Quantum de Finetti theorem under fully-one-way adaptive measurements

# Quantum de Finetti theorem under fully-one-way adaptive measurements

Ke Li IBM TJ Watson Research Center, Yorktown Heights, NY 10598, USA Center for Theoretic Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA    Graeme Smith IBM TJ Watson Research Center, Yorktown Heights, NY 10598, USA
July 10, 2019
###### Abstract

We prove a version of the quantum de Finetti theorem: permutation-invariant quantum states are well approximated as a probabilistic mixture of multi-fold product states. The approximation is measured by distinguishability under fully one-way LOCC (local operations and classical communication) measurements. Our result strengthens Brandão and Harrow’s de Finetti theorem where a kind of partially one-way LOCC measurements was used for measuring the approximation, with essentially the same error bound. As main applications, we show (i) a quasipolynomial-time algorithm which detects multipartite entanglement with amount larger than an arbitrarily small constant (measured with a variant of the relative entropy of entanglement), and (ii) a proof that in quantum Merlin-Arthur proof systems, polynomially many provers are not more powerful than a single prover when the verifier is restricted to one-way LOCC operations.

Consider random variables representing the color of a sequence of balls drawn without replacement from a bag of red balls and blue balls. These variables are not independent, since the probability of withdrawing a red ball on the th withdrawl depends on the number of balls of each color remaining. They are, however, exchangeable: the probability of removing a particular sequence of balls is equal to the probability of removing any reordering of that sequence for permuatation . Remarkably, the de Finetti theorem tells us that any such exchangeable random variables can be represented by independent and identically distributed ones deFinetti37 (); Diaconis-Freedman80 (), yeilding a profound result in probability theory and a powerful tool in statistics.

A series of works have established analogues of this theorem in the quantum domain Stormer69 (); Hudson-Moody76 (); Raggio-Werner89 (); CFS02 (); Konig-Renner05 (); CKMR07 (); Renner07 (); Brandao-Harrow12 (), where a classical probability distribution is replaced by a quantum state and the situation is more complicated and interesting, due to entanglement and the existence of many different ways to distinguish states of multipartite systems. These quantum de Finetti theorems are appealing not only due to their own elegance on the characterization of symmetric states, but also because of the successful applications in many-body physics Raggio-Werner89 (); Fannes-Vandenplas06 (); LNR13 (), quantum information Renner07 (); Brandao-Plenio-10 (); Christandl-Renner12 (), and computational complexity theory BSW11 (); BCY11 (); Brandao-Harrow12 ().

More precisely, a quantum de Finetti theorem concerns the structure of a symmetric state that is invariant under any permutations over the subsystems note0 (). It tells how the reduced state on a smaller number of subsystems could be approximated by a mixture of -fold product states, namely, de Finetti states of the form . Here is a probability measure over density matrices. Using the conventional distance measure, trace norm, Ref. CKMR07 () proved a standard de Finetti theorem with an essentially optimal error bound for the approximation ( denotes the dimension of the subsystems). However, in many situations this bound is too large to be applicable. Luckily it is possible to circumvent this obstruction. For example, Renner’s exponential de Finetti theorem employs the “almost” de Finetti states and has an error bound that decreases exponentially in  Renner07 (), being very useful in dealing with cryptography or information theory problems Renner07 (); Brandao-Plenio-10 (); Christandl-Renner12 ().

In a beautiful work Brandao-Harrow12 () Brandão and Harrow recently proved an LOCC (local operations and classical communication) de Finetti theorem, generalizing a similar result for the case  BCY11 (). Both Brandao-Harrow12 () and BCY11 () have overcome the limitation of the standard de Finetti theorem regarding the dimension dependence. The basic idea is to relax the measure of approximation by replacing the trace norm with a kind of one-way LOCC norm. This gives an error bound  note1 (), scaling polynomially in instead of polynomially in as in earlier de Finetti results, which is crucial to the complexity-theoretic applications.

