OPERATOR SPACE APPROACH TO STEERING INEQUALITY

OPERATOR SPACE APPROACH TO STEERING INEQUALITY MICHAL HORODECKI, MARCIN MARCINIAK, AND ZHI YIN

arXiv:1405.1945v1 [quant-ph] 8 May 2014

Abstract. In this paper, we use operator space theory to study the steering scenario. We obtain a a bipartite q steering inequality F = {Fx ∈ Mn , x, a = 1, . . . , n} with unbounded largest violation n . We can also construct all ingredients explicitly. Moreover, we study the role of of order log n maximally and partially entangled state play in the violation of steering inequality. Finally, we focus on the bipartite dichotomic case, we can always find a steering inequality with unbounded largest violation. This is different to the Bell scenario. Since by any bipartite dichotomic Bell inequality, only bounded largest violation can be obtained.

1. Introduction The violation of local realism, called usually nonlocality, plays an important role in quantum information science. It was first studied in 1935 by Einstein, Podolsky and Rosen [6]. In their paradoxical paper, Einstein, Podolsky, and Rosen (EPR) argued that quantum mechanics is incomplete because it does not provide a complete description of the elements of reality. Moreover, they predicted either quantum mechanics develops to a complete theory or quantum mechanics is replaced by another complete theory. In the paper, EPR wrote: “We believe that such (complete) a theory is possible.” Theories compatible with the EPR’s ideas are called “local-realistic (LR) theories”. Although Bohr rebutted shortly after the EPR’s ideas, their arguments, without observational consequences, did not suggest a clear conclusion, hence the debate has then subsided. The EPR arguments were resurfaced by J. S. Bell in 1964 when he derived a constraint for correlation between two remote subsystems, known as Bell’s inequality which is satisfied by all LR theories. He proved that it is violated by quantum correlations of two spin 1/2 particles in a singlet state, i.e., an entangled state. States that violate some Bell inequalities form strict subset of the set of entangled states [27]. In [28] the authors proposed an intermediate form of quantum correlations between Bell nonlocality and entanglement, by use of quantum steering. The latter concept was introduced by Schr¨odinger in 1935 to reply the EPR paradox [22]. Wiseman, Jones and Doherty reformulated this concept in a rigorous way [28] and have, in particular, shown that the set of states admitting steering is a strict subset of entangled states on one hand and a strict superset of states violating Bell inequalities. Since then, quantum steering has attracted more and more attention both in theory [5, 15, 20, 28] and experiment [23, 24]. The simplest example of quantum steering is the following one, which was a basis for famous EPR paradox [6] (in Bohm version [3]). Namely, when Alice and Bob share a pair of particles in singlet state, Alice, by choosing one of two measurements can create at Bob’s site one of two ensembles: one consisting of basis states |0i and |1i with equal probabilities and the other, consisting of complementary states √12 (|0i + |1i) and √12 (|0i − |1i), again with equal probabilities. It turns out that this would be impossible, if Bob particle were in some well defined state, perhaps unknown to him – so called ”local hidden state” (LHS), and Alice merely used her knowledge about the state. Thus, existence of the above Alice’s measurements proves that the shared state is entangled. Remarkably, Alice can in this way convince Bob, that the shared state is entangled even if Bob does not trust her. Indeed, Bob can ask Alice to create one of the two ensembles at random, and upon receiving message from Alice telling which outcome she obtained, he can verify that indeed she created (or: ”steered” to) the above states with the mentioned probabilities, provided many runs of the experiment are performed. More generally, bipartite states for which there exist measurements Date: May 9, 2014. 2000 Mathematics Subject Classification. Primary: 46L50, 46L07. Secondary: 58L34, 43A55. Key words and phrases. Steering inequality, Unbounded largest violation, Operator spaces.

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M. Horodecki, M. Marciniak and Z. Yin

of Alice steering to ensembles, which cannot come from local hidden state are called steerable (or admitting quantum steering). In the steering scenario, one can study “steering inequalities” which are analogs of Bell inequalities. The violation of a steering inequality provides a natural way to quantify the deviation from a local hidden state description. However it is not easy to compute the violation for a given steering inequality analytically; one usually uses here numerical method, called semi-definite program [20, 25]. In this paper, we will use the operator space theory, which has been widely developed after the pioneering and fundamental work of Effros-Ruan and Blecher-Paulsen [7, 18], to study the violation of steering inequality. Our work is motivated by a series works by Junge, Palazuelos, P´erez-Garc´ıa, Villanueva and Wolf [9, 10, 21], where they used operator space theory to analyze Bell inequalities. Before stating the results, let us recall formal definition of steering inequalities. Suppose Alice can choose among n different measurement settings labeled by x = 1, . . . , n, each of which can result in one of m outcomes, labeled by a = 1, . . . m. Suppose further that Bob has a quantum system described by a d-dimensional Hilbert space Hd . Define an “assemblage” to be a set of d × d Hermitian matrices σxa . The set is called an assemblage if a i) σ Px ≥ a0 (positivity); ii) a σx is independent of x and has trace 1.

Define a “steering functional” (or ”steering inequality”) F as a set of d × d matrices Fxa , where aPranges from 1 to m and x ranges from 1 to n. F maps an assemblage to a real number | x,a Tr(Fxa σxa )|. We will link the classical bound BC (F ) of this steering inequality to the injective norm of tensor product of two Banach spaces ℓn1 (ℓm ∞ ) ⊗ǫ B(Hd ), as well as the quantum bound BQ (F ) to the norm of minimal tensor product of two operator spaces ℓn1 (ℓm ∞ ) ⊗min B(Hd ). BQ (F ) For any steering inequality F, the largest violation is defined by LV (F ) = BC (F ) . Hence we can transfer the problem of studying the largest violation to the problem of comparing the minimal and injective tensor product. This “min vs ǫ” strategy (see [21]) allows us to obtain our main results. Namely, for every n ∈ N, we can obtain a steering inequality F =q{Fxa ∈ Mn : x, a = 1, . . . , n}, n such that the largest violation of F is greater than the order of log n . We can also construct

this steering inequality with unbounded largest violation explicitly. Similarly as in the Bell scenario, we will study the role of the maximally entangled state in violation of steering inequalities. We provide a steering inequality with unbounded largest violation, but this unbounded largest violation is obtained by the partially entangled state, rather than the maximally entangled state. Moreover, by adding some white noise to this partially entangled state, we can get a family of PPT (Positive Partial Transpose) states. Actually, these states belong to a class of PPT states, which are constructed in [4]. We prove that for any steering inequality, only bounded largest violation can be obtained by these kinds of PPT states. Thus even though there are PPT states violating some steering inequalities [14] (problem posed in [20]), our example provides some evidence, that PPT states cannot provide unbounded violation. Finally, we consider dichotomic case. In this bipartite dichotomic case, the quantum bound of any Bell inequality will be bounded by the classical bound with an universal constant (Grothendieck constant) [19, 21]. However, as reported in a companion paper [13] in the steering scenario, this is not true: there a steering √ inequality F = {Fx ∈ Mn , x = 1, . . . , n} is provided with unbounded largest violation of order log n. Here we put the example of [13] into the framework of operator spaces and show how the inequality arises from a bounded but not completely bounded map from ℓn∞ to Mn . We finish this introduction by setting the following convention: throughout this paper, we will use & and . to denote the inequality up to an universal constant irrelevant to n ∈ N, and we also use the Dirac symbol |iihj|, i, j = 1, . . . , n to denote the canonical basis of Mn . 2. Mathematical tools

2.1. Tensor product of Banach spaces. Here we recall some general properties of tensor product of Banach spaces [26]. Let X ⊗ Y be the algebraic tensor product of two Banach spaces X and Y over the field of complex numbers. It is useful to identify X ⊗ Y with a subspace of B(X ∗ , Y ).

