Decoupling with unitary approximate two-designs

Consider a bipartite system, of which one subsystem, A, undergoes a physical evolution separated from the other subsystem, R. One may ask under which conditions this evolution destroys all initial correlations between the subsystems A and R, i.e. decouples the subsystems. A quantitative answer to this question is provided by decoupling theorems, which have been developed recently in the area of quantum information theory. This paper builds on preceding work, which shows that decoupling is achieved if the evolution on A consists of a typical unitary, chosen with respect to the Haar measure, followed by a process that adds sufficient decoherence. Here, we prove a generalized decoupling theorem for the case where the unitary is chosen from an approximate two-design. A main implication of this result is that decoupling is physical, in the sense that it occurs already for short sequences of random two-body interactions, which can be modeled as efficient circuits. Our decoupling result is independent of the dimension of the R system, which shows that approximate 2-designs are appropriate for decoupling even if the dimension of this system is large.


I. INTRODUCTION
Consider a joint quantum system AR in a state τ AR . We say that A is decoupled from R if the joint system is in a product state τ A ⊗ τ R . Operationally, this means that the probability distributions obtained upon measuring the A and R systems are statistically independent. Decoupling is the art of producing a decoupled state from an arbitrarily correlated state ρ AR by local operations on the A system.
Arguments that make use of the fact that the systems A and R can be decoupled have proven useful in various applications in quantum information theory. Examples abound in the area of quantum Shannon theory: state merging [17] and state transfer [14]. Other important theorems, such as the best known achievable rates for sending quantum information through a quantum channel [15], can be proven concisely via decoupling. Moreover, such arguments have been used in a more physical context and, for example, deepened our insight into the black hole information paradox [16] and the role of negative conditional entropies in thermodynamics [7].
In [9], a decoupling theorem is derived that generalizes the previous decoupling theorems used in the above papers. There one considers a situation where a subsystem A of a joint system AR undergoes an evolution while R is left unchanged. The mapping describing the evolution of A is conceptually split into two parts: a unitary followed by an arbitrary trace-preserving and completely positive map T A→E . The Decoupling Theorem [9,10] states that if an initial state ρ AR and and a process T = T A→E are fixed and the unitary is either taken from the Haar measure or from a two-design [6], then the expected distance of the resulting state from a decoupled state is bounded in terms of entropic quantities: Here the operator ω only depends on the map T A→E and specifically is independent of the chosen input state, ρ AR . Roughly speaking, the min-entropy, H min (A|R) ρ , (cf. Definition 1) in the above formula quantifies the uncertainty an observer with access to R has about the A subsystem prior to the decoupling operation, while H min (A ′ |E) ω quantifies the uncertainty of an observer who has access to the output E of T A→E about a copy, A ′ , of the input state. The min-entropy can be seen as a generalization of the well-known von Neumann entropy in the following sense. If the min-entropy is evaluated for n identical copies of the same state then in the asymptotic limit of large n it reduces to the von Neumann entropy (cf. Equation 2). Thus an important special case of the above relation arises when we consider the limit of a large number of identical copies of states, ρ AR , and channels, T A→E , applied to them. In this scenario the subsystems decouple if H(A ′ |E) ω + H(A|R) ρ > 0 holds for the conditional von Neumann entropies of ω and ρ.
Often T A→E is chosen in a specific way. If T A→E is the partial trace over a subsystem of A this yields the Fully Quantum Slepian-Wolf (FQSW) Theorem [14]. Another special case is state merging [17], where T A→E represents a measurement of the A system. In the FQSW scenario, the above inequality is known to be tight [14].
In this paper we analyze the decoupling behavior when the random unitary is taken from an almost two-design instead of a two-design. This is motivated by the fact that almost two-designs, as opposed to exact two-designs such as the Clifford group [8], emerge in certain realistic models of physical systems. As an example, we consider a typical quantum mechanical evolution of an A subsystem that is governed by two-particle interactions. More precisely, we follow the lines of [13] and model the internal dynamics of the A subsystem in terms of a quantum circuit and address the question of how well these dynamics decouple. Moreover, our decoupling results open the door to a more efficient implementation of operational tasks such as state transfer and state merging, since one might expect good almost two-designs to outperform exact two-designs in terms of circuit complexity 1 .
Note also that in quantum cryptography, a special case of decoupling -where the A system constitutes a classical random variable that is correlated with a quantum memory, R, in the hands of an adversary -is used to extract private randomness secret from an adversary with quantum memory [25]. In this context, two-universal hash functions [3] replace the unitary two-design used in the Decoupling Theorem and an extension to almost two-universal hash functions is known [28]. Our work can thus be seen as a fully quantum version of this result.
In this paper, we consider finite dimensional systems only. However, the task of extracting private randomness was recently generalized to the case when the adversary holds an infinite dimensional system [11] or a general von Neumann algebra of observables [1], suggesting that such generalizations are also possible for decoupling.
The remainder of the paper is organized as follows. In Section II, we introduce the mathematical framework used to derive our main technical results that are presented in Section III. Finally, in Section IV we discuss applications of our decoupling result.

