannels R for perfectly correctable noise channels provides no advantage.Moreover, according to Theorem 1 (D), to predict if a noise channel is probabilistically correctable, we may consider only single Kraus encoding operations.However, among all conditions presented in Proposition 2 there is no condition, which in general allows us to restrict our attention to an isometry channel realization of S. Indeed, there is a class of noise channels E for which, in order to maximize the probability p of successful error correction, we need to consider a general channel realization of S. Paraphrasing, to obtain the best performance, we have to encode the initial state |ψ ψ| ∈ D(X ) into the mixed state S(|ψ ψ|).In Section V we will present a family of noise channels for which it is necessary to use mixed state encoding.

V. NEED FOR MIXED STATE ENCODING
In this section, we provide an example of a parametrized family of noise channels {E R } R for which the mixed state encoding improves the probability of successful error correction.In our example we assume that X = C 2 and Y = C 4 .For each R ∈ P(C 4 ) satisfying R ≤ 1l C 4 let us define a noise channel E R ∈ C(C 4 ) given by the equation We define the optimal probability p 0 of successful error correction as We also define the optimal probability p 1 of successful error correction restricted to the pure state encoding: Our claim, which we will present later, is that there exists a family of operators R for which p 0 (R) > p 1 (R).
We start with the following lemma, where we show the optimal error-correcting scheme (S, R) and a simplified version of the maximization problem p 0 (R).
Lemma 3. Let R ∈ P(C 4 ) and R ≤ 1l C 4 .Define Π R as a projector on the support of R. For E R defined in Eq. (11) we have the following simplified form of the maximization problem p 0 (R): An optimal scheme (S, R) which achieves the probability p 0 (R), that is RE R S = p 0 (R)I C 2 , can be taken as where P is an argument maximizing p 0 (R) in Eq. (14).Moreover, if there exists another optimal scheme ( S, R), that is RE R S = p 0 (R)I C 2 , then rank(J(S)) ≤ rank(J( S)).
The proof of Lemma 3 is presented in Appendix A 4. Let us separately consider two cases: rank(R) < 4 and rank(R) = 4.The first one will be discussed briefly as it will not support our claim.
Corollary 4. Let us take R ∈ P(C 4 ) such that R ≤ 1l C 4 and rank(R) < 4. Define Π R as a projector on the support of R. For the noise channel defined in Eq. (11) we have p 0 (R) = p 1 (R).Moreover, it holds ) where R −1 denotes Moore-Penrose pseudo-inverse.
The proof of Corollary 4 is presented in Appendix A 5. In the case when the operator R is invertible, the situation is more interesting.Let us focus on p 0 (R) obtained in Eq. ( 14).As Π R = 1l C 4 , the equation Π R (P ⊗ X)Π R = P ⊗ X is always satisfied.We can take P = tr(P )ρ, for ρ ∈ D(C 2 ).The inequality tr(P ) ∞ .Hence, we get To calculate p 1 (R) it will be sufficient to add the constraint S = K ((S)).According to Lemma 3 the optimal S is of the form S(X) Then, we have Proposition 5. Let us define an unitary matrix U ∈ U(C 4 ) which columns form the magic basis [32] Let us also define a diagonal operator D(λ) := diag † (λ), which is parameterized by a 4−dimensional real vector λ = (λ 1 , λ 2 , λ 3 , λ 4 ), for which it holds 0 < λ i ≤ 1.For R = U D(λ)U † and the noise channel E R defined in Eq. (11) we have The proof of Proposition 5 is presented in Appendix A 6. We can clearly see that in the case rank(R) = 4, there are operators R, for which the mixed state encoding improves the probability of successful error correction over the pure state encoding, p 0 (R) > p 1 (R).In general, the maximization problem in Eq. ( 17) intuitively supports the inequality so it is possible, that the minimal value of it will be achieved for some mixed state ρ.We observed such behavior in Proposition 5 for R given in the spectral decomposition R = U D(λ)U † .The introduced family of noise channels is parameterized by a 4−dimensional vector λ = (λ 1 , . . ., λ 4 ), such that λ i ∈ (0, 1].For almost all such λ we have p 0 (R) > p 1 (R).The only exception is the 3−dimensional subset defined by the relation which describes the situation, when the pure state encoding match the mixed state encoding, p 0 (R) = p 1 (R).In an extremal case, e.g.
The family of parameters R introduced in Proposition 5 is not the only one for which the minimum value of Therefore, the value of p 0 (R) is one-to-one related with the maximum value of the output min-entropy of the channel Φ (see for instance [33]).Especially, we can see, if the image of the Bloch ball under Φ is a three dimensional ellipsis and contains the maximally mixed state ρ * 2 in its interior, then the mixed state encoding provides benefits.
Finally, the noise channel E R defined for R from Proposition 5 is perfectly correctable for X = C 2 if and only if R = 1l C 4 .Interestingly, this suggests that perfectly correctable noise channels may constitute only a small subset of probabilistically correctable noise channels.This behavior will be the object of our investigation in the next section.