Operator space approach to steering inequality

Namely, to any u = (2.1)

P

i

3

xi ⊗ yi ∈ X ⊗ Y, we assign an operator Tu : X ∗ → Y given by X hxi , f iyi , f ∈ X ∗ . Tu f = i

If a norm k · kβ on X ⊗ Y satisfies the condition

(2.2)

kx ⊗ ykβ = kxkkyk, x ∈ X, y ∈ Y,

then it is called a cross-norm on X ⊗ Y. The completion of X ⊗ Y under a cross-norm k · kβ is denoted by X ⊗β Y. We define two important cross-norms. Given a Banach space Z we let BZ denote the unit ball of Z. Let X and Y be Banach spaces. The injective norm k · kǫ on X ⊗ Y is defined for any u ∈ X ⊗ Y by ) ( n X f (xi )g(yi ) : f ∈ BX ∗ , g ∈ BY ∗ , (2.3) kukǫ = sup i=1 Pn where u = i=1 xi ⊗ yi . The projective norm of u ∈ X ⊗ Y is (2.4)

kukπ = inf

n X i=1

kxi kkyi k,

Pn where the infimum runs over all decompositions of the form u = i=1 xi ⊗ yi . It is clear that kukǫ ≤ kukπ for any u ∈ X ⊗ Y . The completion X ⊗ǫ Y is called the injective tensor product of X and Y , while X ⊗π Y is called the projective one. If X and Y are finite dimensional Banach spaces then the injective and projective tensor products are dual to each other in the following sense (2.5)

X ⊗ǫ Y = B(X ∗ , Y ) = (X ∗ ⊗π Y ∗ )∗ ,

where the equalities mean isometric isomorphisms. It can be proved by using properties of the embedding X ⊗ Y ∋ u 7→ Tu ∈ B(X ∗ , Y ). 2.2. Operator spaces. We recall some basic notions of the theory of operator spaces. We refer the reader to [7, 18] for more information. An (concrete) operator space is a closed subspace E of B(H) where H is a Hilbert space. Let Mn (E) denote the space of n × n matrices with entries in E. Then E inherits the matricial structure of B(H) via the embedding Mn (E) ⊂ Mn (B(H)). More precisely, Mn (E) is equipped with the norm induced by B(ℓn2 (H)). An abstract matricial norm characterization of operator spaces was given by Ruan. To be more precisely, let E be a complex vector space and assume that a norm k · kn is given on Mn (E) for every n ∈ N. We say that E is an (abstract) operator space if the sequence of norms k · kn satisfies Ruan’s axioms: i) kaxbk n ≤ m kbk.  kakkxk



x 0 ii) = max{kxkm , kykn }

0 y m+n for every n, m ∈ N, x ∈ Mm (E), y ∈ Mn (E), a ∈ Mn,m and b ∈ Mm,n . Let H, K be two Hilbert spaces. Suppose that E ⊂ B(H) and F ⊂ B(K) are two operator spaces. A map u : E → F is called completely bounded (in short c.b.) if (2.6)

sup kidMn ⊗ ukMn (E)→Mn (F ) < ∞, n

and the c.b. norm kukcb is defined to be the above supremum. Completely bounded maps are morphisms in the category of operator spaces. We denote by CB(E, F ) the space of all c.b. maps from E to F , equipped with the norm k · kcb. This is a Banach space. For an operator space E, there exists a natural matricial structure on the Banach dual E ∗ of E so that E ∗ becomes an operator space too. The norm of Mn (E ∗ ) is that of CB(E, Mn ), where Mn = Mn (C). Now assume that E and F are operator spaces. Like in Banach space theory, we consider tensor products of E and F in the category of operator spaces. In order to define a tensor product of E and F one should equip the algebraic tensor product E ⊗ F with a sequence of cross norms satisfying Ruan axioms and take a completion. It turns out that in the category of operator spaces there are analogs of the injective (ǫ) and projective (π) tensor norms from the category of Banach spaces. The analog of the injective tensor product of Banach is called minimal tensor product of operator

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M. Horodecki, M. Marciniak and Z. Yin

spaces. It can be easily defined when E and F are concrete operator spaces, i.e. E ⊂ B(H) and F ⊂ B(K) for some Hilbert spaces H and K. Then we have natural embedding E ⊗ F ⊂ B(H ⊗ K) which defines a structure of a concrete operator space on the closure of the image of E ⊗ F . This closure is denoted by E ⊗min F , while by k · kmin we denote the norm induced by the embedding. The minimal tensor product has the following injectivity property: if E ⊂ X and F ⊂ Y, then E ⊗min F ⊂ X ⊗min Y also holds. The analogue of the π tensor product is called the operator space projective tensor product and is denoted by ∧. The norms are defined as: (2.7)

kukMn(E⊗∧ F ) = inf{kαkMn,lm kxkMl (E) kykMm(F ) kβkMlm,n : u = α(x ⊗ y)β},

where u = α(x ⊗ y)β means the matrix product. These two tensor norms, ∧ and min have the following duality property: for finite dimensional operator spaces we have the natural completely isometric identifications: (E ⊗∧ F )∗ = CB(E, F ∗ ) = E ∗ ⊗min F ∗ .

(2.8)

In this paper, we will use the following operator spaces: i) ℓn∞ . There is aP natural operator space structure on ℓn∞ . We embed ℓn∞ as diagonal maps in Mn . Then for x = i Ai ⊗ ei ∈ Mk ⊗ ℓn∞ we have

X

Ai ⊗ |iihi| = max kAi kMn . (2.9) kxkk = Mkn

i

i

We denote ℓn1 as the operator space dual of ℓn∞ . For any operator space X, we will write ℓn1 (X) = ℓn1 ⊗∧ X and ℓn∞ (X) = ℓn∞ ⊗min X. ii) Row and column operator spaces Rn and Cn . As Banach spaces, they are both equal to the ℓn2 , but they carry different operator space structure. To see this, we embed ℓn2 into Mn as the first row nX o (2.10) Rn = αk |1ihk| : αk ∈ C k

and similarly as the first column

(2.11) For x = (2.12)

Cn = P

nX k

i

Ai ⊗ ei ∈ Mk ⊗

ℓn2 ,

o αk |kih1| : αk ∈ C .

12

12

X

X



Ai A∗i and kxkMk ⊗min Cn = kxkMk ⊗min Rn = A∗i Ai . i

iii) The intersection of row and column space Rn (Rn , Cn ) 21 . (See. [18])

T

i

Cn and the Hibletian operator space OHn =

Now let us recall the following useful fact [7, 18, 19], which will be used in this paper from time to time. Consider a linear map u : E → F. i) If either E = F = Cn or E = F = Rn , then CB(E, F ) = B(E, F ) and kukcb = kuk. ii) If either E = Cn and F = Rn or E = Rn and F = Cn , then u is c.b. if and only if it is Hilbert-Schmidt and, denoting by k · k2 the Hilbert-Schmidt norm, we have kukcb = kuk2 . 3. Main result 3.1. Steering inequality in operator space framework. We deal with following steering scenario [20]. Suppose Alice can choose among n different measurement settings labeled by x = 1, . . . , n. Each of which can result in one of m outcomes, labeled by a = 1, . . . , m. Suppose Bob has a d-dimensional quantum system Hd . Definition 3.1. An assemblage is a set {σxa : x = 1, . . . , n, a = 1, . . . , m} of d × d Hermitian matrices satisfying the following conditions: a i) σ Px ≥ a0 (positivity); ii) a σx is independent of x and trace 1.

Operator space approach to steering inequality

5

We say that an assemblage has a quantum realization, if there exists a Hilbert space H such that σxa = T rA ((Exa ⊗ 1lB )ρ)

(3.1)

a m for every x and a, where ρ ∈ B(H ⊗ Hd ) is a density matrix and {E ⊂ B(H) is a POVM Px }a=1 a measurement on Alice for every x, i.e. Ex ≥ 0 for every x, a, and a Exa = 1l for every x. We denote the set of all assemblages with quantum realizations by Q. On the other hand, we say that an assemblage has a local hidden state (LHS) model, if there P are finite set of indices Λ, nonnegative coefficients qλ such that λ qλ = 1, density matrices σλ in B(H P d ) for λ ∈ Λ, and probability distributions {pλ (a|x)}a for every x and λ (i.e. pλ (a|x) ≥ 0 and a pλ (a|x) = 1 for every x, λ), such that X (3.2) σxa = qλ pλ (a|x)σλ , λ∈Λ

for every x, a. We denote the set of LHS assemblages by L.

