SCIENCE CHINA Physics, Mechanics & Astronomy
. Article
May 2013 Vol. 56 No. 5: 947–951 doi: 10.1007/s11433-013-5045-1
The quantification of quantum nonlocality by characteristic function WEN Wei* State Key Laboratory for Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083, China Received April 19, 2012; accepted May 14, 2012; published online April 15, 2013
We propose a way to measure the strength of quantum nonlocal correlation (QNC) based on the characteristic function, which is defined as a response function under the local quantum measurement in a composite system. It is found that the strength of QNC based on the characteristic function is a half-positive-definite function and does not change under any LU operation. Generally, we give a new definition for quantum entanglement using the strength function. Furthermore, we also give a separability-criterion for 2 × m-dimensional mixed real matrix. This paper proposes an alternative way for QNC further research. quantum nonlocality, characteristic function, strength of QNC, quantum entanglement, Schr¨odinger steering PACS number(s): 03.67.Mn, 03.65.Ud Citation:
Wen W. The quantification of quantum nonlocality by characteristic function. Sci China-Phys Mech Astron, 2013, 56: 947–951, doi: 10.1007/s11433013-5045-1
One of the most subtle phenomena in quantum theory is quantum nonlocal correlation (QNC). Although a large amount of research on QNC has been done, it has mainly arisen from the view that nonlocality cannot be described by any local hidden variable (LHV) theory. Based on this, some new concepts have been proposed, such as Bell nonlocality [1], quantum entanglement [2–4], Schr¨odinger’s steerability [5–8] and so on, which are all defined by different forms of the local joint quantum measurement (LJQM) probability P(a, b|A, B; W) [7,8]. However, the question about what is the QNC is still far from being solved. Till now, although much attention is paid to the information perspective, the physical aspect of the QNC is also worth studying. Recently, some other researchers have paid attention to other unorthodox methods and suggested that nonlocality could be more general [9–13]. For example, Bandyopadhyay [12] presented that the nonlocality could be redefined by local indistinguishability of a set of orthogonal quantum states, and showed that more nonlocality may be with less purity. *Corresponding author (email:
[email protected])
c Science China Press and Springer-Verlag Berlin Heidelberg 2013
Luo and Fu [13] pointed out that the measurement could induce the non-locality. These works all try to find a new way to study quantum nonlocality and inspirit researchers to reinspect the physical action played in the QNC. In this work, we will study the QNC from the other perspective. QNC is drawn out to be a component of a density matrix and then can be quantitative determination in mathematization. We express the relation between QNC and the state of a composite system with the mathematical language as follows: B : ρABC··· → {{ρA , ρB , ρC , · · · }, {QNCs}}.
(1)
The function B is a bijection and every composite state is mapped into the set of subsystems and the QNC between them. Our task is to find a mathematical expression to describe the QNC B : ρABC··· → {{ρA , ρB , ρC , · · · }, FQNC }.
(2)
We define the function F that maps the abstract physical quantity QNC to a mathematica quantity as a characteristic function of QNC. A special function F corresponds to a special composite state once its sub-states are fixed on. phys.scichina.com
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We can use the characteristic function to analyze and classify the composite correlative system. Furthermore, as we can see, the characteristic function is much more fundamental than the previous concepts that are defined in quantum correlation measurement. Hence, it could not only redefine these concepts, such as the Bell nonlocality, quantum entanglement, Schr¨odinger’s steerability, but also show the sublet differences among the types with the same correlative measurement. In this work, we find out a formulation about the characteristic function and discuss a possible quantum entanglement definition based on this formulation.