While Brandao-Harrow12 () showed approximation in the parallel one-way LOCC norm associated with the measurement class , here we prove a de Finetti theorem where the approximation is measured with the fully one-way LOCC norm (or relative entropy) associated with (cf. Fig. 1). The error bound remains essentially the same as that of Brandao-Harrow12 (). This improves Brandão and Harrow’s LOCC de Finetti theorem considerably: it is conceptually more complete and when applied to the problems considered in BCY11 (); Brandao-Christandl11 (); Brandao-Harrow12 () gives new and improved results. For the problem of entanglement detection, so central to quantum information theory and experiment, we present strong guarantees for the effectiveness of the well-known heirarchy of entanglement tests of DPS (). We also consider the power of multiple-prover quantum Merlin Arthur games, which bears directly on the problems of pure-state vs mixed-state -representability LCV07 () as well as the entanglement properties of sparse hamiltonian’s ground states Chailloux-Sattath11 ().

Operational norms as distance measures. We identify every positive operator-valued measure with a measurement operation : for any state , with an orthonormal basis. For simplicity we call them both quantum measurement. Given a class of measurements , the operational norm is defined as MWW09 ()

 ∥ρ−σ∥M=maxM∈M∥M(ρ)−M(σ)∥1.

It measures the distinguishability of two quantum states under restricted classes of measurements. We will be particularly interested in and . Obviously the former is lower bounded by the latter, since . In fact, these two norms can differ substantially: using a recent result obtained in Aubrun-Lancien14 (), we can show for all there are constant and states and such that but (see Appendix).

Improved LOCC de Finetti theorem. Our main result is the following Theorem 1. Besides the improvement with the fully one-way LOCC norm, for the first time we employ relative entropy to measure the approximation, defining .

In the proof, we will use information-theoretic methods similar to Brandao-Harrow12 (), along with some new ideas. In particular, Lemma 2 presented below is a crucial new technical tool, which may be of independent interest. We employ and manipulate entropic quantities to derive the final result: apart from relative entropy, the mutual information of a state is defined as , and the conditional mutual information of a state is defined as .

###### Theorem 1

Let be a permutation-invariant state on . Then for integer there exists a probability measure on density matrices on such that

 DLOCC1(ρA1…Ak∥∥∫σ⊗kdμ(σ))≤(k−1)2log|A|n−k, (1) ∥∥∥ρA1…Ak−∫σ⊗kdμ(σ)∥∥∥LOCC1≤√2(k−1)2ln|A|n−k. (2)

Proof. Eq. (2) follows from Eq. (1) immediately by using the Pinsker’s inequality Fuchs-Graaf99 (), . So it suffices to prove Eq. (1).

Group the subsystems as shown in Fig. 2: except for one subsystem, the others are divided into groups of subsystems each (we discard the possibly remaining qubits, of which there will be fewer than ). So, we have groups. Label the groups as bigger subsystems and the isolated system as . Let the subsystems in be and the system is also identified with .

Obviously the total state is invariant under permutations over . So Lemma 3 applies. Thus there exists a measurement , such that for any measurement we have

 I(A;Y|X)≤log|A|m≤(k−1)log|A|n−k. (3)

effectively decomposes the state on into an ensemble. Specifically, we have , where is the probability of obtaining the measurement outcome and is the resulting state on . Note that since is permutation-invariant, the post-measurement states are also permutation-invariant. Now we rewrite Eq. (3) in terms of the relative entropy: for any measurement on ,

 ∑xpxD(P⊗idAk(ρxA1…Ak)∥∥P(ρxA1…A(k−1))⊗ρxAk) ≤(k−1)log|A|n−k. (4)

Pick a one-way LOCC measurement acting on systems and denote its reduced measurement on the first systems as . Now we apply Lemma 2 to each state and get

 D(Λk(ρxA1…Ak)∥∥Λk(ρxA1⊗…⊗ρxAk)) (5) ≤ k∑ℓ=2D(Λℓ−1⊗id(ρxA1…Aℓ)∥∥Λℓ−1(ρxA1…A(ℓ−1))⊗ρxAℓ) ≤ (k−1)D(Λk−1⊗id(ρxA1…Ak)∥∥Λk−1(ρxA1…A(k−1))⊗ρxAk)),

where for the first inequality we have also applied the monotonicity of relative entropy Lindblad-Uhlmann77 () and for the second inequality we used the monotonicity of relative entropy again as well as the symmetry of the state . Combining Eq. (4) and Eq. (5) we arrive at

 (6)

where the first inequality is due to the joint convexity of relative entropy. At this point we are able to conclude Eq. (1) from Eq. (6), noticing that is picked arbitrarily and is a de Finetti state of the form due to the symmetry of .

