Measurement-induced nonlocality based on the trace norm

Nonlocality is one unique property of quantum mechanics differing from classical world. One of its quantifications can be properly described as the maximum global effect caused by locally invariant measurements, termed as measurement-induced nonlocality (MIN) (2011 \emph{Phys. Rev. Lett.} {\bf 106} 120401). Here, we propose to quantify the MIN by the trace norm. We show explicitly that this measure is monotonically decreasing under the action of completely positive trace-preserving map, which is the general local quantum operation, on the unmeasured party for the bipartite state. This property avoids the undesirable characteristic appearing in the known measure of MIN defined by the Hilbert-Schmidt norm that may be increased or decreased by trivial local reversible operations on the unmeasured party. We obtain analytical formulas of the trace-norm MIN for any $2\times n$ dimensional pure state, two-qubit state, and certain high-dimensional states. As other quantum correlation measures, the new defined MIN can be directly applied to various models for physical interpretations.

The quantum description of nature differs in many aspects from that of our conventional understanding. One of such intriguing difference is the celebrated notion known as nonlocality [1]. It implies that the local operation on one subsystem can alter the overall state of a multipartite system, additionally, the properties of one party may be affected by another spatially separated party. Consider, for instance, the bipartite state ρ AB = 1 i=0 |ii ii|/2, for which there is obviously no quantum correlation between the two qubits. However, a proper choice of the local operation on A, e.g., a completely positive trace-preserving (CPTP) channel Λ with the Kraus operators Λ 0 = |0 0| and Λ 1 = |+ 1| [where |+ = (|0 + |1 )/ √ 2], may create quantum correlation between them. Here, ρ AB is mapped into the state ρ ′ AB = (|00 00| + | + 1 +1|)/2, and the quantum discord (a measure of quantum correlation) [2] for it is given by D ≃ 0.2018. Therefore, state in the Hilbert space H AB is disturbed by the local action of Λ in the space H A .
Historically, nonlocality is widely studied by means of different Bell-type inequalities, and thus was termed as Bell nonlocality [3][4][5]. These inequalities that are derived by the local hidden variable theory may be violated by the quantum measurement outcomes, and can be considered as manifestations of the elusive feature of quantum theory [6]. Although the notion of Bell nonlocality is intimately related to entanglement and sometimes it is taken for granted that the violation of Bell inequalities implies the existence of entanglement in a system [4], it should be note that their relations are in fact very intricate and subtle, and cannot be regarded as synonymous. One reason for this judgement is that when considering the mixed states, they may be entangled but do not possess any Bell-type nonlocality [7]. Other studies further consolidated that there is also nonlocality without entanglement [8], or nonlocality without quantum correlations other than entanglement [9].
The delicate and intriguing features of nonlocality, together with its fundamental role in various quantum communication and computation tasks [10], prompted a huge surge of interest from the quantum community [11][12][13][14]. Considerable efforts have been devoted to the development of this perplexing field in the past few years, with notable progresses being achieved. One such achievement was made by Luo and Fu [15], who presented a new measure of nonlocality which they termed measurement-induced nonlocality (MIN). As the name itself indicates, the MIN characterizes nonlocality from a measurement perspective, and thus is fundamentally different from that of the Bell nonlocality which is featured by Bellinequality violation [6]. It is a manifestation of the global disturbance to the overall state of a system caused by a locally non-disturbing measurement on one subsystem, and is hoped to has potential applications in revealing the advantage of certain quantum tasks [15].
Due to the foregoing reasons, the MIN has been one of current research focuses [16][17][18][19][20][21][22][23][24]. However, the geometric measure of MIN based on the Hilbert-Schmidt norm (Schatten 2norm) [15], while intuitively appealing and conceptually significant, has certain discouraging properties. To see this explicitly, we first recall its definition, which reads [15] for a bipartite state ρ AB in H AB . Here, ||X|| 2 = Tr(X † X) denotes the Hilbert-Schmidt norm, and the maximum is taken over the full set of local projective measurements Π A = {Π A k } that keep the reduced state ρ A = Tr B ρ AB invariant, namely, An analytical formula of MIN for any 2 × n dimensional state can be obtained [15].
