Computing Quantum Discord Is NP-Complete (Theorem 2)

Home , Computational problem, PH (complexity), Separable state, TC (complexity)

arXiv:1305.5941v3 [quant-ph] 28 Mar 2014 hoy Pcmlt rbesaetehreti Pi h sen the in NP in hardest the are problems NP-complete theory. 5 ,1,2,5,5,6,7].Gnrly unu icr ca discord quantum Generally, 76]). 60, 52, 51, 29, 19, 6, [5, ffr,fwaayia eut r nw vnfr‘ipe and ‘simple’ for even known are di quantum results computing analytical Nevertheless, few correlati effort, [66]. quantum years of few tel measures past quantum the related [63], (and coding discord superdense quantum st 80], quantum [63, [26], algori distillation distillation randomness classical common over in speed-up entanglem concept quantum beyond the correlation for quantum of responsible measure (a 69] entanglement [39, little with w computation computation quantum quantum mixed-state deterministic exi [44, instance, also For measures correlation states. entanglement quantum Various nontrivial However, 42]. glement. [11], [41, coding c merging superdense and as state such I operations 4 tasks local correlation. 31, enabling quantum of processing, [21, of notion manifestation nonlocality the prominent quantum most on the based 75], defined 49, 73], 48, [44, not 12, do mechanics [7, quantum in relations concepts fundamental few a Quite Introduction 1 h oino Pcmltns 2]i udmna n remar and fundamental is [22] NP-completeness of notion The ttsaddtcigcasclsae nacne e,poiigev providing intractable. set, computationally convex generically a is linea in states problems: states sical typical classical two infor detecting of and quantum NP-completeness states into the insights prove significant we offer over, may super they distillation, applic teleportation; entanglement directly merging, a are state capacity results quantum distillation, complexity-theoretic Holevo These constrained and compute. to information) info mutual mutual of of sq larization entanglement entanglement, conditional of entanglement, entropy measure squashed relative entanglement syste t some formation, quantum with by-products, of a entanglement As in exponentially discord practice. grow quantum in to computing possible that believed so is space ru discord Hilbert the quantum quantu intractable: computing computing computationally for is that discord prove quantum and Therefore, entanglement), beyond relation esuytecmuainlcmlxt fqatmdsod( meas (a discord quantum of complexity computational the study We eateto hsc,Uiest fClfri,Bree,C 9472 CA Berkeley, California, of University Physics, of Department optn unu icr sNP-complete is discord quantum Computing ac 1 2014 31, March ihnHuang Yichen Abstract 1 prain[3,ec n a established has and etc, [63], eportation hs unu icr sas useful a also is discord Quantum thms. unu eeotto 9 n quantum and [9] teleportation quantum t ncransprbe(o entangled) (not separable certain in sts ti rud[5 htqatmdiscord quantum that [25] argued is It . t n ui DC)[6 samdlof model a is [56] (DQC1) qubit one ith sarsuc nqatminformation quantum in resource a is t nyb optdnumerically. computed be only n t egn 1,6,6] entanglement 63], 62, [17, merging ate cr sdffiut ept considerable Despite difficult. is scord sflsae eg two-qubit (e.g. states useful n;sescin3frisdfiiin is definition) its for 3 section see ent; ,7] t.Qatmentanglement Quantum etc. 73], 4, n sa ciersac oi over topic research active an as on) 3 r eotdt uniyentan- quantify to reported are 73] eta necetagrtmfrany for algorithm efficient an that se asclcmuiain(OC,is (LOCC), communication lassical aecasclaaos uncertainty analogs: classical have al ncmuainlcomplexity computational in kable ahdetnlmn,classical entanglement, uashed dneta okn ihclas- with working that idence mto,adbodatregu- broadcast and rmation, nml nageetcost, entanglement (namely s bei omnrandomness common in able nn ieo n algorithm any of time nning piiainoe classical over optimization r icr sNP-complete. is discord m es oig n quantum and coding, dense ainpoesn.More- processing. mation eNP-hard/NP-complete re fmdrt iei not is size moderate of m r fqatmcor- quantum of ure edmnino the of dimension he ,USA 0, X states NP-complete problem implies efficient algorithms for all problems in NP, and NP-hard problems are at least as hard as NP-complete problems. An NP-hard/NP-complete problem is computationally intractable: the running time of any algorithm for the problem is believed to grow exponentially with the input size. The NP-completeness of the separability problem (detecting whether a given state is separable) was first proved in [35, 36]; see [32, 55] for technical improvements. This may be the reason why a lot of effort is devoted to entanglement criteria [30, 34, 43, 44, 46, 47, 50, 70, 79], which are simple sufficient conditions for entanglement. The classicality problem (detecting whether a given state has zero quantum discord) can be solved in polynomial time [18, 23, 45], but the computational complexity of quantum discord is not known. Here we prove that computing quantum discord is NP-complete (theorem 2). Therefore, the running time of any algorithm for computing quantum discord is believed to grow exponentially with the dimension of the Hilbert space, so that computing quantum discord in a quantum system of moderate size is not possible in practice. As by-products, some entanglement measures (namely entanglement cost [10], entanglement of formation [10], relative entropy of entanglement [84], squashed entanglement [20], classical squashed entanglement, conditional entanglement of mutual information [89], and broadcast regularization of mutual information [72]; theorem 1) and constrained Holevo capacity [78] (corollary 1) are NP-hard/NP-complete to compute. As direct applications (one-way), distillable common randomness, regularized one-way classical deficit, entanglement consumption in extended quantum state merging, and minimum loss due to decoherence of the yield of a family of protocols are also NP-hard/NP-complete to compute; such complexity- theoretic results may offer significant insights into quantum information processing. Moreover, we prove the NP-completeness of the following two typical problems: linear optimization over classical states (proposition 1) and detecting whether there are classical states in a given convex set (proposition 2). The former is the simplest optimization problem over classical states, and the latter is just one step further than the classicality problem. Conceptually, the NP-completeness of these two problems provides evidence that working with classical states is generically computationally intractable. We conclude with some interesting open problems and research directions.