1 The definition of the characteristic function Before expatiating on the characteristic function, we will introduce some intuitive-right but unobvious conclusions first: a set of local unlimited quantum measurements can totally detect an arbitrary density matrix. An unlimited quantum measurement is defined as a physical process in which the projection operator can be arbitrarily chosen and the number of copies of the unknown state measured is sufficient. Considering a general situation, an arbitrary projection opˆ erator in a finite n Hilbert space is expressed as M(Θ, Ψ) = |ψ(Θ, Ψ)ψ(Θ, Ψ)|, where Θ and Φ are the sets of variables {ψk } and {θk }. |ψ(Θ, Φ) is a pure state in n-dimensional space, iφl |ψ(Θ, Ψ) = nk=1 ak |k, where ak = k−1 l=1 sin(θl ) cos(θk )e n−1 when k < n and an = l=1 sin(θl ). We bring out the following lemmas about the unlimited-quantum measurement. ˆ Lemma 1 If ρ1 , ρ2 are density matrixes and M(Θ, Φ) is the projection operator in a finite n-dimensional Hilbert ˆ ˆ space, we say ρ1 = ρ2 when Tr( M(Θ, Φ)ρ1 ) = Tr( M(Θ, Φ)ρ2 ) for ∀θi ∈ (0, π) and ∀φi ∈ (0, 2π). Proof It is known any n × n density matrix ρ x can be decomposed into an orthonormal basis Γ(k) n of traceless generators of group S U(n) [14,15] ⎛ ⎞ 2 n −1 ⎟⎟⎟ 1 ⎜⎜⎜⎜ (k) (k) ρ x = ⎜⎜⎝1 + r x Γn ⎟⎟⎟⎠ , n k=1
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√ (k) n − 1. where r(k) x = Tr(ρ x Γn ) and the length of vector |r| = (i) ( j) The generators Γ(k) n have the property of Tr(Γn Γn ) = nδi j . The n2 − 1 real parameters r(k) x will uniquely determine a density matrix ρ x , and vice versa. According to eq. (3), when n = 2, ρ1 and ρ2 can be replaced with vectors r1 and r2 in the Bloch-Sphere and Mˆ with r M (θ, φ) on the surface of the Bloch-Sphere. Consequently, ˆ 1 ) = Tr( Mρ ˆ 2 ) means that r M (θ, φ) · (r1 − r2 ) = 0. The Tr( Mρ last equation holds iff r1 − r2 = 0, namely ρ1 = ρ2 . For n > 2, we can always reduce it into n(n − 1)/2 2dimensional subsystems and show any corresponding subsystems are equal. To detail this process, we define an operation V(i, j)n,m = δn,m (δn,i + δn, j ) first. It is known that R s (i, j) = V(i, j)RV(i, j)†
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satisfies Ri, j = (R s (i, j))i, j , where R is an n × n matrix. ˆ Moreover, Mˆ s (i, j) = V(i, j) MV(i, j)† is an equivalent 2dimensional projection operator and can be realized by setting some variables φk and θk to zero. Tr( Mˆ s (i, j)ρ1 ) = ˆ s (i, j)ρ2 ) for every i and j, namely V(i, j)ρ1 V(i, j)† = Tr( M V(i, j)ρ2 V(i, j)† . Therefore ρ1 = ρ2 . This theorem is very important in this letter to get the characteristic function. It shows us that any states, mixed or pure, can be distinguished by unlimited quantum measurements. It is anti-intuition because the general opinion is that n2 − 1 parameters are needed to fix an arbitrary n × n density matrix, but we show here that quantum measurements with 2n − 1 parameters are just enough. Following this, the corollary about the local quantum measurement is given. It will be shown that unlimit local quantum measurement can also explore the whole information of the composite system. Corollary 1 If ρAB , ρAB are density matrixes in nA ⊗ nB -dimensional Hilbert space and Mˆ A (Θ, Φ) is the projection operator of system A, we say ρAB = ρAB when ˆ A (Θ, Φ)ρ1 ) = TrA ( M(Θ, ˆ TrA ( M Φ)ρ2 ) for ∀θi ∈ (0, π) and ∀φi ∈ (0, 2π). Proof For a composite system HA ⊗ HB , the orthonormal basis {ΓkAB } = {ΓiA ⊗ ΓBj }; n B · i + j = 1, 2, . . . , nA nB . For convenience, Γ(0) = I here. Therefore, it is shown that nA IA ⊗ I B (i) + r(i) Mˆ A ⊗ IB = M ΓA ⊗ I B , nA i=1 (5) nA ,nB IA ⊗ I B ( j,k) ( j) + rAB ΓA ⊗ Γ(k) . ρAB = B nA n B j+k=1 (i) It is noted that only these terms Γ(0) A Γ B remain when a partial trace is done under the system A, and then we take the form (k) ˆ A ρAB ) − TrA ( Mˆ A ρAB ) = TrA ( M rM · Δr(k) (6) AB Γ B , k
where rM = (1, rM ). Therefore, it is gotten that rM · Δr(k) AB = 0 ˆ A ρAB ) − TrA ( M ˆ A ρ ) = 0. According to the Lemma if TrA ( M AB ( jk) 1, ΔrAB = 0, namely ρAB = ρAB .