Here, we argue that the MIN in Eq. (1), despite being favored for its convenience of calculation, may has certain undesirable properties. More specifically, we will show that it can increase or decrease under trivial local reversible operations on the unmeasured subsystem [B in Eq. (1)]. Consider, for instance, a channel Γ B acting as Γ B (ρ AB ) = ρ AB ⊗ ρ C (i.e., it introduces a local ancilla to B), then by making use of the multiplicativity of the Schatten p-norm under tensor products, we obtain This equality means that N 2 (ρ A:BC ) ≤ N 2 (ρ AB ) as Trρ 2 C ≤ 1. Particularly, if ρ C = I n /n with I n being the n × n identity operator, we will obtain N 2 (ρ A:BC ) = N 2 (ρ AB )/n. This yields N 2 (ρ A:BC ) → 0 when n → ∞, and it differs completely from our intuition that the nonlocal properties of a sys-tem should not be affected by trivially adding or removing an uncorrelated local ancilla.
We remark here that the above perplexity is reminiscent of the perplexity encountered for the geometric measure of quantum discord (GQD) [25][26][27]. In that case, several well-defined measures of GQD have been introduced to remedy this problem [28][29][30][31]. In analogy to the 1-norm GQD [30], and motivated by the work [29], we propose to define the MIN for a bipartite state ρ AB as where ||X|| 1 = Tr √ X † X, and Π A denotes still the projective measurements that satisfy Π A (ρ A ) = ρ A .
The MIN measure defined above can circumvent the problem occurred for the 2-norm MIN as implied by Eq. (2). This can be proved by adopting similar methodologies as obtaining Eq. (2), which gives rise to N 1 (ρ A:BC ) = N 1 (ρ AB )Trρ C = N 1 (ρ AB ), and therefore N 1 (ρ AB ) does not increase under the action of Γ B , namely, it is unaffected by adding or removing a factorized local ancilla on B. Here, we further show a more general and powerful result related to the 1-norm MIN.
is monotonous under the action of any CPTP channel E B on the unmeasured party B, i.e., we always have Proof. Let E B be an arbitrary CPTP channel acting on party B of ρ AB , and {Π A k } be the optimal projection-valued measurement operator on party A that maximizes the Schatten 1norm on the right-hand side of Eq.
. We first show that for any ρ AB and E B , one can always construct a state σ AB in the space where the first inequality comes from the fact that σ AB is not necessarily the optimal state to ρ AB , and the second inequality is due to the contractivity of the Schatten 1-norm under CPTP map (Theorem 9.2 of Ref. [32]). This completes the proof. We now list some other basic properties of the 1-norm MIN, which applies also to the 2-norm MIN [15]. (i) N 1 (ρ AB ) = 0 for all the product states ρ AB = ρ A ⊗ ρ B , as well as for the classical-quantum states ρ CQ , which is obvious since the Schatten 1-norm (trace norm) is preserved under unitary transformations [32].
The maximization in Eq. (3) over the set of locally invariant measurements on party A can be obtained for certain family of quantum states, and in turn the 1-norm MIN can be evaluated analytically. We present them via the following theorems.
Theorem 2. For any 2 × n dimensional pure state |ψ with the Schmidt decomposition |ψ Proof. By denoting ρ ψ = |ψ ψ|, and ρ ψ Then as one can always find a unitary operator U A such that U A |φ A k = |k , and as N 1 (ρ ψ ) is locally unitary invariant, we obtain N 1 (ρ ψ ) = 2 √ λ 1 λ 2 = 1 after a similar analysis as that performed for the nondegenerate case, and this completes our proof.
For general m×n dimensional pure state in the Schmidt ex- A is nondegenerate, but a closed form of its singular values and therefore N 1 (ρ Ψ ) cannot be derived for m 3.

For degenerate ρ Ψ
A , an analysis similar as that for m = 2 yields We point out here that the 2-norm MIN for any pure state |Ψ is given by N 2 (|Ψ Ψ|) = 1 − k λ 2 k [15], and therefore we have N 1 (|ψ ψ|) = 2N 2 (|ψ ψ|) for any 2 × n dimensional pure state |ψ , which means that for this special case, both N 1 (|ψ ψ|) and N 2 (|ψ ψ|) give qualitatively the same characterizations of nonlocality. Moreover, as for |ψ the entanglement of formation was given by E f = − i λ i log 2 λ i [33], the 1-norm MIN N 1 (|ψ ψ|) constitutes also an entanglement monotone. But this is not the case for general states, even when they are pure.