Remark 3.2. Schr¨odinger [22] (and later [8]) has shown that any assemblage satisfying conditions i) and ii) of the above definition has a quantum realization. It is known [28] that L ( Q. Like in the Bell scenario, our aim is to quantify the difference between sets Q and L. Our strategy is to analyse some steering inequalities to see how much value can they obtained on the quantum assemblages comparing with values on LHS assemblages. For given natural n, m and d, define ([20]) a steering functional or inequality F as a set {Fxa : x = 1, . . . , n, a = 1, . . . , m} of d × d matrices. The functional maps an assemblage σ to a real number n X m X (3.3) hF, σi = Tr(Fxa σxa ). x=1 a=1

Given an assemblage σ ∈ Q we now define the largest steering violation that σ may attain as:

(3.4)

ν(σ) = sup {|hF, σi| : F is a steering inequality such that |hF, τ i| ≤ 1 for τ ∈ L.} .

Definition 3.3. We say that σ = {σxa } is an incomplete quantum assemblage, if there exists a Hilbert space H such that (3.5)

σxa = TrA ((Exa ⊗ 1lB )ρ)

⊂ B(H) is an incomplete for every x, a, where ρ ∈ B(H ⊗ Hd ) is a density matrix and {Exa }a P POVM measurement on Alice for any x, i.e. Exa ≥ 0 for every x, a and a Exa ≤ 1l for every x. On the other hand, we say that an assemblage σ has an incomplete local hidden state (LHS) model, if X (3.6) σxa = qλ pλ (a|x)σλ λ∈Λ

P for every x, a, where Λ is a finite set of indices, qλ are nonnegative numbers such that λ qλ = 1, σλ ∈ P B(Hd ), σλ ≥ 0, Tr(σλ ) ≤ 1 for all λ, and pλ (a|x) are nonnegative numbers for all a, x, λ, such that a pλ (a|x) ≤ 1 for all x, λ. We denote the set of incomplete quantum assemblage by Qin and the set of incomplete LHS by in L . Now we can define Definition 3.4. Given a steering inequality F = {Fxa ∈ B(Hd ) : x = 1, . . . , n, a = 1, . . . , m}, we define the classical bound of F as the number (3.7) and the quantum bound of F as (3.8)

BC (F ) = sup{|hF, σi| : σ ∈ Lin }, BQ (F ) = sup{|hF, σi| : σ ∈ Qin }.

We define the largest quantum violation of F as the positive number BQ (F ) . (3.9) LV (F ) = BC (F )

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3.2. Large Bell violation implies large steering violation. It is well known that there is a very close relation between Bell inequality and steering inequality. M. Junge, C. Palazuelos, D. P´erez-Garc´ıa, I. Villanueva and M. M. Wolf studied the following largest violation of bipartite Bell inequality [9, 10]: (3.10)

LV (M ) =

BQ (M ) , BC (M )

Pn Pm a,b n m where M = x,y=1 a,b=1 Mx,y ex ⊗ e′a ⊗ ey ⊗ e′b ∈ ℓn1 (ℓm ∞ ) ⊗ ℓ1 (ℓ∞ ), and (3.11) n X o a,b BQ (M ) = sup Mx,y Tr((Exa ⊗ Eyb )ρ) : Exa , Eyb are incomplete POVMs, ρ is a state, x,y,a,b

(3.12) X n X o X X a,b BC (M ) = sup Mx,y ρ(λ)P(a|x, λ)P(b|y, λ) : P(a|x, λ) ≤ 1, P(b|y, λ) ≤ 1. x,y,a,b

a

λ

b

Now we have the following proposition:

Pn Pm a,b ′ Proposition 3.5. Suppose there is a Bell inequality M = x,y=1 a,b=1 Mx,y ex ⊗ ea ⊗ ey ⊗ ′ n m n m eP ℓ1 (ℓ∞ ) ⊗ ℓ1 (ℓ∞ ), such that LV (M ) & C, then we can find a steering inequality F = b ∈ P n m ′ a n m x=1 a=1 (ex ⊗ ea ) ⊗ Fx ∈ ℓ1 (ℓ∞ ) ⊗ B(Hd ), such that (3.13)

LV (F ) ≥ LV (M ) & C.

Proof. Suppose there exist incomplete POMVs Exa , Eyb and density matrix ρ ∈ B(Hd ) ⊗ B(Hd ), such that X a,b (3.14) BQ (M ) = Mx,y Tr((Exa ⊗ Eyb )ρ) . x,y,a,b

Let Fxa =

P

a,b b y,b Mx,y Ey

(3.15)

Moreover, we have

and σxa = TrA ((Exa ⊗ 1l)ρ). Then X a,b a b BQ (M ) = Mx,y TrB ((TrA ((Ex ⊗ 1l)ρ)Ey ) x,y,a,b    X X a,b b  a    = TrB Mx,y Fy TrA ((Ex ⊗ 1l)ρ) x,a y,b X = Tr(Fxa σxa ) ≤ BQ (F ), x,a

BC (F ) =

X

Tr

Fxa

x,a

(3.16) =

X

x,a,y,b

Thus we complete the proof.

X λ

a,b Mx,y

X λ

ρ(λ)P(a|x, λ)σλ

!

ρ(λ)P(a|x, λ)Tr(Eyb σλ ) ≤ BC (M ). 

The physical reason of this proposition is simple: Since Bell nonlocality is stronger than steerability [28], large Bell violation always lead to large steering violation. By Pnthe proposition and Junge-Palazuelos’ work [9] one can easily obtain a steering inequality F = x,a=1 (ex ⊗ e′a ) ⊗ Fxa ∈ ℓn1 (ℓn∞ ) ⊗ B(Hn ) such that √ n . (3.17) LV (F ) & log n In next subsection, by direct computation we will improve the above estimation of the largest steering violation.

Operator space approach to steering inequality

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3.3. Unbounded largest violation of steering inequality. Our main result concerns the case when all the numbers n, m and d are equal. It can be stated as follows: Theorem 3.6. For every n ∈ N, there exists an assemblage σ = {σxa ∈ B(Hn+1 ) : x = 1, . . . , n, a = 1, . . . , n + 1} ∈ Q such that √ n . (3.18) ν(σ) & √ log n By letting Fb as the extension of F for which Fbxm+1 = 0, we have following lemma, which is analogue to the lemma 1 of [10]:

Lemma 3.7. Assume we have a steering inequality F = {Fxa ∈ B(Hd ) : x = 1, . . . , n, a = 1, . . . , m}, such that LV (F ) = C. Then, there exists another steering inequality Fb = {Fbxa ∈ B(Hd ) : x = 1, . . . , n, a = 1, . . . , m + 1}, such that (3.19)

supσ∈Q |hFb , σi| = C. supσ∈L |hFb , σi|

Remark 3.8. By the definition of ν, the conclusion of theorem 3.6 is equivalent to the following statement: For every n ∈ N, there exists a steering inequality F = {Fxa ∈ B(Hn+1 ) : x = 1, . . . , n, a = 1, . . . , n + 1}, such that √ supσ∈Q |hF, σi| n (3.20) . & √ supσ∈L |hF, σi| log n By this lemma and remark, we will deal with the following reformulation of Theorem 3.6. Theorem 3.9. For every n ∈ N, we can find a steering inequality F = {Fxa ∈ B(Hn ) : x, a = 1, . . . , n}, such that √ n (3.21) LV (F ) & √ . log n Now, let us introduce some notations. For k ∈ N we denote by (e1 , . . . , ek ) the canonical basis of ℓk1 and by (e′1 , . . . , e′k ) its dual basis in ℓk∞ . Then the system (ex ⊗ e′a )x=1,...,n; a=1,...,m forms a ′ n m basis of the space ℓn1 (ℓm ∞ ) as well as the system (ex ⊗ ea )x=1,...,n a=1,...,m is a basis of ℓ∞ (ℓ1 ). Let a F = {Fx }x,a be a steering inequality. We will identify F with the following element of the tensor product ℓn1 (ℓm ∞ ) ⊗ B(Hd ) (3.22)

n X m X

(ex ⊗ e′a ) ⊗ Fxa .