2 NP-hardness/NP-completeness of computing entanglement measures

Let us briefly recall the definitions of some entanglement measures (see the review papers [44, 73] for details). Entanglement cost EC (ρ) [10] is the minimum rate j/k to convert j copies of the two-qubit maximally entangled state ψ = ( 00 + 11 )/√2 to k copies of the bipartite state ρ by LOCC with | i | i | i vanishing error in the asymptotic limit j, k + . Conversely, distillable entanglement ED(ρ) [10] → ∞ is the maximum rate j/k to convert ρ⊗k to ψ ⊗j by LOCC with vanishing error in the asymptotic | i limit. Entanglement of formation [10] is defined as

i EF (ρAB)= inf X piS(ρA), (1) {pi,|ψii} i where the inﬁmum is taken over all ensembles of pure states pi, ψi satisfying ρAB = pi ψi ψi , { | i} Pi | ih | and S(ρi )= tr(ρi log ρi ) (2) A − A A i is the von Neumann entropy of the reduced density matrix ρ = trB ψi ψi or the entanglement A | ih | entropy of ψi . The relative entropy of entanglement [84] | i ER(ρ)= inf S(ρ σ)= inf tr(ρ log ρ ρ log σ) (3) σ∈S k σ∈S −

2 quantiﬁes the distance from the state ρ to the set of all separable states, where S(ρ σ) is the S k quantum relative entropy. The regularized relative entropy of entanglement is given by

∞ ⊗k ER (ρ) = lim ER(ρ )/k. (4) k→+∞ Squashed entanglement [20] is defined as 1 Esq(ρAB)= inf S(ρAC )+ S(ρBC ) S(ρC ) S(ρABC ) , (5) 2 ρABC { − − } where the infimum is taken over all states ρABC in an extended Hilbert space satisfying ρAB = C trC ρABC , and ρAC = trBρABC , etc. Classical squashed entanglement Esq(ρAB) is given by (5) where the infimum is taken with the additional restriction that ρAB|C is quantum-classical (24) across the cut AB C. Conditional entanglement of mutual information [89] is defined as | 1 EI (ρAB)= inf I(ρAA′|BB′ ) I(ρA′B′ ) , (6) 2 ρAA′BB′ { − } where the infimum is taken over all states ρAA′BB′ satisfying ρAB = trA′B′ ρAA′BB′ , and

I(ρA′B′ )= S(ρA′ )+ S(ρB′ ) S(ρA′B′ ) (7) − ⊗k k is the quantum mutual information. A state ρ ⊗k with X = Xi is a k-copy broadcast state X ⊗i=1 k of ρX if ρX = trX1X2···Xi−1Xi+1···Xk ρX⊗ for any i = 1, 2,...,k. Broadcast regularization of mutual information [72] is given by

∞ 1 Ib (ρAB)= lim inf I(ρA⊗k B⊗k )/k, (8) k k | 2 k→+∞ ρA⊗ |B⊗ where the inﬁmum is taken over all k-copy broadcast states of ρAB.

Lemma 1. (a) The deﬁnition (1) of EF remains the same if the number of states in the ensemble is restricted to be less than or equal to m2n2 [67, 81], where m n is the dimension of the bipartite × state ρAB. ∞ ⊗k (b) EC (ρ)= EF (ρ) = limk→+∞ EF (ρ )/k [38]. C ∞ (c) EF (ρ) ER(ρ) [83], EC (ρ) Esq(ρ) [20], Esq(ρ) Ib (ρ) EI (ρ) Esq(ρ) [72]. ∞ ≥ 2 ≥ ≥ ≥ ≥2 (d) E (ρ) infσ ρ σ /(2mn log 2) [71], Esq(ρ) infσ ρ σ /(2448 log 2) [14, 15, R ≥ ∈S k − k1 ≥ ∈S k − k2 65], where X 1 = tr√X†X and X 2 = √trX†X are the trace norm and the Frobenius norm, k k k k respectively. Accounting for the ﬁnite precision of numerical computing, hereafter, every real number is as- sumed to be represented by a polynomial number of bits, and the formulation of each computational problem is approximate. Indeed, we will prove that the problems are computationally intractable even if small errors are allowed. We begin by recalling the following lemma. Lemma 2 (NP-completeness of the separability problem). Given a bipartite quantum state ρ of dimension m n with the promise that either (Y) ρ or (N) infσ ρ σ 2 δ, it is NP- × ∈ S ∈S k − k ≥ complete to decide which is the case, where δ = 1/poly(m,n) is some inverse polynomial in m,n.

Remark 1. The NP-completeness of the separability problem with δ = exp( O(m,n)) is proven − in [55], and the NP-hardness of the separability problem with δ = 1/poly(m,n) is proven in [32]. The separability problem can be solved in exp(O((log m)(log n)/δ2)) time (a quasi-polynomial-time algorithm for δ = 1/poly(log m, log n)) [16].