###### Lemma 2

Let be a one-way LOCC measurement on quantum systems . Denote its reduced measurement corresponding to the first steps on as . Then for any state we have

 D(Λk(ρA1…Ak)∥Λk(ρA1⊗…⊗ρAk))=k∑ℓ=2D(Λℓ(ρA1…Aℓ)∥Λℓ(ρA1…A(ℓ−1)⊗ρAℓ)).

Proof. It suffices to show

 D(Λk(ρA1…Ak)∥Λk(ρA1⊗…⊗ρAk))=D(Λk−1(ρA1…Ak−1)∥Λk−1(ρA1⊗…⊗ρAk−1))+D(Λk(ρA1…Ak)∥Λk(ρA1…Ak−1⊗ρAk)), (7)

because applying this relation recursively allows us to obtain the equation claimed in Lemma 2. Write and . Let be realized as follows. We first apply on . Then depending on the measurement outcome we apply a measurement on . Thus we can write

 Λk(ρA1…Ak) =∑xpx|x⟩⟨x|⊗Mx(ρxAk), Λk(ρA1…Ak−1⊗ρAk) =∑xpx|x⟩⟨x|⊗Mx(ρAk), Λk(ρA1⊗…⊗ρAk) =∑xqx|x⟩⟨x|⊗Mx(ρAk),

where is the state of when is applied on and outcome is obtained. With these, we can confirm by direct computation that

 (8)

and

 D(Λk(ρA1…Ak)∥Λk(ρA1…Ak−1⊗ρAk))=∑xpxD(Mx(ρxAk)∥Mx(ρAk)). (9)

Eq. (8) and Eq. (9) together lead to Eq. (7) and this concludes the proof.

Remark. The quantity is sometimes denoted as and called the multipartite mutual information. It is easy to see that . Using this repeatedly we can write the multipartite mutual information as a sum of bipartite mutual information quantities. This decomposition can be done in many different ways depending on how we split the subsystems. Lemma 2 is a similar result. However, with the one-way LOCC measurement , the decomposition only works for our special choice of splitting.

The following lemma, which is a statement of the monogamy of entanglement, is adapted from Brandao-Harrow12 (). For completeness we give a proof in the Appendix.

###### Lemma 3

Let be a state that is invariant under any permutation over . Let and be measurement operations performed on systems and , respectively. We have

 minQmaxPI(A;Y|X)idA⊗P⊗Q(ρAB1…Bm)≤log|A|m.

Applications. By replacing the (or Bell) measurements in Brandao-Harrow12 () with measurments from , we obtain a couple of interesting results as follows, for which technical proofs are given in the Appendix.

Detecting multipartite entanglement. Deciding whether a density matrix is entangled or separable is one of the most basic problem in quantum information theory, with both theoretical and practical significance Horodeckis07 (). Despite the existence of many entanglement criteria, up to date the only complete ones that detect all entangled states are infinite hierarchies Horodeckis07 (). Among them searching for symmetric extensions is probably the most useful DPS (). This is exactly the scenario where quantum de Finetti theorems could be expected to be useful.

We consider the situation where a small error is permitted, meaning that we must detect all the entangled states except for those very weak ones that are -close to separable (at the same time all the separable states should be detected correctly). This is equivalently formulated as the Weak Membership Problem for separability: given a state that is either separable or -away from any separable state, we want to decide which is the case. It has been shown that this problem is -hard when is of the order no larger than inverse polynomial of local dimensions (in trace norm) Gurvits03 (); Gharibian08 (); Beigi08 (). Surprisingly, Brandão, Christandl and Yard found a quasipolynomial-time algorithm for constant in one-way LOCC norm for bipartite states BCY11 (). This algorithm was generalized to multipartite states in Brandao-Christandl11 (), then in Brandao-Harrow12 () using a stronger method. These algorithms are all based on the searching for symmetric extensions of DPS (). Along these lines, we present the following result, which is obtained by applying Theorem 1 to bound the distance between properly extendible states and separable states.

###### Corollary 4

Testing multipartite entanglement of a state with error can be done via searching for symmetric extensions in time

 exp⎛⎝c(k∑i=1log|Ai|)2k2f(ϵ)⎞⎠, (10)

where if the error is measured by the norm and if it is measured by the relative entropy .