A. Notation
Let H be a finite dimensional, complex Hilbert space. The set of linear operators on H will be denoted by L(H), the set of Hermitian operators by L † (H) and the set of positive-semidefinite operators is given by P(H). The set of quantum states is given by S = (H) := {ρ ∈ P(H) | tr ρ = 1} and the set of sub normalized quantum states is S ≤ (H) := {ρ ∈ P(H) | tr ρ ≤ 1}. For the Lie group of unitary matrices we write U. A subscript letter following some mathematical object denotes the physical system to which it belongs. However, when it is clear which systems are described we might drop the subscripts to shorten the notation.
Bipartite systems AB are represented by a tensor product space H A ⊗ H B =: H AB . We will denote by 1 A the identity operator on H A and by π A := 1 A /d A the completely mixed state on A, where d A = dim H A . Linear maps from L(H A ) to L(H B ) will be denoted by calligraphic letters, e.g. T A→B . Quantum operations are in one to one correspondence with the trace preserving completely positive maps (TPCPMs). The TPCPM we will encounter most often is the partial trace (over the system B), denoted tr B (·), which is defined to be the adjoint mapping of T A→AB (ξ A ) = ξ A ⊗ 1 B for ξ A ∈ L † (H A ) with respect to the Schmidt scalar product A, B := tr(A † B). This means tr((ξ A ⊗ 1 B )ζ AB ) = tr(ξ A tr B (ζ AB )) for any ζ AB ∈ L † (H AB ). Given a multipartite state ξ AB , we write ξ A := tr B ξ AB for the reduced density operator on A and ξ B := tr A ξ AB , respectively, on B.
For isomorphic H A and H A ′ , we denote by Φ AA ′ the completely entangled state on AA ′ , i.e. Φ AA ′ : orthonormal bases. The Choi-Jamio lkowski representation [4,19] of T A→E ∈ Hom(L(H A ), L(H E )) is given by the operator ω A ′ E := (T A→E ⊗ I A ′ )(Φ AA ′ ). Here, I A ′ denotes the operator identity on A ′ , which we will only write explicitly if it is not clear from context.
The metric induced on L(H) via the Schatten 1-norm is D(ρ, σ) := ρ − σ 1 . Another measure of distance on P(H) is the fidelity, F (ρ, σ) := √ ρ √ σ 1 . We also require a norm for linear maps Given such a map, its diamond norm is defined to be [20]: Note that the diamond norm is the dual of the well-known norm of complete boundedness [22].

B. Smooth Entropies
Entropies are used to quantify the uncertainty an observer has about a quantum state. Moreover, conditional entropies quantify the uncertainty of an observer about one subsystem of a bipartite state when he has access to another subsystem. The most commonly used quantity is the von Neumann entropy. Given a state ρ AB ∈ S = (H AB ), we denote by H(A|B) ρ := H(ρ AB ) − H(ρ B ) the von Neumann entropy of A conditioned on B.
While the von Neumann entropy is appropriate for analyzing processes involving a large number of copies of an identical system, the smooth min-entropy is the relevant quantity when a single system is considered [24]. Its definition is based on the following quantity.
Definition 1 (Min-Entropy [24]). Let ρ AB ∈ S ≤ (H AB ), then the min-entropy of A conditioned on B of ρ AB is defined as More precisely the smooth conditional min-entropy is defined as the largest conditional minentropy one can get within a distance of at most ε from ρ. Here closeness is measured with respect to the purified distance, P (ρ, σ), which is defined to be whereF (ρ, σ) is the generalized fidelity;F (ρ, σ) := F (ρ, σ)+ (1 − tr ρ)(1 − tr σ) for ρ, σ ∈ S ≤ (H). In [26] it is shown that P constitutes a metric on S ≤ (H) and the following inequalities are derived We say that ρ is ε-close toρ, denotedρ ≈ ε ρ, if P (ρ,ρ) ≤ ε.
The fully quantum asymptotic equipartition property (QAEP) states that in the limit of an infinite number of identical states the smooth min-entropy converges to the von Neumann entropy [27]: Let ε > 0 and let ρ AB ∈ S = (H AB ), then In that sense, the smooth conditional min-entropy can be seen to be a one-shot generalization of the von Neumann entropy.