3 Theorem 1 (NP-hardness/NP-completeness of computing entanglement measures). Given a bipartite quantum state ρ of dimension m n and a real number a with the promise that either × (Y) EF (ρ) a or (N) EF (ρ) a + ǫ, it is NP-complete to decide which is the case, where ǫ = ≤ ≥ ∞ C ∞ 1/poly(m,n). In the same sense, computing ER is NP-complete and computing EC , ER , Esq, Esq, EI ,Ib is NP-hard.

Proof. Computing EF , ER is in NP: the certiﬁcates of the yes instances (Y) are the optimal ensemble of pure states pi, ψi and the closest separable state σ, respectively. The NP-hardness { | i} of computing entanglement measures is totally expected, as computing entanglement measures is more diﬃcult than just detecting entanglement. Indeed, the hardness proof is a reduction from lemma 2. Set a = 0 and ǫ = δ2/(2448mn log 2) = 1/poly(m,n). (Y) If ρ is separable, then

∞ C ∞ EC (ρ)= EF (ρ)= ER(ρ)= ER (ρ)= Esq(ρ)= Esq(ρ)= EI (ρ)= Ib (ρ) = 0. (9)

(N) If infσ ρ σ 2 δ, then ∈S k − k ≥ ∞ C ∞ EF (ρ) ER(ρ) E (ρ), EF (ρ) EC (ρ) Esq(ρ), E (ρ) I (ρ) EI (ρ) Esq(ρ)(10), ≥ ≥ R ≥ ≥ sq ≥ b ≥ ≥ ∞ 2 2 2 ER (ρ) inf ρ σ 1/(2mn log 2) inf ρ σ 2/(2mn log 2) δ /(2mn log 2) ǫ, (11) ≥ σ∈S k − k ≥ σ∈S k − k ≥ ≥ 2 2 Esq(ρ) inf ρ σ 2/(2448 log 2) δ /(2448 log 2) ǫ. (12) ≥ σ∈S k − k ≥ ≥

Remark 2. The computational problem in theorem 1 requires a guess of EF (ρ) as an input. This formulation is reasonable: if there is an eﬃcient subroutine for the problem, a binary search for EF (ρ) can be done by calling the subroutine O(log(log(mn)/ǫ)) = O(log m, log n) times to achieve the precision ǫ = 1/poly(m,n). The hardness proof does not apply to ED, as ED(ρ) can be zero ∞ C ∞ for an entangled state ρ. It is an open problem whether computing EC , ER , Esq, Esq, EI ,Ib is in NP. For instance, it is not clear how large the dimension of ρABC should be so that the right-hand side of (5) is optimal (or suﬃciently close to optimal).

3 NP-completeness of computing quantum discord

As a measure of quantum correlation (beyond entanglement), quantum discord [69]

D(ρAB B)= I(ρAB) J(ρAB B) (13) | − | is the difference between total correlation (quantified by quantum mutual information) and classical correlation [39] i J(ρAB B)= S(ρA) inf X piS(ρA), (14) {Πi} | − i where Πi is a measurement on the subsystem B; pi = tr(ρABΠi) is the probability of the ith { } i measurement outcome; ρA = trB(ρABΠi)/pi is the post-measurement state. The infimum is taken over either all von Neumann measurements or all generalized measurements described by positive- operator valued measures (POVM); the corresponding notations are JN , DN and JP , DP , respectively. See [68] for an introduction to von Neumann measurements and POVM measurements. The definitions of JP , DP remain the same if the number of operators in the POVM is restricted to be less than or equal to n2, where n is the dimension of the subsystem B. This is because the

4 optimal POVM must be extremal [37], and an extremal POVM contains at most n2 operators [24]. Regularized classical correlation and quantum discord are given by

∞ ⊗k ⊗k ∞ ∞ ⊗k ⊗k J (ρAB B) = lim J(ρAB B )/k, D (ρAB B)= I(ρAB) J (ρAB B) = lim D(ρAB B )/k. | k→+∞ | | − | k→+∞ | (15)

Theorem 2 (NP-completeness of computing quantum discord). Given a bipartite quantum state ρAB of dimension m n and a real number b with the promise that either (Y) DP (ρAB B) b or × | ≤ (N) DP (ρAB B) b + ǫ, it is NP-complete to decide which is the case, where ǫ = 1/poly(m,n). In | ≥ ∞ ∞ the same sense, computing DN , JN,P is NP-complete and computing DN,P , JN,P is NP-hard.

Proof. Computing DN,P is in NP: the certificates of (Y) are the optimal measurements Πi on the { } subsystem B. The hardness proof is basically a reduction from theorem 1 via the Koashi-Winter relation [57] between EF and DP , and technically we derive a similar relation between EF and DN (note that the Koashi-Winter relation is between EF and DP rather than between EF and DN ). Given a bipartite state ρAB of dimension m n, by diagonalizing ρAB we construct a tripartite 2 2× pure state ΨABC of dimension m n m n satisfying ρAB = trC ΨABC ΨABC (note that such | i × × | ih | a tripartite pure state of dimension m n mn exists, but a larger dimension of the subsystem × × C will be useful later). (i) A POVM measurement Πi on C produces an ensemble pi, ρi { } { } satisfying ρAB = piρi, where pi = tr( ΨABC ΨABC Πi) and ρi = trC ( ΨABC ΨABC Πi)/pi. Pi | ih | | ih | (ii) For any ensemble pi, ρi satisfying ρAB = piρi, a POVM measurement Πi exists on C { } Pi { } such that pi = tr( ΨABC ΨABC Πi) and ρi = trC ( ΨABC ΨABC Πi)/pi; moreover, such a von | ih | | ih | Neumann measurement on C exists if the dimension of C is greater than or equal to the number of states in the ensemble [53] (this condition is satisfied due to lemma 1(a)). As the definition (1) of EF remains the same if the infimum is taken over all ensembles of possibly mixed states pi, ρi { } satisfying ρAB = Pi piρi, the relation

