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Quadratic Entanglement Criteria for Qutrits K Vol. 137 (2020) ACTA PHYSICA POLONICA A No. 3 Quadratic Entanglement Criteria for Qutrits K. Rosołeka, M. Wieśniaka;b;∗and L. Knipsc;d aInstitute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics, and Informatics, University of Gdańsk, 80-308 Gdańsk, Poland bInternational Centre for Theories of Quantum Technologies, University of Gdańsk, Wita Swosza 57, 80-308 Gdańsk, Poland cMax-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany dDepartment für Physik, Ludwig-Maximilians-Universität, D-80797 München, Germany (Received September 10, 2019; revised version December 2, 2019; in final form December 5, 2019) The problem of detecting non-classical correlations of states of many qudits is incomparably more involved than in the case of qubits. One reason is that for qubits we have a convenient description of the system by the means of the well-studied correlation tensor allowing to encode the complete information about the state in mean values of dichotomic measurements. The other reason is the more complicated structure of the state space, where, for example, different Schmidt ranks or bound entanglement comes into play. We demonstrate that for three-dimensional quantum subsystems we are able to formulate nonlinear entanglement criteria of the state with existing formalisms. We also point out where the idea for constructing these criteria fails for higher-dimensional systems, which poses well-defined open questions. DOI: 10.12693/APhysPolA.137.374 PACS/topics: quantum information, quantum correlations, entanglements, qutrits 1. Introduction practical reasons. The task is very simple for pure states, which practically never occur in a real life. However, for For complex quantum systems, we distinguish between mixed states, it is still an open question. One method classical correlations of separable states, i.e., those, which is to apply a positive, but not a completely positive can be obtained by a prearranged preparation of each map to one of subsystems [6, 7]. This should drive an subsystem individually, and entanglement, which goes entangled state out of the set of physically admissible beyond this scheme. While separable states can be per- density operators. By the Jamiołkowski–Choi isomor- fectly correlated in one way at a time, entangled ones may phism [8] we can equivalently use an entanglement wit- reveal perfect correlations, say, whenever the same quan- ness, a composite observable taking negative mean val- tity is measured by two observers. This observation has ues only for entangled states. In this manner, we can led to a serious debate about the most fundamental as- certify all forms of entanglement, but we do not know pects of nature. First, Einstein, Podolsky, and Rosen [1] all the non-completely positive maps. In order to make have asked if quantum mechanics can be supplemented entanglement detection schemes more efficient, nonlinear with additional, hidden parameters, and later it was an- criteria were introduced. They appeared also in partic- swered that if it was indeed so, these parameters would ular context of necessary conditions on states to violate need to violate certain reasonable assumptions, such as Bell inequalities [9–11]. A state can violate the Werner– locality [2] or noncontextuality [3]. Wolf–Weinfurter–Żukowski–Brukner inequalities only if The Bell theorem [2] has consequences of not only (but not necessarily if) certain of its squared elements of philosophical nature, but has also found applications in the correlation tensor add up to more than 1. A sim- certain communication tasks. In particular, having a Bell ilar condition appeared in the context of so-called geo- inequality violated by a quantum state is equivalent to metrical inequalities [12], which treat correlations of the an advantage in a distributed computing [4]. Specifi- system as a multidimensional vector not belonging to a cally, if protocol users share an entangled state, they can convex set of local realistic models. This approach re- achieve a higher probability of locally getting the cor- sulted in geometrical entanglement criteria [13], which rect value of a certain function than when they are al- are highly versatile, and quadratic ones, particularly easy lowed only to communicate classically. The role of the to construct [14–16]. Bell theorem has been also pointed out in the context of, e.g., cryptography [5]. Only recently, Pandya, Sakarya, and Wieśniak have Therefore, schemes of entanglement detection have discussed using the Gilbert algorithm [17] to classify gathered a lot of attention for both fundamental and states as entangled and construct state-tailored entangle- ment witnesses for, in principle, any system [18]. How- ever, this method requires the full knowledge of the state. In contrast, with quadratic criteria, we attempt to certify ∗corresponding author; e-mail: [email protected] entanglement with only two measurements