x=1 a=1

We still use F to denote this element. Now, as in [10], we can link the problem of largest violation of steering inequality to the “min vs ǫ” problem for F . P Proposition 3.10. Given F = x,a (ex ⊗ e′a ) ⊗ Fxa ∈ ℓn1 (ℓm ∞ ) ⊗ B(Hd ), we have the following equivalence: i) Classical bound: (3.23)

BC (F ) ≤ kF kℓn1 (ℓm ≤ 16BC (F ). ∞ )⊗ǫ B(Hd )

ii) Quantum bound: (3.24)

BQ (F ) ≤ kF kℓn1 (ℓm ≤ 4BQ (F ). ∞ )⊗min B(Hd )

Proof. In the proof of the classical bound we use the duality between injective and projective tensor products for finite dimensional Banach spaces. For a Hilbert space H we denote by Sp (H) the p-th Schatten class for H, where p is a number such that p ≥ 1. Any element σ ∈ ℓn∞ (ℓm 1 ) ⊗π S1 (Hd ) = ∗ ∗ n m ) ⊗ B(H ). Its action on F is )) ⊗ (B(H )) can be considered as a functional on ℓ (ℓ (ℓn1 (ℓm π d ǫ d ∞ 1 ∞ given by n X m X (3.25) hF, σi = Tr(Fxa σxa ), x=1 a=1

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M. Horodecki, M. Marciniak and Z. Yin

P where matrices σxa ∈ B(Hd ) are determined by the unique decomposition σ = x,a e′x ⊗ ea ⊗ σxa . Observe that the action of a steering inequality on an assemblage given by (3.3) is a special case of the described above duality. Given a Banach space X we let BX denote the unit ball of X. Thus, by the duality, we have  (3.26) kF kℓn1 (ℓm = sup |hF, σi| : σ ∈ Bℓn∞ (ℓm . ∞ )⊗ǫ B(Hd ) 1 )⊗π S1 (Hd ) Now, observe that

Lin ⊂ Bℓn∞ (ℓm . 1 )⊗π S1 (Hd )

(3.27)

P Indeed, if σ ∈ Lin , then it follows λ qλ Pλ ⊗ σλ , where qλ are P from Definition 3.3 that σ = = 1, σλ ∈ B(Hd ) are such that σλ ≥ 0 and Tr(σλ ) ≤ 1, and nonnegative numbers such that Pλ qλ P n m m ′ Pλ ∈ ℓn∞ (ℓP 1 ) are given by Pλ = x=1 a=1 pλ (a|x)ex ⊗ ea for some nonnegative numbers pλ (a|x) such that a pλ (a|x) ≤ 1 for every x, λ. Therefore (3.28) X X X X kσkℓn∞ (ℓm ≤ qλ kPλ kℓn∞ (ℓm kσλ kS1 (Hd ) = qλ sup pλ (a|x) Tr(σλ ) ≤ qλ = 1. 1 )⊗π S1 (Hd ) 1 ) λ

λ

x

a

λ

So, σ ∈ Bℓn∞ (ℓm . The first inequality in (3.23) follows from inclusion (3.27). To show 1 )⊗π S1 (Hd ) the second inequality, let us remind that the ball Bℓn∞ (ℓm is the convex hull of the set 1 )⊗π S1 (Hd ) {P ⊗ τ : P ∈ Bℓn∞ (ℓm , τ ∈ B }. Consider an element σ = P ⊗ τ from this set. Then S1 (Hd ) 1 ) P P P = x,a px,a e′x ⊗ ea for some complex numbers px,a such that a |px,a | ≤ 1 for every x, and τ ∈ B(Hd ) is such that Tr(|τ |) ≤ 1. Each number px,a can be decomposed as a sum px,a = p(1) (a|x) − p(2) (a|x) + ip(3) (a|x) − ip(4) (a|x) where 0 ≤ p(α) (a|x) ≤ |px,a | for α = 1, 2, 3, 4 and every x, a. Similarly, τ = τ (1) −P τ (2) + iτ (3) − iτ (4) for positive matrices such that Tr(τ (β) ) ≤ Tr(|τ |) for (α) β = 1, 2, 3, 4. Let P = x,a p(α)(a|x)e′x ⊗ ea for every i = 1, 2, 3, 4, and let σ (α,β) = P (α) ⊗ τ (β) P (α,β) where for every α, β = 1, 2, 3, 4. Then σ (α,β) ∈ Lin for every α, β and σ = α,β ωα,β σ |ωα,β | = 1 for every α, β. Thus 4 4 X X (3.29) |hF, σi| = ωα,β hF, σ (α,β) i ≤ |hF, σ (α,β) i| ≤ 16BC (F ). α,β=1 α,β=1 Thus, the second inequality in (3.23) follows from the fact that Bℓn∞ (ℓm is the closure of 1 )⊗π S1 (Hd ) convex combinations of simple tensors of the form P ⊗ τ . For the quantum bound, we first recall the following fact from [10]: Fact 3.11. Given a set of incomplete POVMs {Exa }a=1,...,m , x = 1, . . . , n, on B(H), the operator a ′ u : ℓn1 (ℓm ∞ ) → B(H) defined by u(ex ⊗ ea ) = Ex , is a complete contraction. Now, let us recall that the minimal tensor norm of F can be expressed as follows ([18]) (3.30)

kF kℓn1 (ℓm = sup k(u ⊗ id)(F )kB(H)⊗min B(Hd ) , ∞ )⊗min B(Hd ) H,u

where the sup taken over all possible Hilbert spaces H and complete contractions u : ℓn1 (ℓm ∞) → B(H). Let σ ∈ Qin , i.e. there is a Hilbert space H, a density matrix ρ ∈ B(H) ⊗ B(Hd ), and incomplete POVMs {Exa } on B(H) such that σxa = TrA ((Exa ⊗ 1l)ρ) for every x, a. Let u be a map a u : ℓn1 (ℓm ∞ ) → B(H) defined by u(ex ⊗ ea ) = Ex . By the aforementioned fact u is a complete contraction. Thus, in order to show the first inequality in (3.24) it is enough to prove the following claim: (3.31)

hF, σi = Tr((u ⊗ id)(F )ρ).

We use the following notation. On the algebra B(H) ⊗ B(Hd ) we consider partial traces TrA = TrB(H) ⊗ idB(Hd ) and TrB = idB(H) ⊗ TrB(Hd ) . Observe that
(3.32) Tr = TrB(H) ⊗ TrB(Hd ) = TrB(Hd ) ◦ TrB(H) ⊗ idB(Hd ) = TrB(Hd ) ◦ TrA .

Let us remind also that the partial trace TrA has the following conditional expectation property: (3.33)

TrA ((1l ⊗ X)Y (1l ⊗ Z)) = XTrA (Y )Z

Operator space approach to steering inequality

9

for any X, Z ∈ B(Hd ) and Y ∈ B(H ⊗ Hd ). We have Tr ((u ⊗ id)(F )ρ) = X X = Tr ((u ⊗ id)(ex ⊗ e′a ⊗ Fxa )ρ) = Tr ((Exa ⊗ Fxa )ρ) x,a

= = =

(3.34)

X

x,a

Fxa )(Exa

x,a

TrB(Hd ) (TrA ((1l ⊗

x,a

TrB(Hd ) (Fxa TrA ((Exa ⊗ 1l)ρ))

x,a

TrB(Hd ) (Fxa σxa ) = hF, σi.