EF (ρAB)= DN,P (ρBC C)+ S(ρA) S(ρAB) (16) | − follows immediately from the deﬁnitions of EF and DN,P (note that in the present case DN = DP , though generically DN = DP ). Set b = a S(ρA)+ S(ρAB). We complete the reduction from EF 6 − to DN,P by taking ρBC as the input to the computational problem in theorem 2. The regularized relation ∞ EC (ρAB)= D (ρBC C)+ S(ρA) S(ρAB). (17) N,P | − ∞ implies a reduction from EC to DN,P . These reductions are polynomial-time reductions.

4 NP-completeness of computing constrained Holevo capacity

A quantum channel Φ is a completely positive trace-preserving linear map [68] from states of dimension ni to states of dimension no. The constrained Holevo capacity [78] is deﬁned as

χΦ(ρ)= S(Φ(ρ)) inf X piS(Φ( ψi ψi )), (18) {pi,|ψii} − i | ih | where the inﬁmum is taken over all ensembles of pure states pi, ψi satisfying ρ = pi ψi ψi . { | i} Pi | ih | The deﬁnition (18) of χΦ(ρ) remains the same if the number of states in the ensemble is restricted 2 to be less than or equal to ni [68]. The regularized constrained Holevo capacity is given by

∞ ⊗k χΦ (ρ) = lim χΦ⊗k (ρ )/k. (19) k→+∞

5 The Holevo capacity χΦ = supρ χΦ(ρ) is the maximum rate at which classical information can be transmitted through the quantum channel Φ using product states as codewords [40, 77]. The regularized Holevo capacity is given by

∞ ∞ χΦ = lim χΦ⊗k /k = sup χΦ (ρ). (20) k→+∞ 6 ρ

Corollary 1 (NP-completeness of computing constrained Holevo capacity). Given a quantum channel Φ, a quantum state ρ of dimension ni, and a real number c with the promise that either (Y) χΦ(ρ) c or (N) χΦ(ρ) c ǫ, it is NP-complete to decide which is the case, where ǫ = ≥ ≤ − ∞ 1/poly(ni,no). In the same sense, computing χΦ (ρ) is NP-hard.

Proof. Computing χΦ(ρ) is in NP: the certiﬁcate of (Y) is the optimal ensemble of pure states pi, ψi . The hardness proof is a reduction from theorem 1 via the relation [64, 78] between EF { | i} and χΦ(ρ). Given a bipartite state σAB of dimension m n, let U be a unitary embedding such × that σAB = U(ρ) for a state ρ of dimension rank(σAB). The quantum channel Φ is deﬁned as ′ ′ Φ(ρ )=trBU(ρ ), where ni = rank(σAB)= O(mn) and no = m. Then

EF (σAB)= S(Φ(ρ)) χΦ(ρ). (21) −

Set c = S(Φ(ρ)) a. We complete the reduction from EF (σAB) to χΦ(ρ). The regularized relation − ∞ EC (σAB)= S(Φ(ρ)) χΦ (ρ) (22) − ∞ implies a reduction from EC (σAB) to χΦ (ρ). These reductions are polynomial-time reductions. Lemma 3 (NP-completeness of computing Holevo capacity). Given a quantum channel Φ and a real number c with the promise that either (Y) χΦ c or (N) χΦ c ǫ, it is NP-complete to ≥ ≤ − decide which is the case, where ǫ = 1/poly(ni,no). Remark 3. This is one of the main results of [8], in which, however, the scaling of ǫ is not discussed. Indeed, additional work is needed to establish the NP-completeness of computing χΦ with ǫ = 1/poly(ni,no). We are not going to present the complete proof here. The computational complexity ∞ of χ remains an open problem. The set of all states of dimension ni is convex, and χΦ(ρ) is a Φ − convex function as S(Φ(ρ)) is concave and the inﬁmum in (18) is convex. Thus an alternative proof of the NP-hardness of computing χΦ(ρ) is a polynomial-time reduction from lemma 3 via convex 1 optimization [13]; moreover, the NP-hardness of computing EF can be proved based on lemma 3.

5 Applications

Common randomness is a resource in information theory and cryptography [3, 4]. One-way distillable common randomness Dcr(ρAB B) is the maximum rate at which common randomness can be | extracted from the bipartite state ρAB by local operations and one-way classical communication in ∞ the asymptotic limit. It is equal to regularized classical correlation J (ρAB B) [26], and also equal P | to regularized one-way classical deﬁcit [27]. Thus Dcr and regularized one-way classical deﬁcit are NP-hard to compute. In quantum state merging, Alice and Bob share a bipartite state, and the goal is to transfer Alice’s part of the state to Bob by entanglement-assisted LOCC [41, 42]. The minimum amount of entanglement that must be consumed in extended quantum state merging (a variant of quantum

1Mark M Wilde, private communication.