VI. ADVANTAGE OF PQEC PROCEDURE
The goal of this section is to show that pQEC procedure corrects a wider class of noise channels than the QEC procedure based on Knill-Laflamme conditions Eq. ( 6).For any Euclidean spaces X , Y let us define two families of noise channels; these which are probabilistically correctable for X as ξ(X , Y), and these which are correctable perfectly for X as ξ 1 (X , Y): We begin our analysis with some observations.Proposition 6.For any X , Y we have the following properties: The proof of Proposition 6 is presented in Appendix A 7. We see that if dim(X ) = dim(Y), then there is no need to consider pQEC procedure.The situation changes if we encode the initial information into a larger space, dim(Y) > dim(X ).In the following theorem, we will show that ξ 1 (X , Y) ξ(X , Y) for dim(Y) > dim(X ).Theorem 7. Let X and Y be Euclidean spaces for which dim(X ) < dim(Y).Then, the set ξ 1 (X , Y) is a nowhere dense subset of ξ(X , Y).
The proof of Theorem 7 is presented in Appendix A 8.

A. Choi rank of correctable noise channels
Intensity of a noise channel E can be connected with its Choi rank r = rank(J(E)).Given E in the Stinespring form, the Choi rank describes the dimension of an environment system which unitarily interacts with the encoded information.If the interaction is the weakest (r = 1) we deal with unitary noise channels, which are always perfectly correctable.The strongest interaction (r = dim(Y) 2 ) is a property of hardly correctable noise channels.For example, the maximally depolarizing channel E(Y ) = tr(Y )ρ * Y , which can not be corrected, has the maximal Choi rank.In the following theorem, we investigate the maximum Choi rank of probabilistically correctable noise channels ξ(X , Y) and compare it with the maximum Choi rank for ξ 1 (X , Y). Theorem 8. Let X and Y be some Euclidean spaces such that dim(Y) ≥ dim(X ).The following relations hold: The proof of Theorem 8 is presented in Appendix A 9. In Proposition 6 we showed that if dim(X ) = dim(Y), then the pQEC procedure gives us no advantage.Indeed, the only reversible noise channels, in this case, are unitary noise channels.In the language of Choi rank, that means, if the Choi rank of a noise channel is equal to one, then it can be corrected.We can ask, what is the maximum value of r ∈ N, such that all noise channels which Choi rank is less or equal r, can be corrected perfectly or probabilistically, respectively.Formally speaking, for any X and Y we define the following quantities: The quantity r 1 (X , Y) for a general noise model was studied in [34,35].The authors of [34] calculated a lower bound for r 1 (X , Y) by using a technique of noise diagonalization along with Tverberg's theorem.They obtained the following result It implies that 4 dim(Y) dim(X ) ≤ r 1 (X , Y).On the other hand, by using the Quantum packing bound [35] we may gain some insight of the upper bound for r 1 (X , Y).If we assume that we are allowed to use only non-degenerated codes, then for perfectly correctable E we have a bound of the form rank(J(E)) ≤ dim(Y) dim(X ) .In the next part of this section, we will improve the upper bound of r 1 (X , Y) without putting any additional assumptions.We also will estimate the behavior of r(X , Y).In the particular case X = C 2 and Y = C 4 , we will also show that r 1 (X , Y) < r(X , Y).
Let us start with the following simple, but important properties, required to study r(X , Y).We will notice, that for a constant Choi rank of the noise, it is easier to construct error-correcting scheme, if the dimension of Y is large.