series, which brings the necessary experimental effort to minimum. (374) Quadratic Entanglement Criteria for Qutrits 375 Up to date, these methods turn out to be successful distributed in such a way that their product adds up to mainly for collections of qubits, as their states are con- zero. Finally, when they do not commute, they anticom- veniently described by the means of the correlation ten- mute and their eigenbases can be chosen to be unbiased, sor. The deficit of the Bell inequalities and entanglement i.e., all scalar products between any vector from one basis criteria for higher-dimensional constituents of quantum with any one from the other are equal in modulo. systems follow also from our inability to generalize this For a given state ρ, let the correlation tensor be a set tool. The Pauli matrices, the foundation of this achieve- of averages fT¯ig = fTrρo¯ig. To fulfill the normalization ment, have many interesting properties, each contribut- constraint, T00:::0 ≡ 1, but also for a single qubit we have ing to the success. They are Hermitian, unitary, trace- the pronounced complementarity relation [19]: less, and for individual subsystems their measurements 3 X 2 are complete (except for the unit matrix), meaning that hσii ≤ 1: (2) the individual mean values contain the full information i=1 about the statistics of outcomes, and they have unbi- This relation can be straightforwardly generalized to any ased bases as their eigenbases. In contrast, one of the set of mutually anticommuting operators (where Z is straight-forward generalizations, the Gell-Mann matri- some set of superindex values): ces, do not satisfy any commutativity relations, which X 2 fo¯; o¯g¯ ¯ / δ¯ ¯ ) T ≤ 1 (3) significantly complicates formulating a correlation ten- i j i;j2Z i;j ¯i sor. While they can be chosen to be orthogonal with ¯i2Z respect to the Hilbert–Schmidt product, they are highly with anticommutation brackets fo¯i; o¯jg = o¯iok¯ + o¯jo¯i. degenerate, which makes it complicated to find comple- Notice that operators o¯i and o¯j anticommute if su- mentarity relations. perindices differ on odd number of positions, ex- In this contribution we show that the notions known cluding those, where one superindex has “0”. In from the formalism of the tensor product for multiqubit Ref. [14] this property was further generalized to cut- states can be almost straight-forwardly applied to qutrits, anticommutativity. Namely, consider two operators, [A] [B] [A] [B] when we associate complex roots of infinity to local mea- o1 = o1 ⊗ o1 and o2 = o2 ⊗ o2 . We say that they surement outcomes. We choose to generalize quadratic anticommute with respect to cut AjB if they anticom- entanglement criteria, as they are particularly simple to mute on either of the subsystem. Consequently, 2 2 derive (no optimization over the full set of product states) fo1; o2gAjB = 0 ) ho1i + ho2i ≤ 1 (4) and experimentally friendly (only two measurements se- for states, which are factorizable with respect to the cut. ries necessary to certify entanglement for some states). Due to convexity, the same bound applies to separable In particular, this generalized tensor product is a subject states. to linear and quadratic bounds. Basing on these bounds, Using Eq. (4) quadratic entanglement criteria were we can derive quadratic (and geometrical, i.e., Cauchy– formulated in the following way in Ref. [14]. We choose Schwarz) entanglement criteria. However, for systems a set of the Pauli matrix products fo¯jg¯j with arbitrarily with subsystem dimension larger than 3, this is still an many elements. We create a graph of commutativity, in open challenge. which vertices denote operators from the set, which are connected by an edge if they anticommute. We now find 2. Formalism of many-qubit states the independence number for the graph and repeat the procedure for all relevant cuts into subsystems. If the As we have already mentioned, the success of describ- independence number is largest for the first graph, the ing and analyzing the states of many qubits is due to set can be used to certify entanglement. The criterion is the particularly convenient representation through a cor- that entanglement is certified when the square of sums of relation tensor. Its elements are mean values of ten- mean values of the operators is larger than independent sor products of the Pauli matrices, T¯i = ho¯ii, o¯i = numbers for all cuts. In Ref. [15] this method was further σ[1] ⊗ σ[2] ⊗ ::: ⊗ σ[N], ¯i = (i ; i ; : : : ; i ), and the Pauli developed. First, the authors limited themselves to only i1 i2 iN 1 2 N matrices two operators for each entanglement witness, and sec- ! ! ond, a different entanglement criterion is used to certify 1 0 0 1 σ0 = ; σ1 = ; entanglement with respect to every cut, but still, they 0 1 1 0 can be measured jointly in two measurements series. We ! ! will now take this path for a collection of qutrits. 0 −i 1 0 σ = ; σ = : (1) 2 i 0 3 0 −1 3.
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