X

X

⊗ 1l)ρ))

The proof of the second inequality in (3.24) is similar to the proof of Theorem 6 in [10]. Given m ′ ′ a complete contraction u : ℓn1 (ℓm ∞ ) → B(H) define ux : ℓ∞ → B(H) as ux (ea ) = u(ex ⊗ ea ) for every x = 1, . . . , n. It is easy to check that every ux is a complete contraction. Using Wittstock’s factorization theorem, we can decompose every complete contraction ux = u1x − u2x − i(u3x − u4x ), where each component ujx is a completely positive map. Moreover, it can be checked ([10]) that (ujx (e′a ))a is an incomplete POVM for every x and j. This lead to the constant 4 in the assertion.  Combine this proposition with the definition of largest violation of steering inequality, we have the following corollary. Pn Pm Corollary 3.12. Given F = x=1 a=1 (ex ⊗ e′a ) ⊗ Fxa , we have that (3.35)

LV (F ) ≃

kF kℓn1 (ℓm ∞ )⊗min B(Hd ) . kF kℓn1 (ℓm ∞ )⊗ǫ B(Hd )

Remark 3.13. In the Bell scenario, M. Junge, C. Palazuelos, D. P´erez-Garc´ıa, I. Villanueva and M. M. Wolf link the problem of largest violation of Bell inequality to the “min vs ǫ” problem Pn [10]P m a,b n m ex ⊗ e′a ⊗ ey ⊗ e′b ∈ ℓn1 (ℓm of M = x,y=1 a,b=1 Mx,y ∞ ) ⊗ ℓ1 (ℓ∞ ). The result which we will prove is

Theorem 3.14. For every n ∈ N, there exists an element F = B(Hn ) such that √ kF kmin n √ (3.36) . & kF kǫ log n

Pn

x,a=1 (ex

⊗ e′a ) ⊗ Fxa ∈ ℓn1 (ℓn∞ ) ⊗

Theorem 3.9 follows from this theorem and Corollary 3.12. Proof of Theorem 3.14. Firstly, we recall the following fact (see [9, Theorem 3.2]): there exist [δn] δ ∈ (0, 21 ) and a universal constant C such that, for every n, there are maps V : ℓ2 → ℓn1 (ℓn∞ ) √ [δn] and V ′ : ℓn1 (ℓn∞ ) → ℓ2 satisfying kV k ≤ C log n, kV ′ k ≤ 1 and V ′ V = idℓ[δn] . Moreover, the 2 map V ′ is completely bounded from ℓn1 (ℓn∞ ) to R[δn] and kV ′ kcb ≤ KL kV ′ k . 1, where KL is the constant in little Grothendieck theorem. [δn] Further, define a map W : ℓ2 → B(Hn ) by: (3.37)

W (ek ) = |1ihk|, k = 1, . . . , [δn].

It is easy to check W is a contraction, i.e. kW k ≤ 1. Finally, we consider the element: (3.38) where a = (3.39)

P[δn]

k=1 ek

F = (V ⊗ W )(a) ∈ ℓn1 (ℓn∞ ) ⊗ B(Hn ), ⊗ ek . On one hand, we have

kF kℓn1 (ℓn∞ )⊗ǫ B(Hn ) = kV ⊗ W (a)kℓn1 (ℓn∞ )⊗ǫ B(Hn ) ≤ kV kkW kkakℓ[δn]⊗ǫ ℓ[δn] . 2

2

p log n.

10

M. Horodecki, M. Marciniak and Z. Yin

On the other hand, the formula (3.30) implies (3.40)

kF kℓn1 (ℓn∞ )⊗min B(Hn ) &

& k(V ∗ ⊗ id)(F )kR[δn] ⊗min B(Hn ) = k(V ∗ ⊗ id)(V ⊗ W )(a)kR[δn] ⊗min B(Hn ) = k

[δn] X

k=1

ek ⊗ |1ihk|kR[δn] ⊗min B(Hn ) = k

Combining equations (3.39) and (3.40) we get

X k

1

|1ihk|kih1|k 2 &

√

n.

√ kF kmin n &√ kF kǫ log n

(3.41) and the proof is finished.



It turns out that the specific F which appeared in the above proof, we have the following upper bound. Proposition 3.15. The element F = (V ⊗ W )(a) ∈ ℓn1 (ℓn∞ ) ⊗ B(Hn ) defined in (3.38) verifies p (3.42) kF kℓn1 (ℓn∞ )⊗min B(Hn ) . n log n. [δn]

Proof. Let us consider the map W : ℓ2 we have

→ B(Hn ) defined in (3.37). By the result of Pisier ([18]),

kW : OHδn → B(Hn )k2cb (3.43)

1



X

[δn]

= |1ihk| ⊗ |1ihk|

k=1



21

[δn]

X

≤ |1ihk|kih1|

k=1

√ . n.

B(Hn )⊗min B(Hn )

21

[δn]

X

|1ihk|1ihk|

k=1

Hence kW : OH[δn] → B(Hn )kcb . n 4 . On the other hand, Junge and Palazuelos ([9]) showed that 1p (3.44) kV : OH[δn] → ℓn1 (ℓn∞ )kcb . n 4 log n. Therefore, we obtain:

kF kℓn1 (ℓn∞ )⊗min B(Hn ) (3.45)



X

= (V ⊗ W )( ek ⊗ ek )

n k ℓ (ℓn )⊗min B(Hn )

1 ∞

X

p

. n log n ek ⊗ ek

k OH[δn] ⊗min OH[δn]



X p p

= n log n ek ⊗ ek = n log n.

[δn] [δn] k

ℓ2

⊗ǫ ℓ 2



Remark 3.16. In [9, 10] M. Junge, C. Palazuelos, D. P´erez-Garc´ıa, I. Villanueva and M. M. Wolf studied the violation of bipartite Bell inequality. They compared the minimal operator space tensor norm with the injective Banach space tensor norm on ℓn1 (ℓn∞ ) ⊗ ℓn1 (ℓn∞ ). In the steering scenario we consider the assemblages instead of the joint probabilities, so now we consider the minimal and injective tensor norms on ℓn1 (ℓn∞ ) ⊗ B(Hn ). To see the unbounded largest violation, we can easily construct a specific steering inequality just by using their argument. However, there exists a steering√inequality which with even larger violation. In [13] an inequality with largest violation of order n is constructed. Anyhow, we made a first try to apply the operator space theory to calculate the largest violation in steering scenario. We will see later many advantages of this approach.

Operator space approach to steering inequality

11

3.4. Explicit form of the violation. Up to now, we have constructed a steering inequality with an unbounded largest violation. Similar to the Bell scenario, we can easily construct all ingredients explicitly. Let ǫkx,a , x, a, k = 1, . . . , n be independent Bernoulli sequences. For a constant K we define. i) Steering inequality Fxa ∈ B(Hn+1 ):

 n+1 X  k−1 1 ǫx,a |1ihk| a Fx = n k=2   0

(3.46)

x, a = 1, . . . , n, a = n + 1.

ii) POVMs measurements ([9]) {Exa }n,n+1 x,a=1 in B(Hn+1 ) as       1     nK



ǫ1x,a 1 .. .

1

ǫ1x,a   ..  . a Ex = ǫnx,a ǫnx,a ǫ1x,a     n  X    Exa 1−

(3.47)

··· ··· ···

 ǫnx,a ǫ1x,a ǫnx,a   ..  . . 

a = 1, . . . , n,

1

a=n+1

a=1

for x = 1, . . . , n. iii) States: If (αi )n+1 i=1 is a decreasing and positive sequence then set (3.48)

|ϕα i =

n+1 X i=1

αi |iii.