C. Almost Two-Designs and Quantum Circuits
Heuristically, a unitary two-design is a finite subset D of U that has the property that averaging any polynomial of degree 2 over D gives the same result as integrating this polynomial over U with respect to the Haar measure, U . .
..,n be a set of pairs, where the U i are unitary matrices on a Hilbert space H and the p i ≥ 0 with i p i = 1 are probabilities. We define the maps We will denote an integral over the unitary group with respect to the normalized Haar measure by E U (·) and an average over a unitary almost two-design by E D (·) for notational convenience.
For the applications that we are interested in, the most relevant approximate designs are generated by random quantum circuits [13]. A quantum circuit is a set of wires on which gates are applied. Each wire corresponds to a qubit evolving in time, and each gate on the wire corresponds to some unitary operation being applied to the qubit. A k-qubit gate is given by an element of U(2 k ). For us it will be sufficient to think of the circuit as a sequence of unitaries that are applied in a certain order: W = W t ·...·W 2 ·W 1 , where we call t the time of the circuit. We call a set of gates universal for n qubits if any operation that can be performed on n qubits can be approximated to arbitrary precision using operations from the universal gate set only.

III. DECOUPLING WITH δ-ALMOST UNITARY TWO-DESIGNS
In this section we state and prove our main result. We generalize the decoupling theorem for two-designs to the case where the average is taken over a unitary δ-almost two-design. Theorem 1 is the core technical result of this paper.
Theorem 1. (Decoupling with δ-almost unitary two-designs) Let ρ AR ∈ S ≤ (H AR ) be a subnormalized density operator and let T A→E be a linear map with Choi-Jamio lkowski representation where D constitutes a δ-approximate two-design.

Remark 2.
It should be noted that the factor d 4 A in the above formula can be compensated for by making δ accordingly smaller. For the example discussed in Section IV, where the almost twodesign is created by a random circuit, this can be achieved by increasing the length of the circuit by a constant factor.
We also remark that in the particular case of averaging over a two-design, the proof of our decoupling result includes a shorter derivation of the decoupling theorem for exact two-designs as opposed to the original proof in [9,10] (see Section III B).
The rest of this section is structured in four subsections. First, we prove a lemma that quantifies decoupling in terms of Schatten 2-norms. Then, in Section III B, we derive the decoupling formula for perfect two-designs using that lemma (see Theorem 6). Section III C is devoted to a derivation and an analysis of the decoupling formula for δ-almost two-designs (see Theorem 1). And lastly, in Section III D we reformulate the upper bound given by the decoupling formula for δ-almost two-designs in terms of smooth conditional min-entropies (see Theorem 8). This enables us to make statements about independent, identically distributed states via the QAEP, Equation (2).