We can always find local unitary operation U A ⊗ V B which transforms τ AB into ρ AB = 1 4 4 i,j=0 r ij σ i ⊗ σ j such that the correlation tensor R = (r mn ) (m, n = 1, 2, 3) is diagonal [34]. Then, by denoting x = (x 1 , x 2 , x 3 ), y = (y 1 , y 2 , y 3 ), and c = (c 1 , c 2 , c 3 ), with x i = r i0 , y i = r 0i , and c i = r ii , ρ AB can be written explicitly as The local unitary invariance of the 1-norm MIN N 1 enables N 1 (τ AB ) = N 1 (ρ AB ), and therefore it suffices to consider the representative family of states in Eq. (7), for which N 1 (ρ AB ) can be evaluated analytically.
Theorem 3. For any two-qubit state of the form of Eq. (7), the 1-norm MIN can be obtained as where , and the summation runs over all the cyclic permutations of {1, 2, 3}.
It is interesting to note that for the Bell-diagonal states ρ BD , that is, for the special case of x = y = 0 in Eq. (7), the 2norm MIN [15], as well as the 1-norm GQD [30] and the 2norm GQD [26,36], can also be evaluated analytically, which are given respectively by and therefore we have N 1 (ρ BD ) = 4N 2 (ρ BD ) − D 2 1 (ρ BD ). This implies that the two different measures of MIN may impose different orderings of nonlocality, as when c + keeps unchanged, N 1 (ρ BD ) also keeps unchanged, while N 2 (ρ BD ) increases (decreases) with the increasing (decreasing) value of c 0 in the region of c − c 0 c + . Thus there is no oneto-one correspondence between the well-defined 1-norm MIN and the original 2-norm MIN in general, and we hope this simple example may provide some intuition about the subtle issue concerning the appropriateness of using the Hilbert-Schmidt norm as a distance for quantifying nonlocality, just as the appropriateness of using it for defining GQD [27].
Moreover, it is also worthwhile to point out that when ρ BD being subject to the $ (i) channel (with i = 1, 2, 3 representing respectively, the bit flip, bit-phase flip, and phase flip channels [37]), we have c i (t) = c i (0), and c j,k (t) = c j,k (0)p(t) (i = j = k), where p(t) = e −γt for the one-sided channel $ (i) ⊗ I 2 or I 2 ⊗ $ (i) , and p(t) = e −2γt for the two-sided channel $ (i) ⊗ $ (i) , with γ being the decay rate. As a consequence, if |c i (0)| = max{|c i (0)|, |c j (0)|, |c k (0)|} at the initial time, we obtain N 1 [$ (i) (ρ BD )] = |c i (0)| by Eq. (8), which is not destroyed by the $ (i) noise during the whole time region, and it is in sharp contrast to other quantum correlation measures which remain constant only for a finite time interval. This unique and novel characteristic of the 1-norm MIN is not only conceptually significant, but is also very important for potential quantum algorithms relying on it.
So far we have obtained analytical formulas of the 1-norm MIN for the 2 × n dimensional pure states and a general twoqubit state, and discussed several interesting implications of them. We will now consider some high-dimensional quantum states with symmetry [7]. They are the so-called Werner states and the isotropic states, and the analytical formulas of different quantum correlation measures (see Ref. [38] for a review) for them have already been obtained [26,29,39,40].