Now we have the following Theorem 3.17. There exist universal constants C and K such that for every n ∈ N we can choose with “high probability” a system {ǫkx,a }nx,a,k=1 verifying that {Exa }n,n+1 x,a=1 define POVMs, ( n n+1 ) X X p a a a BC (F ) = sup T r(Fx σx ) : σx ∈ L ≤ C log n

(3.49)

x=1 a=1

and (3.50)

BQ (F ) ≥

n n+1 X X

x=1 a=1

n+1

Tr(Fxa TrA (Exa ⊗ 1l|ϕα ihϕα |)) =

X 1 α1 αk . K k=2

f n+1 → B(Hn+1 ) as follows: Proof. For (3.49), we define two maps Ve : ℓn+1 → ℓn1 (ℓn+1 ∞ ) and W : ℓ2 2  0 k=1   n n (3.51) Ve (ek ) = 1 X X k−1  ǫx,a ex ⊗ e′a k = 2, . . . , n + 1 n x=1 a=1 and

f (ek ) = W

(3.52)

(

0 |1ihk|

k=1 k = 2, . . . , n + 1

By [9, Lemma 3.5] and the following contraction principle ([17]) (3.53)



r n X n X n n X n X n

X

X π



k E gx,a ek ⊗ (ex ⊗ e′a ) E ǫkx,a ek ⊗ (ex ⊗ e′a ) ≤

n n n

2 x=1 a=1 x=1 a=1 k=1

ℓ2 ⊗ǫ ℓ1 (ℓ∞ )

k=1

n n ℓn 2 ⊗ǫ ℓ1 (ℓ∞ )

,

12

M. Horodecki, M. Marciniak and Z. Yin

we get

(3.54)

EkVe :

ℓn+1 2

→

ℓn1 (ℓn+1 ∞ )k

n n n+1

1

X X X k−1 ′ ǫx,a ek ⊗ (ex ⊗ ea ) = E

n+1 n n+1 n x=1 a=1 k=2 ℓ ⊗ǫ ℓ1 (ℓ∞ )

2

n n n

X X X 1

ǫkx,a ek ⊗ (ex ⊗ e′a ) = E

n n n x=1 a=1 k=1 ℓ2 ⊗ǫ ℓ1 (ℓn ∞) p ≤ C log n,

where we have used the injectivity of ǫ-norm for the second equation. Then by Chebyshev’s √ inequality, with “high probability” we can choose {ǫkx,a } such that: kVe k ≤ C log n. Moreover, it f : ℓn+1 → B(Hn+1 ) is a contraction, i.e. kW f k ≤ 1. is easy to see the map W 2 Hence, by Proposition 3.10, we have

n n+1

X X

ex ⊗ e′a ⊗ Fxa BC (F ) ≤

n n+1

x=1 a=1 ℓ (ℓ∞ )⊗ǫ B(Hn+1 )

1 n+1

X

f( = Ve ⊗ W ek ⊗ ek ) (3.55)

n n+1 k=1 ℓ (ℓ∞ )⊗ǫ B(Hn+1 )

1

n+1

X p p

≤ C log n ek ⊗ ek ) = C log n.

n+1 n+1

k=1

ℓ2

⊗ǫ ℓ 2

As regards inequality (3.50) we have BQ (F ) ≥

(3.56)

n n+1 X X

x=1 a=1

Tr(Fxa TrA (Exa ⊗ 1l|ϕα ihϕα |))

  n n n+1 n+1 X 1 X X  X k−1 Tr αi αj hj|Exa |ii|iihj| = ǫx,a |1ihk| n x=1 a=1 i,j=1 k=2

=

n X n n+1 X X

1 α1 n2 K x=1 a=1

k=2

n+1

k−1 k−1 αk ǫx,a ǫx,a =

X 1 α1 αk . K k=2

Pn+1 q 1−α2



Remark 3.18. Given α ∈ (0, 1) let us consider |ϕα i = α|11i + i=2 n |iii. It follows from √ √ √ 1 2 the above theorem that BQ (F ) ≥ K α 1 − α n & n. So, we have constructed explicitly a √ 1 steering inequality F such that LV (F ) & √logn n . On the other hand, if we take αi = √n+1 for each i = 1, . . . , n + 1, i.e. |ϕα i is the maximally entangled state, then we can not obtain this unbounded largest violation. We will focus on this problem in next subsection. 3.5. Largest steering violation by maximally entangled state. Let d ∈ N and ρ = |ψd ihψd | Pd be the maximally entangled state acting on ℓd2 ⊗2 ℓd2 , i.e. |ψd i = √1d i=1 |iii. In order to estimate the bound of steering violation with respect to this maximally entangled state, we can consider the following tensor norm on two operator spaces (see [9, 11]). Given two operator spaces E and F , for any a ∈ E ⊗ F we define its ψ − min norm: (3.57)

kakψ−min = sup |hψd |(u ⊗ v)(a)|ψd i|,

where the supremum runs over all d and all complete contractions u : X → B(Hd ) and v : Y → B(Hd ). The next lemma follows directly from Proposition 3.10 (or see [10]). P P ′ a n m Lemma 3.19. Given an element F = nx=1 m a=1 (ex ⊗ ea ) ⊗ Fx ∈ ℓ1 (ℓ∞ ) ⊗ B(Hd ), we have: (3.58)

sup

Θmax ∈Q|ψd i

|hF, Θmax i| . kF kψ−min,

Operator space approach to steering inequality

13

where Q|ψd i = {(σxa ) = (T rA (Exa ⊗ 1lB )|ψd ihψd |) : {Exa }n,m x,a=1 is a POVMs}., i.e. Q|ψd i is the set of all assemblages constructed with the d-dimensional maximally entangled state. In [9] the authors provided an example of a Bell inequality which gives Bell violations of order but only bounded violations can be obtained with any maximally entangled state. It is not surprising that we have similar conclusion in the steering scenario. √

n log n

Theorem 3.20. There exist a steering inequality 2 Fe = {Fexa ∈ B(Hn+1 ) : x = 1, . . . , 2n , a = 1, . . . , n + 1},

(3.59) such that: i) LV (Fe ) &

√ √ n ; log n

ii) supΘmax ∈Q|ψn i |hFe , Θmax i| .

√ log n.

Proof. In [9, Theorem 5.1] T the authors proved that there are linear maps S : Rn Cn such that and S ∗ : ℓk1 (ℓDn ∞ ) → Rn p ∗ (3.60) S S = idℓn2 , kS ∗ kcb ≤ C, and kSkcb ≤ C log n, 2

T

Cn → ℓk1 (ℓDn ∞ )

2

where k ≤ 2D n . Consider a map W : ℓn2 → B(Hn ) given by W (ek ) = |1ihk| for k = 1, . . . , n. It is easy to check that W is a complete contraction from Rn to B(Hn ). Indeed, X kW : Rn → B(Hn )kcb = k ek ⊗ |1ihk|kCn ⊗min B(Hn ) (3.61)

k

=k

X k

1

|kih1|1ihkk 2 = 1

P As in the proof of Theorem 3.14, we can show that the element: F = (S ⊗ W )( k ek ⊗ ek ) ∈ ℓk1 (ℓDn ∞ ) ⊗ B(Hn ) satisfies: p √ (3.62) kF kǫ . log n and kF kmin & n. Moreover,





X

X



kF kψ−min = S ⊗ W ( ek ⊗ ek ) ≤ kSkcb kW kcb ek ⊗ ek



k k ψ−min ψ−min (3.63)

X

p p

ek ⊗ ek . log n. ≤ C log n

k ǫ T Here we have used the fact that for every a ∈ (Rn Cn ) ⊗ Rn

kak(Rn T Cn )⊗ψ−min Rn ≤ kak(Rn T Cn )⊗ψ−min (Rn T Cn ) ≤ Ckakℓn2 ⊗ǫ ℓn2 . P Now, we can represent F as F = x,a ex ⊗ e′a ⊗ Fxa , then define Fexa ∈ Mn+1 , such that the left-top n × n corner of Fexa is Fxa and other coefficients of Fexa are zeros. If we combine equations (3.62), (3.63) and Lemma 3.19, we get the conclusion of the theorem.  (3.64)

Conclusion. Here we get a steering√inequality Fe. By using operator space methods, its quantum bound is greater than the order of n, mean while the quantum bound obtained from the n + 1√ dimensional maximally entangled state is bounded by the order of log n. It means we have found √ an example of steering inequality which will give an unbounded violation of order √logn n . But this unbounded violation can never been obtained by the maximally entangled state. 3.6. Steering violation by partially entangled states including PPT states. Here we will consider the role of the partially entangled state inP violation of steering inequalities. P Recall the n-dimensional pure partial entangled state: |ψα i = ni=1 αi |iii, where αi > 0 and i α2i = 1. By the Remark 3.18, we know for the given steering inequality Fxa (see (3.46)), through this partially √ entangled state, we can obtain an unbounded largest violation of the order √logn n . For any given

14

M. Horodecki, M. Marciniak and Z. Yin

steering inequality Fxa , x, a = 1, . . . , n, the quantum bound obtained from the partial entangled state is: n n X o BQ|ψα i (F ) = sup T r(Fxa T rA ((Exa ⊗ 1l)|ψα ihψα |)) : Exa be POVMs x,a=1

(3.65)

o n X X αi αj hj|Exa |iihj|Fxa |ii : Exa = sup

≤ ≤ ≤

X i,j

X

x,a i,j

n X o αi αj sup | T r(Exa |iihj|)T r(Fxa |iihj|)| : Exa x,a

αi αj sup{k

i,j

X i,j

αi αj k

X x,a

X x,a

Exa ⊗ Fxa kMn ⊗ǫ Mn : Exa }

ex ⊗ ea ⊗ Fxa kℓn1 (ℓn∞ )⊗ǫ Mn .