A. Decoupling with Schatten 2-Norms
For a map T ∈ Hom(L(H A ), L(H E )) with Choi-Jamio lkowski representation ω A ′ E ∈ L † (H EA ′ ) and an operator ρ AR ∈ L † (H AR ), we prove that For our application and the proof of (3) it is convenient to reformulate the argument of the expectation value in a more symmetric way. We introduce the map EÃ →R , which we define to be the unique Choi-Jamio lkowski preimage of the state ρ AR i.e. EÃ →R (Φ AÃ ) = ρ AR , whereÃ is just a copy of A. Note that E is not trace-preserving in general. We can write for any unitary U A : where we have introduced the decoupling operator ξ AÃ := Φ AÃ − π A ⊗ πÃ. Equation (4), uses the fact that an arbitrary map acting exclusively on the A subsystem of Φ AÃ commutes with any map that only acts onÃ. In Equation (5) the linearity of the maps is used. Analogously one has that Thus the stated result, Equation 3, can be rewritten equivalently in terms of the decoupling operator.
Lemma 3. Let ξ AÃ = Φ AÃ − πÃ ⊗ πÃ and let T A→E ∈ Hom(L(H A ), L(H E )) and EÃ →R ∈ Hom(L(HÃ), L(H R )) be linear maps that preserve hermiticity, then Proof. We have that We introduced two further copies A ′ andÃ ′ of A when using the swap trick (see Appendix A) in Equation (6), i.e. (ξ AÃ ) ⊗2 = ξ AÃ ⊗ ξ A ′Ã′ . In Equation (7) we used the definition of the adjoint of the mapping (T ⊗Ẽ) ⊗2 with respect to the Schmidt scalar product. We have from [9], Lemma 3.4, that with the coefficients α and β satisfying Similar integrals were evaluated in the context of decoupling already in [17]. Using the above we get In Equation (8) we used that tracing out one of the subsystems A,Ã of ξ AÃ gives the zero state. The last line above makes use of the definition of the adjoint of E, the swap trick and the definition of the Schatten 2-norm. Rewriting β we find that Substituting this into Equation (9) yields which proves the lemma.

B. Decoupling with Perfect Two-Designs
In this subsection we show two additional lemmas that we require for the derivation of our main result, Theorem 1. Taking these lemmas together with Lemma 3, we also obtain a concise derivation of the decoupling theorem for the Haar measure (cf. Theorem 6).

Lemma 5. For any
Proof. Choose ζ R such that it saturates the bound in the definition of the H min -entropy. Without loss of generality ζ R is invertible (otherwise, redefine R such that it corresponds to the support of ρ AR ). Then Taking the trace on both sides of (11) proves Lemma 5.
Before proving our main theorem, it will be useful for the sake of completeness to first state and prove the decoupling theorem of [9] in the formulation which is given in [10]: Theorem 6. (Decoupling Theorem, [9]) Let ρ AR ∈ S ≤ (H AR ) be a subnormalized density operator and let T A→E be a linear map with Choi-Jamio lkowski representation ω A ′ E ∈ S ≤ (H EA ′ ), then Proof. Note first that for a proof of Theorem 6 it suffices to show that holds and to apply the Jensen Inequality. To prove Equation (12), we work with the integrand in terms of the decoupling operator (Lemma 3). We use Lemma 4 to bound the Schatten 1-norm of the integrand with the Schatten 2-norm: Introducing the positive and normalized operators σ E and ζ R , we have One can abbreviate the notation using the completely positive mapsT A→E andẼÃ →R by defining which yields By Equation (10) we have that In Equation (16) we used the Cauchy-Schwarz inequality (Lemma 3.5 in [9]) to infer that both bracket terms are smaller than one. The derivation is valid for any positive and normalized operators σ E and ζ R , therefore one can chooseσ E andζ R such that they minimize the expression in (16). An application of Lemma 5 then shows that