It is worth mentioning that the run time in Eq. (10) is quasipolynomial, for constant particle number and constant error . The algorithm in Brandao-Christandl11 () using -norm behaves exponentially slower than ours with respect to the number of particles , while the algorithm of Brandao-Harrow12 () has the same run-time as ours but works only for -norm rather than our -norm approximation. Thus our result has bridged the gap between these two works. Furthermore, here for the first time we catch the importance of the amount of entanglement in this problem. The quantity , introduced in Piani09 (), is asymptotically normalized since for maximally entangled state of local dimension  Li-Winter14 (). Corollary 4 shows that, detecting all the -partite entangled states such that can be done in quasi-polynomial time in local dimensions. This is a stronger statement than using -norm as the error measure. We point out that for the bipartite case this result can also be obtained by combining the algorithm of BCY11 () with the “commensurate lower bound” for squashed entanglement of Li-Winter14 ().

QMA proof system with multiple proofs. , the quantum analogue of the complexity class , is the set of decision problems whose solutions can be efficiently verified on a quantum computer, provided with a polynomial-size quantum proof Watrous08 (). In recent years there have been significant advances on the structure of QMA systems, where multiple unentangled proofs and possibly locally restricted measurements in the verification were considered KMT03 (); ABDFS08 (); Harrow-Montanaro10 (); BCY11 (); Brandao-Harrow12 (). It has been proven that many natural problems in quantum physics are characterized by QMA proof systems (see, e.g., QMA-physics (); LCV07 (); Chailloux-Sattath11 (); GHMW13 ()).

To solve a problem, the verifier performs a quantum algorithm on the input along with the quantum proofs. The algorithm then returns “yes” or “no” as the answer to the instance . This procedure of verification can be effectively described as a set of two-outcome measurements on the proofs. In the definition below, a problem is formally identified with a “language”.

###### Definition 5

A language is in if there exists a polynomial-time implementable verification with each measurement from the class such that

• Completeness: If , there exist states as proofs , each of size qubits, such that

 Tr(Mx(ω1⊗…⊗ωk))≥c.
• Soundness: If , then for any ,

 Tr(Mx(ω1⊗…⊗ωk))≤s.

We are also interested in QMA systems with multiple symmetric proofs. is defined in a similar way but here we replace independent proofs with identical ones in both completeness and soundness parts. As a convention, we set to be (the class of all measurements), , , and as defaults note2 (). We can now state our application of Theorem 1 to these complexity classes.

###### Corollary 6

We have

 QMA=QMALOCC1(poly)=SymQMALOCC1(poly). (11)

In particular,

 (12) QMALOCC1(k)m,c,s⊆QMA0.6m2k4ϵ−2,c,s+ϵ (13)

It has been proven in BCY11 () that for constant . Our result generalizes this statement to a polynomial number of proofs. It is also a generalization of the results in Brandao-thesis (); Brandao-Harrow12 () which prove the reduction of to ( denotes local measurements). On the other hand, Ref. Brandao-Harrow12 () proved that, assuming ETH (exponential time hypothesis for 3-IPZ98 (), any multi-prover QMA protocol with symmetric proofs and Bell verification for 3-, can not bring better than the square-root reduction of Chen-Drucker10 () to the proof size. Eq. (12) implies that, this is still true even if adaptively local verification (one-way LOCC measurement) is permitted.

Arguably the biggest open question in the study of QMA proof systems is whether (note that Harrow and Montanaro have proved that for any polynomial  Harrow-Montanaro10 ()). On the one hand, there are natural problems from quantum physics that are in but not obviously in  LCV07 (); Chailloux-Sattath11 (); GHMW13 (). On the other hand, Harrow and Montanaro showed that if the first equality in Eq. (11) holds for a kind of separable measurements (even only for the case of two proofs), then . Our result here, although does not touch this open question directly, is a step towards a larger measurement class compared to Brandao-Harrow12 () and we hope it will stimulate future progress in solving this open question.

Polynomial optimization over hyperspheres. Theorem 1 also gives some improved results on the usefulness of a general relaxation method, called the Sum-of-Squares (SOS) hierarchy Lasserre01 (); Parrilo00 (), for polynomial optimization over hyperspheres (see, e.g., Brandao-Harrow12 (); BKS14 ()). The relevance in physics is that pure states of a quantum system form exactly a hypersphere and hence some computational problems in quantum physics are indeed to optimize a polynomial over hyperspheres. See Appendix for the details.