Let us return to the QNC. A local quantum measurement is under a subsystem ρA of a composite system ρAB spanning in the nA × n B Hilbert space. After a local quantum measurement, subsystem ρA will collapse into |ψ(Θ, Φ)A , and ρB will correspondingly change into ρBM =
TrA ( Mˆ A ρAB ) . ˆ A ρAB ) Tr( M
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ρBM is a functional of Mˆ A . Considering the projection operaˆ + δ Mˆ A (which just like tor has a minimal variety Mˆ A → M the stimulation input), the sub-state ρBM will correspondingly change into ρBM → ρBM + δρBM (which is the response output). Basing on these, we define the following equation about the ˆ : ratio of change δρBM /δ M F(Θ, Ψ; ρAB)A→B =
n−1 Tr(|δφi ρBM |) Tr(|δθi ρBM |) eθ i + eφi . (8) θi ˆ Tr(|δφi Mˆ A |) i=1 Tr(|δ MA |)
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According to eq. (3), we get that Tr(|ρ1 − ρ2 |) = |r1 − r2 |. This equation above can be further rewritten as: F(Θ, Ψ; ρAB )A→B =
n−1 θi M |δ r | B
i=1
|δθi r M |
eθ i +
n−1 φi M |δ r | B
i=1
|δφi r M |
eθ i ,
(9)
where δ x f = (∂ f /∂x)δx. FA→B is a vector in (2n − 1)dimensional space. It forms a surface in this spaces when Φ, and Θ each change from 0 to 2π. FA→B satisfies the formulation of eq. (2) and we name it the characteristic function. Every special QNC corresponds to a unique characteristic function. This character of FA→B can be clearly seen in the following theorem. ( j) Theorem 1 Any two density matrixes, ρ(i) AB and ρAB with the same characteristic function FA→B can be transformed into each other by a local unitary transformation under sys(i j) ( j) (i j) † tem B, namely, ρ(i) AB = (IA ⊗ U B )ρAB (IA ⊗ U B ) . The proof of the corollary is not complicated when one notes that Tr(|(δρBM )(i) |) = Tr(|(δρBM )( j) |), which is equivalent (i j) (i j) ( j) to (ρBM )(i) = U B (ρBM )( j) (U B )† , if ρ(i) AB and ρAB have the same characteristic function. Based on the conclusion of corollary 1, we get the theorem above. The definition of FB→A is analogous with the FA→B and will not be repeated. Either FB→A or FA→B can act as the characteristic function in a composite state. According to this theorem, we can also conclude that |F(Θ, Ψ; ρAB )A→B | = |F(Θ , Ψ ; ρAB )A→B| if ρAB = U A ⊗ U B ρAB U A† ⊗ U B† , where ˆ , Ψ ). This is because ρ M = ˆ U A M(Θ, Ψ)U A† = M(Θ B TrA (MA ρAB ), where MA = U A† MA U A . Moreover, Tr(|δMA |) = Tr(|δMA |) because δMA = U(δQ − δS )U † (δQ and δS are infinitesimal positive operators with orthogonal support). Hence Tr(|δMA |) = 2Tr(δQ) = Tr(|δMA |). Let us show examples. For a pure qubit system, |ψAB = cos α|0, 0 + sin α exp(iγ)|1, 1, the characteristic function can be expressed as: F(θ, φ; |ψAB )A→B =
2| sin 2α|(eθ + eφ ) . (10) 2 + cos 2(θ − α) + cos 2(θ + α)
To show that local transformation cannot change √ the shape of |F|, we let |ψAB = U A |ψAB , where U A |0 = 3/2|0+1/2|1, √ and U A |1 = − 3/2|1 + 1/2|0. We show the pictures of the characteristic function of the state |ψAB and |ψAB in the Bloch sphere. As can be seen in Figure 1, these characteristic functions can be transformed into each other through a rotation.
2 The strength of QNC According to this character, we definite a physical quantity that is independent of the form of unlimited quantum measurement as: √ nA G(ρAB)A→B = |FA→B |Tr(ρA Mˆ A )dR2n−2 Ω, (11) Ω Ω
Figure 1 (Color online) The absolute value of the characteristic function in Bloch Sphere. (a) shows the shape of |F(θ, φ; |ψAB )A→B | and (b) explores the shape of |F(θ, φ; |ψAB )A→B |. We choose α = π/3 here.
where dR2n−2 Ω is dR2n−2 Ω =
n−1
sin(θl )n−l−1 dθl dφl .