6 state merging) is an operational interpretation of quantum discord [17], and thus NP-complete to compute. Quantum discord quantifies the effect of decoherence in a family of protocols. It is the minimum difference between the yield of the fully quantum Slepian-Wolf (FQSW) protocol [1] in the presence and absence of decoherence [63]. The same holds for all descendant protocols of FQSW, where ‘yield’ refers to the amount of entanglement consumed in quantum state merging [62], the amount of distilled entanglement in entanglement distillation [80], the amount of classical information encoded in superdense coding, and the number of teleported qubits in quantum teleportation (see [63] for details). Thus computing the minimum loss due to decoherence of the yield of all aforementioned protocols is NP-complete.

6 Computational complexity of classical states

A bipartite state ρAB is separable if it can be expressed as

A A B B ρAB = X pi ψi ψi ψi ψi , (23) i | ih | ⊗ | ih |

A B where ψ , ψ are pure states in the subsystems A, B, respectively, and pi 0 satisﬁes pi = 1. | i i | i i ≥ Pi ρAB is quantum-classical if A B ρAB = X piρi Πi , (24) i ⊗ where ρA’s are normalized, possibly mixed states in A, and ΠB is a von Neumann measurement i { i } on B. ρAB is quantum-classical if and only if D(ρAB B) = 0 [69] (note that DN (ρAB)= 0 if and | only if DP (ρAB) = 0). ρAB is classical-classical if

A B ρAB = X pijΠi Πj , (25) i,j ⊗ where pij 0 satisﬁes pij = 1. ≥ Pi,j Lemma 4 (NP-completeness of linear optimization over separable states [55]). Given an operator O on a bipartite Hilbert space of dimension m n and a real number d with the promise that either × (Y) maxρAB tr(ρABO) d or (N) maxρAB tr(ρABO) d ε, it is NP-complete to decide which ∈S ≥ ∈S ≤ − is the case, where ε = 1/poly(m,n).

Let ( ) be the set of all quantum-classical (classical-classical) states. QC CC Proposition 1 (NP-completeness of linear optimization over classical states). Given an operator O on a bipartite Hilbert space of dimension m n and a real number d with the promise that either × (Y) maxρAB tr(ρABO) d or (N) maxρAB tr(ρABO) d ε, it is NP-complete to decide ∈CC ≥ ∈CC ≤ − which is the case, where ε = 1/poly(m,n). The same holds for linear optimization over . QC Proof. Linear optimization over is in NP: the certiﬁcate of (Y) is the optimal state ρAB. CC CC ⊆ implies QC ⊆ S max tr(ρABO) max tr(ρABO) max tr(ρABO). (26) ρAB ∈CC ≤ ρAB ∈QC ≤ ρAB ∈S A A B B For any separable state σAB = pi ψ ψ ψ ψ , Pi | i ih i | ⊗ | i ih i | A A B B tr(σABO)= X pitr( ψi ψi ψi ψi O) X pi max tr(ρABO)= max tr(ρABO), (27) ρAB ∈CC ρAB ∈CC i | ih | ⊗ | ih | ≤ i

7 A A B B as ψ ψ ψ ψ and pi = 1 [74]. Thus, | i ih i | ⊗ | i ih i |∈CC Pi

max tr(ρABO)= max tr(ρABO)= max tr(ρABO). (28) ρAB ∈CC ρAB ∈QC ρAB ∈S

Lemma 5 ([28, 58]). A bipartite quantum state ρAB is separable if and only if there exists a state ρ ′ ′ in an extended Hilbert space such that ρAB = trA′B′ ρAA′BB′ , or if and only if a state AA |BB ∈ CC ρ ′ exists such that ρAB = tr ′ ρ ′ . A|BB ∈ QC B ABB Remark 4. The deﬁnition (23) of separability remains the same if the number of terms in the summation is restricted to be less than or equal to m2n2 [43], where m n is the dimension of the × bipartite state ρAB. By slightly modifying the original proofs in [28, 58], the dimensions of ρAA′|BB′ 3 2 2 3 2 3 and ρ ′ can be required to be m n m n and m m n , respectively. A|BB × × Proposition 2 (NP-completeness of detecting classical states in a convex set). Given a convex set K of bipartite quantum states (K is given by a polynomial-time algorithm outputting whether a state is in K) with the promise that either (Y) K = or (N) infρ K,σ ρ σ 1 δ, it ∩ CC 6 ∅ ∈ ∈CC k − k ≥ is NP-complete to decide which is the case, where δ = 1/poly(m,n). The same holds for detecting quantum-classical states in K.

Proof. Detecting classical-classical states in K is in NP: the certiﬁcate of (Y) is an element in K = . The hardness proof is a polynomial-time reduction from lemma 2. Given a bipartite ∩ CC 6 ∅ state ρAB, deﬁne the convex set

K = ρ ′ ′ ρAB = trA′B′ ρAA′BB′ . (29) { AA |BB | }

(Y) If ρAB is separable, then K = . (N) If infσAB ρAB σAB 2 δ, then for any ∩ CC 6 ∅ ∈S k − k ≥ ρAA′BB′ K and σAA′BB′ , ∈ ∈ CC

ρAA′BB′ σAA′BB′ 1 trA′B′ (ρAA′BB′ σAA′BB′ ) 1 = ρAB σAB 1 ρAB σAB 2 δ, (30) k − k ≥ k − k k − k ≥ k − k ≥ as 1 is non-increasing under partial trace [59] and σAB = tr ′ ′ σ ′ ′ is separable. The k · k A B AA BB NP-completeness of detecting quantum-classical states in K can be proved analogously.