Consider first the celebrated Werner state on C d ⊗ C d [7], which can be written as which admits the local unitary invariance, i.e., and therefore one can choose the optimal measurement basis asΠ A i = |i i|, which yields As i =j |ij ji| constitutes a permutation matrix (a binary matrix with exactly one entry 1 in each row and each column and zeros elsewhere), the singular values of ρ W −Π A (ρ W ) can be evaluated directly as |dx−1|/(d 3 −d) with multiplicity d(d − 1). Therefore, by the definition (3) we obtain from which one can see that N 1 (ρ W ) vanishes only when x = 1/d. Meanwhile, the 2-norm MIN (coincides with the 2-norm GQD) is given by , and this means that both N 1 (ρ W ) and N 2 (ρ W ) give qualitatively the same descriptions of MIN for ρ W with finite d. But it should be note that their asymptotic behaviors are different because lim d→∞ N 1 (ρ W ) = |x|, and lim d→∞ N 2 (ρ W ) = 0. The second high-dimensional state we want to consider is the d × d dimensional isotropic state expressed as follows with |Φ = 1 √ d i |ii , and |i denotes the computational basis on C d . For this state, letΠ A k = |k k | be the measurement basis that maximizes the Schatten 1-norm in Eq. (3), then due to the symmetry of |Φ , one can always find unitary operation k |kk , and the local unitary invariance of N 1 enables N 1 (ρ I ) = N 1 [U ρ I U † ]. Therefore, by denoting ρ I U = U ρ I U † , we obtain the singular values of which can be evaluated analytically as ] with multiplicity d − 1, and this yields Here, the 1-norm MIN N 1 (ρ I ) = 0 only when x = 1/d 2 . Moreover, the 2-norm MIN (coincides with the 2-norm GQD) has been obtained as [14], thus N 1 (ρ I ) = 2 (d − 1)N 2 (ρ I )/d, while their asymptotic values are given by lim d→∞ N 1 (ρ I ) = 2x, and lim d→∞ N 2 (ρ I ) = x 2 , respectively. This implies that the two measures of MIN still give qualitatively the same characterizations of nonlocality for the isotropic state.
It is remarkable that for the special case x = 1, i.e., when ρ I reduces to the maximally entangled state, we have N 1 (ρ I ) = 2(d − 1)/d, which is just twice that of the 2-norm MIN.
To summarize, we have introduced a well-defined measure of MIN by making using of the Schatten 1-norm. It can remedy the undesirable property of the 2-norm MIN which can be changed arbitrarily and reversibly by trivial local action on the subsystem. We proved explicitly that the proposed 1-norm MIN is monotonous under the action of general CPTP quantum channels on the unmeasured subsystem. This property has by itself a conceptual significance, as it has already been proved that the Schatten 1-norm is the only p-norm that can be used to give a well-defined quantum correlation measure [30].
Here the fascinating properties of the 1-norm MIN show again the ubiquitousness and intrinsic significance of the Schatten 1-norm for MIN. We hope this may shed some new light on the issue concerning the characterization and quantification of nonlocality from a measurement perspective.
We have also presented analytical formulas of the 1-norm MIN for any 2 × n dimensional pure state, two-qubit state, as well as the Werner states and isotropic states on C d ⊗C d which possess high symmetry. We revealed through these results that the 1-norm MIN captures the nonlocal property of a system more intrinsically than that of the 2-norm MIN. Particularly, the two measures of MIN may impose different orderings of nonlocality, as there are ρ 1 nd ρ 2 such that N 1 (ρ 1 ) N 1 (ρ 2 ), while N 2 (ρ 1 ) N 2 (ρ 2 ). There even exists a limiting case for which N 1 (ρ W ) = |x| ∈ [0, 1] and N 2 (ρ W ) = 0.
Finally, we remark that the entropic measure of MIN based on the von Neumann entropy [16], or its equivalent form based on the relative entropy [17], is also monotonous due to the monotonicity of the quantum mutual information under channels on B (see Ref. [27] for a detailed proof). Moreover, it has already been pointed out that one can remedy the MIN via the square root of the considered density matrix [29]. Here, we mention that it is also natural to define the MIN as N B (ρ AB ) = 2 max Π A {1− F [ρ AB , Π A (ρ AB )]} via the Bures distance [31], with Π A being the locally invariant measurement and F (ρ, σ) = [Tr( √ ρσ √ ρ) 1/2 ] 2 denotes the Uhlmann fidelity. By using the monotonicity of the Bures distance [32] and after a similar analysis as that for proving Theorem 1, one can show directly that N B is also monotonous under general CPTP channels. But its evaluation may be intractable and further investigation is needed.