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l=1
For a two-particle pure state, it can be proven that G(ρAB )A→B = G(ρAB )B→A, but for an arbitrary state, these two terms are not necessarily equal. Hence we define the strength of QNC as the average value of these two terms: G(ρAB) =
1 (G(ρAB )A→B + G(ρAB )B→A ). 2
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Theorem 2 These density matrixes which can be transformed into each other by local operation have the same nonlocal strength. Namely, G(ρAB ) = G(U A ⊗ U B ρAB U A† ⊗ U B† ). Theorem 2 is obvious because according to the analysis above |F(Θ, Ψ) | = |F(Θ , Ψ )| and the integrating range of eq. (11) is S U(2n − 2) by symmetry. In terms of the definition of eq. (13), some separable states, such as ρAB = cos(α)2 |0000|+sin(α)2 |1111|) have a nonlocal correlation although without entanglement (In fact, G(ρAB ) = 1/2 sin(2α) here). It can be also seen that G(ρAB ) is not monotonic under LOCC, but it is monotonic under local trace-preserving quantum operation. Lemma 2 Suppose ε p is a partial local trace-preserving quantum operation and let ρ be a density operator. Then G(ρAB) G(εPx (ρAB)).
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G(ρAB ) G(ργ⊗ AB ).
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εAp (ρAB ) = i λi U A ⊗ IB ρAB U A† ⊗ IB and εBp (ρAB ) = i λi IA ⊗ U B ρAB IA ⊗ U B† . To prove this lemma, we should use the previous conclusion D(ρ, σ) D(ε(ρ), ε(σ)), where D(ρ, σ) is the trace distance [16]. Based on this lemma, we will get a more important conclusion as follows. Corollary 2 Suppose ρAB = i γi ρ(i) AB is a pure state de (i) (i) (i) composition of ρAB and ργ⊗ = γ ρ i i A ⊗ ρ B , where ρA = AB (i) (i) TrB (ρ(i) AB ) and ρ B = TrA (ρAB ). Then,
The mark γ⊗ here is just an illustration of a direct product decomposition of ρAB and we call ργ⊗ AB the productization of ρAB in brief. It is not difficult to understand this corollary because the ρAB contains more information than ργ⊗ AB . We can
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obtain ργ⊗ AB from ρAB but not the reverse. Then, we define a half-positive-definite quantity E(ρAB) = C inf{γ : G(ρAB ) − G(ργ⊗ AB )},
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where γ = {γi , ρ(i) AB } is a symbol of the set of productization. We can determine if ρAB is separable, E(ρAB ) = 0 and else E(ρAB ) > 0. Let us look at a special example. The nonlocal correlation strength of a pure qubit state |ψAB = cos α|0, 0 + sin α exp(iγ)|1, 1 is G(|ψ) = | sin 2α|. Hence, the maximum of strength of QNC appears where the maximum of entanglement appears. This is not surprising because for an arbitrary two-partite pure state, the form of the productization is determined, equals to zero. Therefore E(|ψAB ) = G(|ψAB), namely the strength of QNC can be used as the measurement form of quantum entanglement in two-partite pure state. For a totally mixed state ρAB = 1/2(|ψ+ψ+ | + |ψ− ψ− |), where |ψ+ and |ψ− are the Bell states, we get ργ⊗ AB
= 1/2(|0000| + |1111|). (17)
γ⊗ Therefore, G(ρAB ) = G ρAB = 1/2 and the entanglement E(ρAB ) = 0. We should note that eq. (16) is usually hard to calculate because we still have not an efficient way to find out the supremum of G(ργ⊗ AB ) for a general state ρAB . However, it does not mean this definition is useless. Historically, the entanglement of formation had been also hard to calculated initially until the concurrence was proposed. Eq. (16) supports an alternate way to research the quantum entanglement and its values need further studies. In fact, it is different from previous theories that this definition is the formulation of the function integral and clearly shows the relationship between QNC and quantum entanglement. The reason why we take eq. (2) is that we are inspired by the words of Schr¨odinger. According to Schr¨odinger, in a composite correlated state, a subsystem will be steered or piloted into one or the other type of state if a local quantum measurement is done on the other subsystem [17]. We think this “steering” can be seen as the corresponding change of one sub-state when the other one is locally measured. It is namely the trace of rBM . In fact, the trace of r BM will form a surface in a (n2B − 1)-dimensional space when the projecˆ tion operator M(Θ, Φ) ranges through the parameter-space Θ| = {[0, π]}, Φ = {0, 2π}. Studying the surface of rBM can result in some new conclusions. For example, consider a (i) (i) nA ⊗ 2 composite separate state ρAB = m i ai ρA ⊗ ρ B and (i) ( j) M with ψA |ψA = δi j . r B will form a m-polyhedron in the Bloch sphere. We give some examples shown in Figure 2. The states {ρ(i) B } are symmetrically distributed on the surface of Bloch sphere. Then we would show rBM forms a regular polyhedron when states {|ψ(i) A } are chosen as the basis-states |iA . This is very interesting, because for an inseparable state, r BM is usually smooth. In fact, the converse result is also