Discussions. The advantage of our method, inherited from Brandao-Harrow12 (), is that it tells us more information than that of BCY11 (); Li-Winter14 () about the valid de Finetti (separable) state that approximates the symmetric (extendible) state. As a result, we obtain a huge improvement over Brandao-Christandl11 () on the particle-number dependence, and we are able to strengthen the relation from the constant of BCY11 () to polynomial . We hope that the de Finetti theorem presented in this letter will find more applications in the future.

We ask whether Theorem 1 can be further improved, to work for two-way LOCC or even separable measurements. This would accordingly give stronger applications, and possibly, solve the vs puzzle due to the result of Harrow-Montanaro10 (). Another open question is, for a state supported on the symmetric subspace (aka Bose-symmetric state), whether its reduced states have pure-state approximations of the form with pure that are not worse than the mixed-state approximations given by our theorem. We notice that this is indeed the case for the de Finetti theorem of CKMR07 () and a similar statement holds for Renner07 (). However, our method, as well as that of Brandao-Harrow12 () seems to require that the state must be generally mixed.

Acknowledgements. KL is supported by NSF Grant CCF-1110941 and CCF-1111382. GS acknowledges NSF Grant CCF-1110941. We thank Charles Bennett, Fernando Brandão, Aram Harrow and John Smolin for interesting discussions, and the anonymous referees for helping improve the manuscript.

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Appendix

Inequivalence of and . Here we show the following: for all there are constant and states and such that but .

To see this, notice that for states of the form and we have

 ∥ρABC−σABC∥LOCC1=∥ρAB−σAB∥LOCC1, ∥ρABC−σABC∥LOCC∥1=∥ρAB−σAB∥LO,

where denotes the set of local measurements. We can then apply the existence of bipartite states with and as shown in Theorem 2.4 of Aubrun-Lancien14 ().

Proof of Lemma 3. Let , with being measurement operations. Due to the chain rule of mutual information,

 I(A;Z1…Zm)=I(A;Z1)+I(A;Z2|Z1)+…+I(A;Zm|Z1…Zm−1). (14)

Now we fix a special choice of . Let maximize . Then under this choice of , we choose that maximizes . Repeat this procedure and at last we pick that maximizes under the previously fixed measurements . As a result, for each conditional mutual information we have

 I(A;Zℓ|Z1…Zℓ−1)≥minQ′maxP′I(A;Yℓ|Xℓ)idA⊗P′⊗Q′(ρAB1…Bℓ), (15)

where and are measurement operations. Further relax the minimization to allow the measurement to be performed on all the systems except for . Then we can set to be without changing the value due to the symmetry of the state. Thus Eq. (15) is further lower bounded by

 minQmaxPI(A;Y|X)idA⊗P⊗Q(ρAB1…Bm)

with measurement operations and . This, combined with Eq. (14) lets us conclude that

 log|A|≥I(A;Z1…Zm)≥mminQmaxPI(A;Y|X)idA⊗P⊗Q(ρAB1…Bm),

and we are done.

Proof of Corollary 4. We first prove this result with the error measured by the fully one-way LOCC norm. Then we explain that a slight adaptation works for the case of relative entropy.

Let be an integer. Introduce quantum systems with , for all and . We search for an state such that is permutation-invariant and . If such a state exists, we feed back “separable”. Otherwise we conclude that it is entangled. For our purpose we set .

This search can be done using semidefinite programming in time polynomial of the total dimension of , which coincides with the claimed result.

To analyze the correctness, first assume that such an extension exists. Then we apply Theorem 1 to see that there is certain probability measure such that

 ∥∥∥~ρ¯A1…¯Ak−∫σ⊗kdμ(σ)∥∥∥LOCC1≤ϵ.

By definition, if we restrict the measurement to be performed only on systems then the above inequality implies that there exists a separable state such that . So if is -away from any separable state, the required extension can not exist. But if is separable, it is obvious that such an extension does exist. As a result in both cases the above procedure works correctly.

The above argument works as well if the error is measured by the relative entropy. The small modification needed is just to replace by and here we set .