C. Decoupling with δ-Almost Two-Designs
This subsection is devoted to a proof of the core theorem of this paper: of Theorem 1. Due to the Jensen Inequality it suffices to show that holds. To prove (17), we proceed in a similar fashion to our proof of Theorem 6. As before, we introduce the map EÃ →E which we define to be the unique Choi-Jamio lkowski preimage ρ AR and the state ξ AÃ = Φ AÃ − π A ⊗ πÃ and write for any unitary: . To upper bound the left-hand side of (17), we apply Lemma 4. We introduce positive, normalized operators σ E and ζ R and the mapsT andẼ as defined in equations (13) and (14) respectively and find Applying the swap trick (Appendix A) and using the definitions of the adjoint mappings ofT and E gives With the relations For now we fix our attention on the first term of Equation (18). Bounding this term gives where inequality (19) uses the explicit form of ξ AÃ = Φ AÃ − π A ⊗ πÃ and the definition of the δ-almost two-design. In the following steps we upper bound the term ||( be the projector corresponding to the biggest absolute eigenvalue of (T † ) ⊗2 [F E ]. The ∞-norm can then be rewritten as To be able to apply the swap trick, we decompose P + AA ′ into some basis: Without loss of generality we choose the coefficients c ij to be real. This gives: We rewriteT (σ i A ) using the Choi-Jamio lkowski representation ofT To obtain an upper bound for Equation (22) we apply Lemma 7. The proof of this lemma will be given after concluding the proof of Theorem 1. Using Lemma 7 and the fact that (P + AA ′ ) ⊺ is a rank one projector we get This gives the bound And identically we find that Thus we obtain the desired bound for the first term of (18) using (19): tr ω 2 The only thing left is to evaluate the second term of (18), but this term was already calculated as part of the proof of the decoupling theorem. It equals the term on the right hand side of (15) and can be bounded using (16): tr ω 2 An application of Lemma 5 on (24) and (25) gives (17) and thus concludes the proof of the decoupling theorem with almost two-designs.
It remains to show Lemma 7, which was used to obtain the inequality in (23).
. Moreover we choose them to be orthonormal with respect to the Schmidt scalar product (i.e. tr(σ i A σ j A ) = δ ij and likewise for the E system). Hence, the product states {σ i A ⊗σ E j } i=1,...,d A ; j=1,...,d E also form an orthonormal basis for L † (H AE ) with respect to the Schmidt scalar product: We write the operators ω AE , ω A ′ E and ρ AA ′ in that basis: Since all matrices in the above statements are hermitian, the coefficients a ij and c ij are real. Moreover the coefficients in the expansion of ω AE and ω A ′ E are the same, because the corresponding matrices are the same. Substituting the expansions into the left-hand side of the lemma gives: We now introduce the matrices A := (a ij ) and C := (c ij ) and use Equation (26) to find that We calculate the Schatten 2-norm of C using that ||C| | 2 2 = ij |c ij | 2 ([2]) and the explicit formula for the c ij : The trace term in (27) can be calculated similarly. We use the explicit formula for the coefficients: Taking (28) together with (29) and substituting them into (27) concludes the proof of Lemma 7.

D. A Smoothed Decoupling Formula for Almost Two-Designs
In order to achieve a tighter bound in the decoupling formula for almost two-designs (Theorem 1), we now introduce a modified upper bound stated in terms of smooth conditional minentropies (see Definition 2). We refer to [10] for a discussion of the optimality of decoupling in terms of these quantities. The smooth conditional min-entropy has the additional advantage that it reduces to the von Neumann entropy in the important special case where the state is a tensor product of many identical states, as shown by the Fully Quantum Asymptotic Equipartition Theorem (see Equation 2). Theorem 8. (Smoothed decoupling formula for δ-almost unitary two-designs) Let ρ AR ∈ S ≤ (H AR ) be a subnormalized density operator and let T A→E be a linear map with Choi-Jamio lkowski representation ω A ′ E ∈ S ≤ (H EA ′ ) and let ε be such that min { tr(ρ), tr(ω)} > ε ≥ 0. Then be the state that saturates the bound in the definition of H ε min , i. e. P (ω A ′ E ,ω A ′ E ) ≤ ε and H min (A ′ |E)ω = H ε min (A ′ |E) ω . Analogouslyρ AR is defined to be an operator with P (ρ AR , ρ AR ) ≤ ε and H min (A|R)ρ = H ε min (A|R) ρ . Using inequality (1), we find that: We decomposeω − ω andρ − ρ into positive operators with orthogonal support writinĝ and conclude from (30) that LetT , D + and D − be the unique Choi-Jamio lkowski preimages ofω A ′ E , ∆ + and ∆ − respectively. We apply Theorem 1 onρ andω to find For any unitary, we have with an application of the triangle inequality In the same wayρ R is eliminated from the product term and we get in total The first term of Equation (31) corresponds to the unsmoothed decoupling formula. For the remaining two terms and we need to find upper bounds. We treat them separately beginning with the first one. To perform the calculation we writeρ − ρ = Γ + − Γ − and use the linearity of T . We get Inequality (34) used that an almost two-design constitutes an almost 1-design automatically. This can be seen straight from the definition by considering states that are given by the identity operator on one of the systems on which the unitaries act. The last inequality (35) can be seen by choosing the eigenvalue of T † (1 E ) which is the biggest in absolute value and defining P A to be the projector corresponding to this eigenvalue. One then has T † (1 E ) ∞ ≤ d A .
Bounding the term (33) is done similarly. We decomposeT − T = D + − D − in accordance with the decompositionω − ω = ∆ + − ∆ − . We then get Combining the expressions (35) and (36) and substituting them into (31), we obtain Finally this yields which proves the smoothed decoupling formula for δ-approximate two-designs.