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correct under this situation. Namely, if r BM forms a mpolyhedron, the ρAB must be expressed by the formula above (To get this conclusion we should use corollary 1). Additionally, a more general conclusion is shown as follows. Theorem 3 The sufficient and necessary condition for the 2 ⊗ nB composite state ρAB to be decomposed into ρAB = i i i i ai ρA ⊗ ρ B , where ρA is a real density matrix, is that the ˆ M is constant in n B -dimensional main normal line of Tr(ρA M)r B space. ˆ M = k ak λk r(k) , where λk = Proof Let rb = Tr(ρA M)r B B ˆ If ρAB = i ai ρi ⊗ ρi , λk can be expressed Tr(|ψkA ψkA | M). A B as: λk = cos(θ − αkA )2 + sin 2θ sin 2αkA sin(ϕ/2 − ϕkA /2)2 . (18) Moreover, ρA is real, ϕkA = 0, hence ψ ( j) sin 2(αi − α j )r(i) rθb × rb = B × r B ,
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i, j
where r means the normalization of r. Because, r(k) B is independent of θ and φ, rθb × rψb =cons. Namely, the direction of the main normal line does not change with the variables φ and θ. Conversely, if the main normal line rn is constant, it ( j) can be rewritten as rn = i, j sin 2(αi − α j )r(i) B × rB when αi and α j are appropriately chosen. Therefore, rb can be determined. consequently, according to corollary 1, ρAB is a separable state.
3 Summary and discussion In conclusion, we define the characteristic function of QNC. We regard the characteristic function as a corresponding fun-
M Figure 2 (Color online) The surfaces of r M B . We show the r B of ρAB = m (i) a |i i | ⊗ ρ forms the m-polyhedron in the Bloch sphere. In this i=1 i A A B figure, the states {ρ(i) } are symmetrically distributed on the surface of Bloch B sphere (corresponding to the vertices of the regular polyhedrons). m are chosen as 4, 8, 12 and 20 in (a)–(d), respectively.
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ction and through the trace distance, a differential theory is brought into QNC research. It is explored that once a characteristic function is given, the state of a composite system, with just a local trace-preserving quantum operation uncertainty, will be determined. In this paper, we support an alternate way to analyze and categorize the QNC. In fact, we show quantum entanglement, as a special sort of QNC, can be redefined with the characteristic function. This redefinition, as can be seen in eq. (16), shows a more definite and clear relationship between the quantum entanglement and general QNC. It is worth reminding that the definitions of eqs. (11) and (16) are convergent in mathematics for any dimensional space and the maximum is 1. This is because Ω−1 Ω dR2n−2 Ω = 1 √ ˆ A ) 1. Therefore, it is different from and nA |FA→B |Tr(ρA M the other definitions in measurement of quantum entanglement. As we know, for an arbitrary pure state, any definition based on the Shannon entropy is equivalent to the method of partial entropy of entanglement. However, the partial entropy of entanglement will be divergence when the dimension of the matrix ρAB trends to infinity owning to the term of Tr(ρA ln(ρA )). Additionally, we could extend this characteristic function method to the successive dimension. For example, using this method, we could calculate the strength of the entanglement for state φ(x, y) = δ(x − y). It is shown E(φ(x, y)) = 1.
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We also show a way to distinguish whether a real mixed composite density matrix is separable or not in this paper. We can preliminarily kick out some separable mixed state using the geometrical shape of r BM . In fact, the r BM of some separable mixed state can form an m-polyhedron in the Bloch sphere. We further show that the sufficient and necessary condition for the separability-criterion for a real composite density maˆ M should be a trix is that the main normal line of Tr(ρA M)r B constant. The productization density matrix ργ⊗ AB is brought into this paper. It only possesses the “non-entanglement” correlation of ρAB . Therefore, the strength of ρAB that subtracts the strength of ργ⊗ AB is the quantum entanglement. It is can be seen that the definition E s (ρAB ) = inf{γ : S (ργ⊗ AB ) − S (ρAB )}, is also a possible entanglement measurement definition, where S (ρAB ) is the Shannon entropy of ρAB . However, it should be reminded that the monotonic character of our new definition of quantum entanglement under LOCC has not been rigor-
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ously proven yet although our computation shows that it is correct. This paper is exploratory to exhibit what the QNC in physics is and the value of it should be further explored.
We would like to especially thank Prof. BAI YanKui for useful comments and discussions. This work was supported by the National Basic Research Program of China (Grant No. G2009CB929300).
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