B. Schur noise channels
In this subsection, we restrict our attention to a particular family of noise channels whose Kraus operators are diagonal in the computational basis.In the literature, these channels are referred to as Schur channels [30,Theorem 4.19].We use them to study an upper bound for r(X , Y) and r 1 (X , Y). Lemma 10.Let X and Y be Euclidean spaces such that dim(Y) ≥ dim(X ).Then, there exists a Schur channel The proof of Lemma 10 is presented in Appendix A 10. The bounds obtained in Lemma 10 are asymptotically tight for Schur noise channels with dim(Y) → ∞.To prove the tightness of the bound for perfectly correctable noise channels, we may use the construction provided in [34].Hence, if we take a Schur channel In the following proposition we will prove the tightness for probabilistically correctable Schur noise channels.
Proposition 11.Let X and Y be Euclidean spaces and dim(X ) ≤ dim(Y).For any Schur channels The proof of Proposition 11 is presented in Appendix A 11.In the case of Schur channels we have a clear separation between probabilistically and perfectly correctable noise channels.

C. From bi-linear to linear problem
In general, the difficulty of finding error-correcting schemes (S, R) comes from bi-linearity of the problem Eq. ( 10).However, there is a particular class of noise channels, for which we can easily rewrite the bi-linear problem as a linear one.In this subsection, we will focus our attention on noise channels E ∈ C(Y), such that rank(E(1l Y )) = dim(X ).Note, that this assumption implies dim(X )rank(J(E)) ≥ dim(Y).
Let E = K ((E i ) i ) and let Π be the projector on the image of E(1l Y ).Consider an associated channel ) is an isometry operator with the image on the subspace defined by Π.It is clear that E is probabilistically correctable for a given space X if and only if there exists a scheme (S, R), such that 0 = RFS ∝ I X .Hence, according to Theorem 1 we need to find S * ∈ M(X , Y), R * ∈ M(X ), such that R * F i S * = c i 1l X and c i0 = 0 for some i 0 .Interestingly, we can combine together an action of S * , R * as just the action of some pre-processing S * ∈ M(X , Y), that is Therefore, we obtained a linear problem equivalent to Eq. ( 10).In the following proposition we will investigate consequences of a such simplification.
Proposition 12. Let X and Y be some Euclidean spaces and dim(X ) ≤ dim(Y).
The proof of Proposition 12 is presented in Appendix A 12. Eventually, it is worth mentioning that the QEC procedure based on Knill-Laflamme conditions works well with this class of noise channels.Consider the situation dim(X )rank(J(E)) = dim(Y).Then, if E ∈ C(Y) and rank(E(1l Y )) = dim(X ), it holds E ∈ ξ 1 (X , Y).To see this, take the Kraus decomposition of E = K ((E i )) and notice that operators E i are orthogonal pieces of some unitary operator.