For the second inequality, we have used the fact that: T r(· |iihj|) ∈ BS1n . Now we consider following density matrix, ρλ = (1 − λ) n1l2 + λ|ψα ihψα |, 0 ≤ λ ≤ 1. By the previous discussion, the quantum bound obtained from ρλ , denote by BQλ (F ), is bounded by X X αi αj )k ex ⊗ ea ⊗ Fxa kℓn1 (ℓn∞ )⊗ǫ Mn . (3.66) (1 − λ + λ i,j

x,a

⊗ ea ⊗ Fxa kℓn1 (ℓn∞ )⊗ǫ Mn . Thus, X αi αj )BC (F ). BQλ (F ) . (1 − λ + λ

On the other hand, we know BC (F ) ≈ k

(3.67)

P

x,a ex

i,j

In [20], the author has presented a stronger version of Peres conjecture: “PPT states can not violate steering inequalites, i.e. the assemblages obtained by measuring them always have LHS models.” The conjecture has been then disproved in [14]. However, one can still ask, whether PPT states can exhibit unbounded violation. Below we consider three classes of PPT states, and show that they allow only for bounded steering violation. Consider first PPT states which is obtained from ρλ . Since we know the eigenvalue of the partial transpose of ρλ is  1−λ   ± λαi αj i 6= j, n2 (3.68)   1 − λ + λα2 i = 1, . . . , n. i n2 Then ρλ is a PPT state if and only if λ ≤ min{ 1+n21αi αj : i 6= j}. From the equation (3.67), we have   X αi αj  BC (F ) BQλ (F ) . 1 − λ + λ i,j

(3.69)

where α = maxi {αi }. Since following conclusion:

P

i

   n−1 : i 6= j BC (F ) ≤ 1 + min 1 + n2 αi αj   n−1  BC (F ), q ≤ 1 + 2 1 + n2 α 1−α n−1 α2i = 1, then α ≥

√1 . n

Thus BQλ (F ) . BC (F ). Now we can make

Conclusion. For any PPT state ρλ = (1 − λ) n1l2 + λ|ψα ihψα |, λ ≤ min{ 1+n21αi αj : i 6= j}, and any given steering inequality F , the quantum bound obtained by ρλ is bounded by the classical bound up to an universal constant. From this point of view, we supply some evidence to support the Peres conjecture (in steering scenario) by analytic method.

Operator space approach to steering inequality

15

In [4], D. Chru´sci´ nski and A. Kossakowski introduced a class of PPT states. This class is invariant under the maximal commutative subgroup of U(n), and it is includes the previous isotropic state ρλ . Briefly speaking, they consider the following two classes of PPT states: Pn Pn i) Isotropic-like state, ρ = i,j=1 aij |iiihjj| + i6=j=1 cij |ijihij|, (aij )i,j ≥ 0, cij ≥ 0, cij cji − P P n n |aij |2 ≥ 0, i=1 aii + i6=j=1 cij = 1; Pn Pn ii) Werner-like state, ρe = bij |ijihji| + i6=j=1 cij |ijihij|, (bij )i,j ≥ 0, cij ≥ 0, cij cji − i,j=1 Pn Pn |bij |2 ≥ 0, i=1 bii + i6=j=1 cij = 1.

Now we will prove that for these two class of PPT states, we can always obtain bounded largest violation. To see this, we will use following proposition, which is analogue to the Theorem 2.1 of [16]. Proposition 3.21. Given an n-dimensional bipartite state ρ ∈ Sn1 ⊗ Sn1 , then for any steering inequality Fxa , x, a = 1, . . . , n, we have (3.70)

Bρ (F ) ≤ kρkSn1 ⊗π Sn1 BC (F ), n P o where Bρ (F ) = sup nx,a=1 Tr(Fxa TrA (Exa ⊗ 1lρ)) : Exa be POVMs .

Proof. The proof is more or less the same with the proof in [16]. By duality and Proposition 3.10, we have: X X Tr(Fxa TrA (Exa ⊗ 1lρ)) = Tr(Exa ⊗ Fxa ρ) x,a

x,a

X

≤ Exa ⊗ Fxa

(3.71)

Mn ⊗ǫ Mn

x,a

kρkSn1 ⊗π Sn1

X

≤ ex ⊗ e′a ⊗ Fxa

n ℓn 1 (ℓ∞ )⊗ǫ Mn

x,a

kρkSn1 ⊗π Sn1

≤ BC (F )kρkSn1 ⊗π Sn1 .

 Now it remains to calculate the projective norm of Isotropic-like state and Werner-like state. It can be proved that the norms of both states are bounded by 2. For instance, for any Isotropic-like state ρ, kρkπ ≤ (3.72)

n X

i,j=1

≤1+

|aij | +

X√ i6=j

n X

i6=j=1

cij ≤ 1 + 

cij cji ≤ 1 + 

X i6=j

X i6=j

|aij |

2

cij  ≤ 2.

Remark 3.22. The PPT states of [4] cover many PPT entangled states known in literature, however it does not describe bound entangled states constructed via unextendible product bases (UPB) [2]. But unfortunately, up to now we can’t estimate the projective norm of PPT states which is constructed by UPB. 4. Dichotomic case In [21] the authors considered the dichotomic setting for Bell scenario in (see also [19]). It is more or less a reformulation of the standard setting for the Bell scenario with two outcomes. It turns out that in the case of the steering scenario there is no longer the case: standard and dichotomic settings are not equivalent. The details of this phenomenon are discussed in [13]. Here we describe its particular exemplification using the operator space techniques. In the dichotomic setting for steering scenario we assume that the measurement for Alice have only two outcomes ±1. Alice prepares two correlated particles sharing with a quantum state ρ ∈ B(HA ⊗ HB ) and send one of them to Bob. Alice wants to convince Bob that ρ is an entangled

16

M. Horodecki, M. Marciniak and Z. Yin

state by doing dichotomic measurement −1l ≤ Ex ≤ 1l, x = 1, . . . , n. After Alice’s measurement has been done, Bob obtains the conditional states (4.1)

σx = TrA (ρ(Ex ⊗ 1l)).

If the nature is described by an LHS model, then X (4.2) σx = p(λ)Ex (λ)σλ , λ

where P (λ) is a probability distribution function, Ex (λ) = ±1 is the deterministic outcome obtained by Alice if she does the measurement Ex , and σλ is a density matrix of B(HB ). For n = dim(HB ) we define a steering inequality as a set of n × n matrices Fx , x = 1, . . . , n. Analogously to the standard case, we can define the classical bound of F as: ( ) X (4.3) BC (F ) = sup Tr(Fx σx ) : σx satisfy (4.1) , x

and the quantum bound of F as (4.4)

( ) X BQ (F ) = sup Tr(Fx σx ) : σx satisfy (4.2) . x

One can also apply the argument of Proposition 3.10 to obtain BC (F ) ≃ kF kℓn1 ⊗ǫ Mn

and BQ (F ) ≃ kF kℓn1 ⊗min Mn .