IV. DECOUPLING IN PHYSICAL SYSTEMS
In this section we explain how our result can be applied to study a typical evolution of a physical system. Consider, as before, a joint system AR in an initial state ρ AR and assume that the A system consists of a large number of interacting particles. In nature the most common type of interaction is a local two-particle interaction. It can be modeled using a two-qubit unitary gate. More generally, one may describe the randomization process induced by the evolution of a many-particle system using a quantum circuit. Such approaches were considered earlier for instance in [5] and [13]. The circuit is constructed in the following way: at each step of the circuit, two qubits from A and an element of a universal gate set for U(4) are chosen uniformly at random. The gate is applied to the qubits and the circuit proceeds to the next step. For a given circuit time t, we consider the set of all possible pairs of unitaries the circuit can produce together with the corresponding probabilities. If t goes to infinity this yields the Haar distribution on the whole unitary group [13]. Unfortunately, it turns out that the convergence rate of the random circuits towards the Haar distribution is exponentially slow in the number of qubits of the underlying system [5,13,21]. Nevertheless, after a time t that grows polynomially in the number of qubits and logarithmically in 1 δ , the above circuit will constitute a δ-almost two-design. More precisely, the authors of [13] (Theorems 2.9 and 2.10) derive the following pivotal theorem.
Theorem 9. (Random quantum circuits are approximate two-designs, [13]) Let µ be the probability distribution corresponding to any universal gate set on U(4) and let W be a random circuit on n qubits obtained by drawing t random unitaries according to µ and applying each of them to a random pair of qubits. Then there exists C and (C = C(µ) only) such that for any δ > 0 and any t ≥ C(n 2 + n log(1/δ)), the set of unitaries produced by W together with the corresponding probabilities forms a δ-approximate unitary two-design.
Following the discussion in [5], we will assume that typical dynamics in nature are given by (short) circuits of the type of Theorem 9. We conclude that in our model the possible evolutions of a many qubit system are given by elements of a unitary almost two-design. Moreover, Theorem 9 states that in order to reach a δ-almost two-design at least a circuit time t := C(n 2 + n log 1 δ ) is required, with C being some constant that only depends on the concrete circuit used.
We can now apply our decoupling theorem for almost two-designs to infer conditions under which typical processes in nature result in decoupling. In this example, we shall assume that the R system is correlated with a subsystem of A and we are interested in how this correlation behaves under a typical evolution. Hence, we decompose A into two parts: A S , which identifies the subsystem of interest, and A E , which corresponds to an environmental system which is uncorrelated with R. Since we are interested in the state of A S we choose T to be the partial trace on the environment system: T (ρ) = tr A E [ρ]. Formally, this implies that H min (A|R) ρ ≥ − log d A S and H min (A ′ |E) ω ≥ log d A E − log d A S . An application of Markov's inequality to the decoupling formula for almost two-designs shows that, for any ǫ > 0, one has This implies that if the environment A E is chosen big enough, decoupling occurs except with small probability. Note, moreover, that the factor d 4 A does not increase the time that is required until decoupling is reached in a significant way. To reach aδ-approximate two-design withδ := δ d 4 A the circuit requires at least a timē t := C n 2 + n log 2 4n δ = C n 2 + 4n 2 + n log 1 δ This means that once the circuit has reached a δ-almost two-design, one has to wait only approximately five times longer until the circuits form aδ-almost two-design. This additional time certainly does not affect our conclusions.
We summarize our discussion with a corollary and give an outlook for possible applications of our results.
Corollary 10. Given a system A which consists of two subsystems A S and A E , assume that A S is correlated with a reference system R. Furthermore assume the A system to consist of interacting particles, whose dynamics can be described with the above circuit model. Then if A E is chosen large enough a typical process reaches decoupling after polynomial time except with small probability.
Finally, note that related results concerning the thermalization of subsystems have been derived in [23] and a generalization of these results using the decoupling approach has recently been proposed in [18].

Appendix A: Swap Operator
For a fixed orthonormal basis {|i } of some Hilbert space H, we introduce the swap operator F acting on the bipartite space H ⊗ H ′ , with H ′ a copy of H.
It is easy to verify that this operator satisfies the following equality.