7 Conclusion and outlook

We have proved that computing quantum discord is NP-complete. Therefore, the running time of any algorithm for computing quantum discord is believed to grow exponentially with the dimension of the Hilbert space so that computing quantum discord in a quantum system of moderate size is not possible in practice. As by-products, some entanglement measures and constrained Holevo capacity are NP-hard/NP-complete to compute. These complexity-theoretic results are directly applicable in quantum information processing, and may offer significant insights. Moreover, we have proved the NP-completeness of two typical problems related to classical states, providing evidence that working with classical states is generically computationally intractable. The NP-completeness of computing quantum discord raises some interesting open problems. Is there an efficient approximation algorithm for computing quantum discord up to a moderate (e.g. constant additive) error? Can quantum discord be efficiently computed for certain important classes of states? What is the computational complexity of other measures of quantum correlation

8 beyond entanglement (e.g. geometric quantum discord [23, 61], quantum deficit)? The computational complexity of quantum correlation in continuous-variable systems is a new research direction. In particular, Gaussian states are of great theoretical and experimental interest [85, 86]. The separability problem for multimode bipartite Gaussian states [87] can be formulated as a semidefinite program [54] and solved efficiently in theory and practice [82] (the analog of lemma 2 for Gaussian states is false). What is the computational complexity of Gaussian entanglement of formation [88] and Gaussian quantum discord [2, 33]?

Acknowledgements

The author would like to thank Sevag Gharibian, Joel E Moore and Mark M Wilde for useful comments. This work was supported by the ARO via the DARPA OLE program.

References

[1] A. Abeyesinghe, I. Devetak, P. Hayden, and A. Winter. The mother of all protocols: re- structuring quantum information’s family tree. Proc. R. Soc. A, 465(2108):2537–2563, Aug. 2009.

[2] G. Adesso and A. Datta. Quantum versus classical correlations in Gaussian states. Phys. Rev. Lett., 105:030501, Jul. 2010.

[3] R. Ahlswede and I. Csiszar. Common randomness in information theory and cryptography. I. Secret sharing. IEEE Trans. Inf. Theory, 39(4):1121–1132, Jul. 1993.

[4] R. Ahlswede and I. Csiszar. Common randomness in information theory and cryptography. II. CR capacity. IEEE Trans. Inf. Theory, 44(1):225–240, Jan. 1998.

[5] M. Ali, A. R. P. Rau, and G. Alber. Quantum discord for two-qubit X states. Phys. Rev. A, 81(4):042105, Apr. 2010.

[6] M. Ali, A. R. P. Rau, and G. Alber. Erratum: Quantum discord for two-qubit X states [Phys. Rev. A 81, 042105 (2010)]. Phys. Rev. A, 82:069902, Dec. 2010.

[7] W. Beckner. Inequalities in Fourier analysis. Ann. Math., 102(1):159–182, Jul. 1975.

[8] S. Beigi and P. W. Shor. On the complexity of computing zero-error and Holevo capacity of quantum channels. arXiv:0709.2090, 2007.

[9] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett., 70(13):1895–1899, Mar. 1993.

[10] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters. Mixed-state entanglement and quantum error correction. Phys. Rev. A, 54(5):3824–3851, Nov. 1996.

[11] C. H. Bennett and S. J. Wiesner. Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett., 69(20):2881–2884, Nov. 1992.

[12] I. Bialynicki-Birula and J. Mycielski. Uncertainty relations for information entropy in wave mechanics. Commun. Math. Phys., 44(2):129–132, Jun. 1975.

9 [13] S. Boyd and L. Vandenberghe. Convex optimization. Cambridge University Press, 2004.

[14] F. G. Brandao, M. Christandl, and J. Yard. Faithful squashed entanglement. Commun. Math. Phys., 306(3):805–830, Sep. 2011.

[15] F. G. Brandao, M. Christandl, and J. Yard. Erratum to: Faithful squashed entanglement. Commun. Math. Phys., 316(1):287–288, Nov. 2012.

[16] F. G. Brandao, M. Christandl, and J. Yard. A quasipolynomial-time algorithm for the quantum separability problem. In Proc. 43rd Ann. ACM Symp. Theory Comput., pages 343–352, 2011.

[17] D. Cavalcanti, L. Aolita, S. Boixo, K. Modi, M. Piani, and A. Winter. Operational interpre- tations of quantum discord. Phys. Rev. A, 83(3):032324, Mar. 2011.

[18] L. Chen, E. Chitambar, K. Modi, and G. Vacanti. Detecting multipartite classical states and their resemblances. Phys. Rev. A, 83(2):020101(R), Feb. 2011.

[19] Q. Chen, C. Zhang, S. Yu, X. X. Yi, and C. H. Oh. Quantum discord of two-qubit X states. Phys. Rev. A, 84(4):042313, Oct. 2011.

[20] M. Christandl and A. Winter. ‘Squashed entanglement’: an additive entanglement measure. J. Math. Phys., 45(3):829–840, Feb. 2004.

[21] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett., 23(15):880–884, Oct. 1969.

[22] S. A. Cook. The complexity of theorem-proving procedures. In Proc. 3rd Ann. ACM Symp. Theory Comput., pages 151–158, 1971.