Remark 4.1. In P [21], the authors considered the Bell scenario in dichotomic setting: for a Bell n inequality M = x,y=1 Mx,y ex ⊗ ey ∈ ℓn1 ⊗ ℓn1 , the classical bound BC (M ) is equivalent to the norm kM kℓn1 ⊗ǫ ℓn1 and the quantum bound BQ (M ) ≃ kM kℓn1 ⊗min ℓn1 . By Grothendieck’s theorem [19], we have ℓn1 ⊗ǫ ℓn1 ≃ ℓn1 ⊗min ℓn1 . So in the dichotomic bipartite case, for any Bell inequality M , there always exists an universal constant K (not depend on the dimension), such that BQ (M ) ≤ KBC (M ). They also proved that for tripartite case, this universal constant doesn’t exist! The situation in steering scenario differs from the features of the Bell scenario described in the above remark. Since ℓn1 ⊗ǫ Mn ≇ ℓn1 ⊗min Mn , one should expect that there is a room for a steering inequality with unbounded largest violation. Such an inequality has been provided in Ref. [13]. Here we restate this result as the following theorem, and provide a proof referring to operator space formalism. n Theorem 4.2. For every n ∈ N, there √ exists a steering inequality (Fx )x=1,...,n ∈ ℓ1 ⊗ Mn with unbounded largest violation of order log n.

Proof. It is enough to prove the following claim: For every n ∈ N, there exists an element F = P n n x=1 ex ⊗ Fx ∈ ℓ1 ⊗ Mn , such that kF kmin p & log n. (4.5) kF kǫ

Since we know that ℓn1 ⊗ǫ Mn = B(ℓn∞ , Mn ) isometrically and ℓn1 ⊗min Mn = CB(ℓn∞ , Mn ) completely isometrically,√it is enough to prove that there exists a map φ from ℓn∞ to Mn , such that kφk . 1 and kφkcb & log n. Here we will use a fact described in [12]: there is a map ϕ : ℓn∞ → M2n , such that ϕ is bounded but not completely bounded. For the reader’s convenience, we just rewrite their proof. Let σi , i = x, y, z be Pauli matrices. Let n−1

z }| { A1 = σx ⊗ 1l ⊗ . . . ⊗ 1l,

n−2

(4.6)

z }| { A2 = σz ⊗ σx ⊗ 1l ⊗ . . . ⊗ 1l, .. .

k−1

n−k

z }| { }| { z Ak = σz ⊗ . . . ⊗ σz ⊗σx ⊗ 1l ⊗ . . . ⊗ 1l, 3 ≤ k ≤ n.

Operator space approach to steering inequality

17

. . , n satisfies Ai = A∗i , Ai Aj + Aj AiP= 2δij 1l2n . It is easy to check that these Ai ∈ M2n , i = 1, .P 2 n ∗ ∗ n n Moreover, they pP a basis of M2 . For any A = i ai Ain∈ M2 , since A A+AA1 = 2 i |ai | 1l2 , √ form 2 √ Ai , i = 1, . . . , n. then kAk ≤ 2 i |ai | . Now we define the map ϕ : ℓ∞ → M2n as ϕ(ei ) = n Then

r s !

1 X

X √ 2 X



ai e i = √ (4.7) |ai |2 ≤ 2 sup |ai |. ai Ai ≤

ϕ

n

n i i i i P Thus kϕk . 1. On the other hand, we let θ = i Ai ⊗ei ∈ M2n ⊗ℓn∞ . Note that kθk = supi kAi k = 1, n n and by using the fact that [12]: there is a unit vector z ∈ C2 ⊗ C2 such that (Ai ⊗ Ai )(z) = z for any i. Then



1 X

Ai ⊗ Ai kϕkcb ≥ k1lM2n ⊗ ϕ(θ)k = √

n i * + (4.8) √ 1 X 1 X ≥ √ (Ai ⊗ Ai )(z), z = √ hz, zi = n. n n i

i

Since for very natural number n ≥ 2, there exists natural number m, such that n ≥ 2m , consider the diagram ϕ

ω

ω

2 1 ℓm ℓn∞ −→ ∞ −→ M2m −→ Mn ,

(4.9)

where ω1 projects ℓn∞ onto the first m coordinates, and ω2 embeds M2m into the top 2m × 2m corner of Mn . Set φ = ω2 ◦ ϕ ◦ ω1 : ℓn∞ → Mn , then p √ (4.10) kφk = kϕk . 1; and kφkcb = kϕkcb ≥ m ≥ log n. √ Pn cb log n. If F = x=1 ex ⊗ φ(ex ), then F Thus we can find a map φ : ℓn∞ → Mn , such that kφk kφk & satisfies the statement of the theorem.  From this theorem, in the dichotomic case, the unbounded largest violation derive from some bounded but not completely bounded map. Now we will discuss for what kind of steering inequality Fx , we can always get bounded largest violation. For any given steering inequality (Fx )x=1,...,n , the classical bound: ! ) ( X X BC (F ) = sup Tr Fx p(λ)Ex (λ)σλ x λ

( ! ) (4.11) X

X X



= sup Tr Fx p(λ)σλ ≤ Fx .



x

x

λ

Mn

On the other hand,



X

ex ⊗ Fx BC (F ) ≈

n x ℓ 1 ⊗ǫ Mn

(4.12) ( ) X

X

= sup f (ex )g(Fx ) : f ∈ Bℓn∞ , g ∈ BS1n ≥ Fx .

x x Mn P Thus BC (F ) ≈ k x Fx kMn . P For the quantum bound, BQ (F ) ≈ k x ex ⊗ Fx kℓn ⊗min Mn . It is known that [10, 18] 1

X n X o



= sup : Ux ∈ Mn , Ux Ux∗ = Ux∗ Ux = 1l. ex ⊗ Fx n Ux ⊗ Fx

(4.13)

x

ℓ1 ⊗min Mn

x

Mn ⊗min Mn

12

12 X o n X

c∗x cx : Fx = bx cx . = inf bx b∗x x

x

If Fx is a positive matrix for every x = 1, . . . , n, then by the lemma 2 of [10], we know X X (4.14) k ex ⊗ Fx kℓn1 ⊗min Mn = k Fx kMn . x

x

18

M. Horodecki, M. Marciniak and Z. Yin

If Fx Fx∗′ = 0 or Fx∗ Fx′ = 0 for every x 6= x′ , then X X X k Ux ⊗ Fx k2min = k Ux Ux∗′ ⊗ Fx Fx∗′ kmin = k Ux Ux∗ ⊗ Fx Fx∗ kmin (4.15)

x,x′

x

=k

X x

x

Fx k2Mn .

We end this section with following remark. RemarkP4.3. If Fx satisfy any onePof following conditions: 1) Fx ≥ 0; 2) Fx Fx∗′ = 0; 3) Fx∗ Fx′ = 0. Then k x ex ⊗ Fx kℓn1 ⊗min Mn ≈ k x ex ⊗ Fx kℓn1 ⊗ǫ Mn . In other words, the quantum bound of these kinds of steering inequalities are always bounded by their classical bound. Acknowledgements We would like to thank Prof. W.A. Majewski for valuable remarks and fruitful discussion. The work is supported by Foundation for Polish Science TEAM project co-financed by the EU European Regional Development Fund, Polish Ministry of Science and Higher Education Grant no. IdP2011 000361, ERC AdG grant QOLAPS and EC grant RAQUEL and a NCBiR-CHIST-ERA Project QUASAR. Part of this work was done in National Quantum Information Center of Gda´ nsk. Part of this work was done when the authors attended the program Mathematical Challenges in Quantum Information at the Isaac Newton Institute for Mathematical Sciences, University of Cambridge. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

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Operator space approach to steering inequality

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[27] R. F. Werner. Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. Phys. Rev. A. 40, (1989) 4277. [28] H. M. Wiseman, S. J. Jones and A. C. Doherty. Steering, Entanglement, Nonlocality, and the EPR Paradox. Phys. Rev. Lett. 98, (2007) 140402. ArXiv:quant-ph/0612147 ´ sk University, Wita Stwosza 57, 80-952 Institute of Theoretical Physics and Astrophysics, Gdan ´ sk, Poland; National Quantum Information Centre of Gdan ´ sk, Andersa 27, Sopot, 81-824, Poland Gdan E-mail address: [email protected] ´ sk University, Wita Stwosza 57, 80-952 Institute of Theoretical Physics and Astrophysics, Gdan ´ sk, Poland Gdan E-mail address: [email protected] ´ sk University, Wita Stwosza 57, 80-952 Institute of Theoretical Physics and Astrophysics, Gdan Gdansk, Poland E-mail address: [email protected]