Closeness of extendible states to being separable. We also show how an extendible multipartite state is close to the set of separable states, under fully one-way LOCC distinguishability. A state is -extendible if there is an extension with , such that for all the state is invariant under any permutations over subsystems and for any we have . Obviously is permutation-invariant and . So similar to the argument in the proof of Corollary 4, a use of Theorem 1 lets us obtain:

 ELOCC1r(ρA1…Ak) ≤(k−1)2∑ilog|Ai|ℓ−k, minσ∈SEP∥∥ρA1…Ak−σA1…Ak∥∥LOCC1 ≤√2(k−1)2∑iln|Ai|ℓ−k

holds for any -extendible state . Here denotes the set of all separable states.

Proof of Corollary 6. Restricting the verification to be performed on the first proof in the multi-prover protocols, we see that

 QMAm,c,s⊆SymQMALOCC1(k)m,c,s, (16) QMAm,c,s⊆QMALOCC1(k)m,c,s. (17)

By definition, Eq. (16) and Eq. (12) imply . Similarly, Eq. (17) and Eq. (13) imply . Note that we can use the amplification of (see Marriott-Watrous05 ()) to keep and .

To prove Eq. (12), we show a way of simulating a protocol in a single-proof system. The prover provides the verifier with a proof of size , which consists of subsystems each of size qubits. Then the verifier makes a uniformly random permutation over the subsystems and then performs the verification on the first (denoted as ) of them. No matter what the initial state of the proof is, Theorem 1 implies that the state on , , can be approximated as with certain probability measure . Let be the one-way LOCC measurements in the protocol for a language . Then the soundness constant in this simulation can be

 s′=maxx∉LmaxρinitialTrMxρA1…Ak≤maxx∉Lmaxμ∫dμ(σ)TrMxσ⊗k+ϵ≤s+ϵ.

One the other hand, suppose the proof for an accepted instance in is . Then in the simulation the state gives the same probability of acceptance. So completeness does not change.

For Eq. (13), we will prove

 QMALOCC1(k)m,c,s⊆SymQMALOCC1(k)km,c,s.

This, together with Eq. (12), leads to Eq. (13). The argument is similar to that in ABDFS08 () (Lemma 38), where the same relation with “” replaced by “” was proved. The strategy is to divide each proof in the system into subsystems of qubits, then simulate the protocol on the subsystem from the proof for all .

Polynomial optimization over hyperspheres. An immediate consequence of Theorem 1 is that we can enlarge in Brandao-Harrow12 () the class of polynomials, for which the optimization over multiple hyperspheres admits efficient SOS approximation. We also provide another class of polynomials whose optimization over the single hypersphere has a similar feature, supplementing a result of BKS14 () on polynomials with nonnegative coefficients.

We use a -dimensional complex vector to encode real variables.

###### Corollary 7

For , let and be identical quantum systems of dimension . Let and be complex vectors. Let be a matrix on such that . The two optimizations

 (O1:) max ⟨α1|⊗…⊗⟨αk|M|α1⟩⊗…⊗|αk⟩ \textupsubjectto ⟨αi|αi⟩=1,i=1,…,k (O2:) max ⟨β|⊗k(\openone⊗M)|β⟩⊗k\textupsubjectto⟨β|β⟩=1

can be solved to within additive error efficiently, via a hierarchy of SDP relaxations (SOS), respectively in time and .

The advantage of Corollary 7 is that, for constant and , the runtime of these two optimizations is only quasi-polynomial of the number of variables, instead of exponential time of exhaustive search.

Proof. The analysis of is the same as that of the polynomial-optimization problem considered in Brandao-Harrow12 (). We only need to employ our Theorem 1 when the de Finetti theorem is used.

It is easy to see that the maximum in equals

 maxσTrMσ⊗k,

with a normalized quantum state. This, in turn, is bounded as

 maxρA1…AℓTr(M⊗\openone)ρA1…Aℓ−√(k−1)2lnd2(ℓ−k)≤ maxσTrMσ⊗k≤maxρA1…AℓTr(M⊗\openone)ρA1…Aℓ,

where and the maximization in the first and last lines are over permutation-invariant state . Note that here the first inequality follows from a direct application of Theorem 1, and the second inequality is by restricting the maximization in the last line to over -fold states of the form . So the problem can be approximated to within additive error , by the lever- SDP hierarchy (SOS hierarchy)

 \textupmaximizeTr(M⊗\openone)ρA1…Aℓ\textupsubjecttoρA1…Aℓ≥0,TrρA1…Aℓ=1,ρA1…Aℓbeing permutation-invariant.

This can be done in time , namely, polynomial of the dimension of . At last, to obtain the claimed result, we choose .

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