[23] B. Dakic, V. Vedral, and C. Brukner. Necessary and suﬃcient condition for nonzero quantum discord. Phys. Rev. Lett., 105(19):190502, Nov. 2010.

[24] G. M. D’Ariano, P. L. Presti, and P. Perinotti. Classical randomness in quantum measurements. J. Phys. A: Math. Gen., 38(26):5979–5991, Jul. 2005.

[25] A. Datta, A. Shaji, and C. M. Caves. Quantum discord and the power of one qubit. Phys. Rev. Lett., 100(5):050502, Feb. 2008.

[26] I. Devetak and A. Winter. Distilling common randomness from bipartite quantum states. IEEE Trans. Inf. Theory, 50(12):3183–3196, Dec. 2004.

[27] I. Devetak and A. Winter. Distillation of secret key and entanglement from quantum states. Proc. R. Soc. A, 461(2053):207–235, Jan. 2005.

[28] A. R. U. Devi and A. K. Rajagopal. Generalized information theoretic measure to discern the quantumness of correlations. Phys. Rev. Lett., 100(14):140502, Apr. 2008.

[29] R. Dillenschneider. Quantum discord and quantum phase transition in spin chains. Phys. Rev. B, 78:224413, Dec. 2008.

[30] L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller. Inseparability criterion for continuous variable systems. Phys. Rev. Lett., 84(12):2722–2725, Mar. 2000.

[31] A. Einstein, B. Podolsky, and N. Rosen. Can quantum-mechanical description of physical reality be considered complete? Phys. Rev., 47(10):777–780, May 1935.

10 [32] S. Gharibian. Strong NP-hardness of the quantum separability problem. Quantum Inf. Com- put., 10(3):343–360, Mar. 2010.

[33] P. Giorda and M. G. A. Paris. Gaussian quantum discord. Phys. Rev. Lett., 105:020503, Jul. 2010.

[34] O. Guhne and G. Toth. Entanglement detection. Phys. Rep., 474(1–6):1–75, Apr. 2009.

[35] L. Gurvits. Classical deterministic complexity of Edmonds’ problem and quantum entanglement. In Proc. 35th Ann. ACM Symp. Theory Comput., pages 10–19, 2003.

[36] L. Gurvits. Classical complexity and quantum entanglement. J. Comput. Syst. Sci., 69(3):448– 484, Nov. 2004.

[37] S. Hamieh, R. Kobes, and H. Zaraket. Positive-operator-valued measure optimization of classical correlations. Phys. Rev. A, 70(5):052325, Nov. 2004.

[38] P. M. Hayden, M. Horodecki, and B. M. Terhal. The asymptotic entanglement cost of preparing a quantum state. J. Phys. A: Math. Gen., 34(35):6891–6898, Sep. 2001.

[39] L. Henderson and V. Vedral. Classical quantum and total correlations. J. Phys. A: Math. Gen., 34(35):6899, Aug. 2001.

[40] A. Holevo. The capacity of the quantum channel with general signal states. IEEE Trans. Inf. Theory, 44(1):269–273, Jan. 1998.

[41] M. Horodecki, J. Oppenheim, and A. Winter. Partial quantum information. Nature, 436(7051):673–676, Aug. 2005.

[42] M. Horodecki, J. Oppenheim, and A. Winter. Quantum state merging and negative information. Commun. Math. Phys., 269(1):107–136, Jan. 2007.

[43] P. Horodecki. Separability criterion and inseparable mixed states with positive partial trans- position. Phys. Lett. A, 232(5):333–339, Aug. 1997.

[44] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki. Quantum entanglement. Rev. Mod. Phys., 81(2):865–942, Jun. 2009.

[45] J.-H. Huang, L. Wang, and S.-Y. Zhu. A new criterion for zero quantum discord. New J. Phys., 13(6):063045, Jun. 2011.

[46] Y. Huang. Entanglement criteria via concave-function uncertainty relations. Phys. Rev. A, 82(1):012335, Jul. 2010.

[47] Y. Huang. Erratum: Entanglement criteria via concave-function uncertainty relations [Phys. Rev. A 82, 012335 (2010)]. Phys. Rev. A, 82(6):069903, Dec. 2010.

[48] Y. Huang. Entropic uncertainty relations in multidimensional position and momentum spaces. Phys. Rev. A, 83(5):052124, May 2011.

[49] Y. Huang. Variance-based uncertainty relations. Phys. Rev. A, 86(2):024101, Aug. 2012.

[50] Y. Huang. Entanglement detection: complexity and Shannon entropic criteria. IEEE Trans. Inf. Theory, 59(10):6774–6778, Oct. 2013.

11 [51] Y. Huang. Quantum discord for two-qubit X states: analytical formula with very small worst- case error. Phys. Rev. A, 88(1):014302, Jul. 2013.

[52] Y. Huang. Scaling of quantum discord in spin models. Phys. Rev. B, 89:054410, Feb. 2014.

[53] L. P. Hughston, R. Jozsa, and W. K. Wootters. A complete classiﬁcation of quantum ensembles having a given density matrix. Phys. Lett. A, 183(1):14–18, Nov. 1993.

[54] P. Hyllus and J. Eisert. Optimal entanglement witnesses for continuous-variable systems. New J. Phys., 8(4):51, Apr. 2006.

[55] L. M. Ioannou. Computational complexity of the quantum separability problem. Quantum Inf. Comput., 7(4):335–370, May 2007.