D. Correctable noise channels with bounded Choi rank
In this subsection we will study the behavior of r(X , Y) and r 1 (X , Y).We will state a lower and a upper bound for both quantities.
Theorem 13.Let X and Y be some Euclidean spaces such that dim(Y) ≥ dim(X ).Then, we have The proof of Theorem 13 is presented in Appendix A 13. Unfortunately, according to this theorem, there is no clear separation of r(X , Y) and r 1 (X , Y) for arbitrary X and Y.The improvement of these bounds will be investigated in the future.
For now, we will calculate explicitly r(X , Y) and r 1 (X , Y) for X = C 2 and Y = C 3 , C 4 .
The proof of Proposition 14 is presented in Appendix A 14. By using Theorem 13 and Proposition 14 we get the following advantage of pQEC protocol for X = C 2 and Y = C 4 .
Corollary 15.For X = C 2 and Y = C 4 we have In particular, it holds E. Random noise channels In the last subsection, we will show the advantage of pQEC procedure for randomly generated noise channels.We will follow the procedure of sampling quantum channels considered in [36][37][38].
Let r ∈ N and let (G i ) r i=1 ⊂ M(Y) be a tuple of random and independent Ginibre matrices (matrices with independent and identically distributed entries drawn from standard complex normal distribution).Define given as This sampling procedure induces the measure P on C(Y) whose support is defined on {E ∈ C(Y) : rank(J(E)) ≤ r}.
Theorem 16.Let E r ∈ C(Y) be a random quantum channel defined according to Eq. (32).Then, the following two implications hold The proof of Theorem 16 is presented in Appendix A 15. To answer this question, observe that the channel E satisfies rank(J(E)) ≤ 2. In Proposition 14 we noticed that such channels are probabilistically correctable for a given input space C 2 , if dim(Y) = 4 (in fact, from monotonicity for dim(Y) ≥ 4).Therefore, to correctly transfer a qubit state through E, we may define an error-correcting scheme with only two physical qubits.
We provide the following pQEC procedure based on Proposition 14.
Run the QEC procedure presented in Figure 2 for |ψ , U S , U R , V R .10 Let σ exp be the output state of the procedure presented in Figure 2. Use the post-processing of the measurements' output (i, j) according to the following table: Figure 2: The circuit representing the pQEC procedure.We have access to two physical qubits.The first qubit is in the state |ψ .This state will be encoded.The second state we set |0 .We implement two-qubit, encoding unitary operator U S .Then, the encoded state U S (|ψ ⊗ |0 ) is affected by the noise channel E. After that, we start the decoding procedure.We implement two-qubit unitary rotation U R .We measure the second qubit in the standard basis and obtain a classical label i ∈ {0, 1}.We prepare a third qubit in the state |0 and implement two qubit unitary rotation V R .We measure the third qubit in the standard basis and obtain a classical label j ∈ {0, 1}.If (i, j) = (0, 0) we accept the output state, otherwise, we reject it.

VIII. GENERALIZATION OF PQEC PROCEDURE
Let us denote by Υ an arbitrary family of noise channels, that is Υ ⊂ C(Y).In this section, we ask if there exists error-correcting scheme (S, R), such that all noise channels E ∈ Υ we have RES = p E I X , for some p E ≥ 0. Note, that p E may differ for different noise channels E, hence, we shall introduce a quantity to "globally" control the effectiveness of (S, R).We propose the following approach.
Let µ be some probability measure defined on the set Υ. We assume that noise channels E ∈ Υ are probed according to µ.The scheme (S, R) will be a valid error-correcting scheme for Υ and µ if in average, the probability of successful error correction is non zero, that is Without loss of the generality we may assume that Υ is convex.Additionally, we assume that the support of µ is equal to Υ. Usually, we can take µ as the flat measure, representing the maximal uncertainty in the process of probing random noise channels E from Υ. Let us define the average noise channel of Υ with respect to µ We will show that we can correct all noise channels from the family Υ, whenever Ē is probabilistically correctable for X .We put this statement as the following proposition.
Proposition 18.Let Υ ⊂ C(Y) be a nonempty and convex family of noise channels.Define µ to be a probability measure defined on Υ and assume that the support of µ is equal to The following conditions are equivalent: (A) For each E ∈ Υ there exists p E ≥ 0 such that RES = p E I X and Υ p E µ(dE) > 0.
(B) It holds that 0 = R ĒS ∝ I X .
The proof of Proposition 18 is presented in Appendix A 16.