[56] E. Knill and R. Laﬂamme. Power of one bit of quantum information. Phys. Rev. Lett., 81(25):5672–5675, Dec. 1998.

[57] M. Koashi and A. Winter. Monogamy of quantum entanglement and other correlations. Phys. Rev. A, 69(2):022309, Feb. 2004.

[58] N. Li and S. Luo. Classical states versus separable states. Phys. Rev. A, 78(2):024303, Aug. 2008.

[59] D. A. Lidar, P. Zanardi, and K. Khodjasteh. Distance bounds on quantum dynamics. Phys. Rev. A, 78(1):012308, Jul. 2008.

[60] X.-M. Lu, J. Ma, Z. Xi, and X. Wang. Optimal measurements to access classical correlations of two-qubit states. Phys. Rev. A, 83(1):012327, Jan. 2011.

[61] S. Luo and S. Fu. Geometric measure of quantum discord. Phys. Rev. A, 82:034302, Sep. 2010.

[62] V. Madhok and A. Datta. Interpreting quantum discord through quantum state merging. Phys. Rev. A, 83(3):032323, Mar. 2011.

[63] V. Madhok and A. Datta. Quantum discord as a resource in quantum communication. Int. J. Mod. Phys. B, 27(1–3):1345041, Jan. 2013.

[64] K. Matsumoto, T. Shimono, and A. Winter. Remarks on additivity of the Holevo channel capacity and of the entanglement of formation. Commun. Math. Phys., 246(3):427–442, Apr. 2004.

[65] W. Matthews, S. Wehner, and A. Winter. Distinguishability of quantum states under restricted families of measurements with an application to quantum data hiding. Commun. Math. Phys., 291(3):813–843, Nov. 2009.

[66] K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral. The classical-quantum boundary for correlations: discord and related measures. Rev. Mod. Phys., 84(4):1655–1707, Nov. 2012.

[67] M. A. Nielsen. Continuity bounds for entanglement. Phys. Rev. A, 61:064301, Apr. 2000.

[68] M. A. Nielsen and I. L. Chuang. Quantum computation and quantum information. Cambridge University Press, 2011.

[69] H. Ollivier and W. H. Zurek. Quantum discord: a measure of the quantumness of correlations. Phys. Rev. Lett., 88(1):017901, Dec. 2001.

12 [70] A. Peres. Separability criterion for density matrices. Phys. Rev. Lett., 77(8):1413–1415, Aug. 1996.

[71] M. Piani. Relative entropy of entanglement and restricted measurements. Phys. Rev. Lett., 103(16):160504, Oct. 2009.

[72] M. Piani, M. Christandl, C. E. Mora, and P. Horodecki. Broadcast copies reveal the quantumness of correlations. Phys. Rev. Lett., 102:250503, Jun. 2009.

[73] M. B. Plenio and S. Virmani. An introduction to entanglement measures. Quantum Inf. Comput., 7(1):1–51, Jan. 2007.

[74] R. Rahimi and A. SaiToh. Single-experiment-detectable nonclassical correlation witness. Phys. Rev. A, 82(2):022314, Aug. 2010.

[75] H. P. Robertson. The uncertainty principle. Phys. Rev., 34(1):163–164, Jul. 1929.

[76] M. S. Sarandy. Classical correlation and quantum discord in critical systems. Phys. Rev. A, 80:022108, Aug. 2009.

[77] B. Schumacher and M. D. Westmoreland. Sending classical information via noisy quantum channels. Phys. Rev. A, 56:131–138, Jul. 1997.

[78] P. W. Shor. Equivalence of additivity questions in quantum information theory. Commun. Math. Phys., 246(3):453–472, Apr. 2004.

[79] R. Simon. Peres-Horodecki separability criterion for continuous variable systems. Phys. Rev. Lett., 84(12):2726–2729, Mar. 2000.

[80] A. Streltsov, H. Kampermann, and D. Bruss. Linking quantum discord to entanglement in a measurement. Phys. Rev. Lett., 106(16):160401, Apr. 2011.

[81] A. Uhlmann. Entropy and optimal decompositions of states relative to a maximal commutative subalgebra. Open Syst. Inf. Dyn., 5(3):209–228, Sep. 1998.

[82] L. Vandenberghe and S. Boyd. Semideﬁnite programming. SIAM Rev., 38(1):49–95, Mar. 1996.

[83] V. Vedral and M. B. Plenio. Entanglement measures and puriﬁcation procedures. Phys. Rev. A, 57(3):1619–1633, Mar. 1998.

[84] V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight. Quantifying entanglement. Phys. Rev. Lett., 78(12):2275–2279, Mar. 1997.

[85] X.-B. Wang, T. Hiroshima, A. Tomita, and M. Hayashi. Quantum information with Gaussian states. Phys. Rep., 448(14):1–111, Aug. 2007.

[86] C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd. Gaussian quantum information. Rev. Mod. Phys., 84:621–669, May 2012.

[87] R. F. Werner and M. M. Wolf. Bound entangled Gaussian states. Phys. Rev. Lett., 86:3658– 3661, Apr. 2001.

[88] M. M. Wolf, G. Giedke, O. Kruger, R. F. Werner, and J. I. Cirac. Gaussian entanglement of formation. Phys. Rev. A, 69:052320, May 2004.

13 [89] D. Yang, M. Horodecki, and Z. D. Wang. An additive and operational entanglement measure: conditional entanglement of mutual information. Phys. Rev. Lett., 101:140501, Sep. 2008.