IX. DISCUSSION
In this work, we analyzed pQEC procedure for a general noise model.We established the conditions to check if a given noise channel is probabilistically correctable.Moreover, we showed that mixed state encoding should be taken into account when maximizing the probability of successful error correction.Finally, we pointed the advantage of the probabilistic error-correcting procedure over the deterministic one.We saw a clear separation especially for a correction of Schur noise channels and random noise channels.We obtained the maximum value of Choi rank of probabilistically correctable noise channels.We also provide a method how to probabilistically correct noise channels with bounded Choi rank.
There are many directions for further study that still remain to be explored.It would be interesting to strengthen Theorem 13 and show the separation between r(X , Y) and r 1 (X , Y) by improving the proposed proof technique in Appendix A 13. We obtained such separation for X = C 2 and Y = C 4 in Corollary 15.Another promising direction is to propose tools for the numerical analysis of pQEC protocols, based on Theorem 1.Such tools would help us estimate the value of r(X , Y) and gain an insight into probabilistically correctable noises that require mixed state encoding.Last but not least, we would like to calculate the worst-case probability of successful error correction for a given noise intensity r ≤ r(X , Y).For example, as we showed in Proposition 14, the errors caused by a unitary interaction with an auxiliary qubit system (r = 2), can be corrected by using only two physical qubits (dim(Y) = 4).We can ask, how many times in average the procedure presented in Algorithm 17 needs to be repeated.Proof.Let L = RES and for any unitary operator U ∈ U(X ) and i = 0, . . ., dim(X ) − 1 define p U,i ∈ [0, 1] by L(U |i i|U † ) = p U,i U |i i|U † .We have L(1l X ) = U ( i p U,i |i i|) U † for any U and hence, there exists p ∈ [0, 1] such that L(1l X ) = p1l X .That means, p U,i = p for any U and i, so L(|ψ ψ|) = p|ψ ψ| for any |ψ ψ| ∈ D(X ).We obtain the thesis by noting that span C (|ψ ψ|) = M(X ).

Proof of Theorem
The following conditions are equivalent: (A) There exist error-correcting scheme (S, R) ∈ sC(X , Y) × sC(Y, X ) and p > 0 such that and there exists i 0 , for which it holds R * E i0 S * = 0.
Moreover, if point (A) holds for S = K ((S k ) k ) and R = K ((R l ) l ), then R ∈ P(Y) from points (B) and (C) can be chosen to satisfy R = l R † l R l .It also holds that R l E i S k ∝ 1l X for any i, k, l.Proof.In order to show that (A) ⇐⇒ (B) ⇐⇒ (C), in all implications presented below, we will use the same encoding S = K ((S k ) k ) ∈ sC(X , Y).Hence, to simplify the proof, we introduce the notation of F := ES given in the form F = K ((F i ) i ).
In the case of p 1 (R), to calculate the largest eigenvalue of tr

Figure 1 :
Figure 1: Schematic realization of pQEC procedure for the noise channel E.
VII. EXAMPLE OF PQEC QUBIT CODE Consider the following scenario.You have a task to transfer a given qubit state ρ ∈ D(C 2 ) through a quantum communication line represented by a noise channel E ∈ C(Y) of the form E(Y ) = tr 2 U (Y ⊗ |ψ ψ|)U † , where |ψ ψ| ∈ D(C 2 ) and U ∈ U(Y ⊗ C 2 ).At this point a natural question arises.What is the minimal size of the communication line dim(Y), which is large enough to recover the state ρ with the pQEC procedure?
) (C) There exist S = K ((S k ) k ) ∈ sC(X , Y), R ∈ P(Y), such that R ≤ 1l Y and a matrix M = [M jl,ik ] jl,ik = 0, for which it holds ∀ i,j,k,l S † l E † j RE i S k = M jl,ik 1l X .(A3)(D)There exist S * ∈ M(X , Y) and R * ∈ M(Y, X ) such that