Physik Department. Matrix Product Formalism


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1 Physik Department Matrix Product Formalism Diplomarbeit von María Gracia Eckholt Perotti Angefertigt an der Technische Universität München und am MaxPlanckInstitut für Quantenoptik Garching, September 2005
2 First Adviser: Prof. Dr. Ignacio Cirac Second Adviser: Dr. Juan José García Ripoll Dr. Michael Wolf
3 Each path is only one of a million paths. Therefore, you must always keep in mind that a path is only a path. If you feel that you must not follow it, you need not stay with it under any circumstances. Any path is only a path. There is no affront to yourself or others in dropping it if that is what your heart tells you to do. But your decision to keep on the path or to leave it must be free of fear and ambition. I warn you: look at every path closely and deliberately. Try it as many times as you think necessary. Then ask yourself and yourself alone one question. It is this: Does this path have a heart? All paths are the same. They lead nowhere. They are paths going through the brush or into the brush or under the brush. Does this path have a heart is the only question. If it does, then the path is good. If it doesn t, then it is of no use. Carlos Castaneda i
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5 Contents 1 Introduction and Motivation 1 2 Entanglement Separability Entanglement Measures Entanglement Detection Positive Maps Entanglement Witnesses Matrix Product States The Basics MPS in the AKLT model MPS from slightly entangled states MPS in DMRG Simple examples of MPS Formal Aspects Expectation values Normal Form Transfer Matrix and Normal Form Some physical properties Decay of correlation functions Entanglement An application: Calculation of groundstates DMRG in brief The Algorithm Normalization Effective Hamiltonian Writing the code Some results Matrix Product Density Operators MPDO from MPS Formal Aspects Expectation values Normal Form iii
6 iv 4.3 Entanglement in MPDO Conclusions and Outlook 47 A Notation and review of selected topics in Linear Algebra 49 A.1 Basic Mathematical Notation A.2 Selected topics in Linear Algebra B Normal Form for MPS from the Schmidt decomposition 51 C Normal Form implies normalized states for OBC 61
7 Chapter 1 Introduction and Motivation Entanglement is one of the most striking features of quantum theory. After playing a significant role in the foundations of quantum mechanics, it has been recently rediscovered as a new physical resource with potential commercial applications such as quantum cryptography, better frequency standards or quantumenhanced positioning, and clock synchronization. The ability to generate entangled states is one of the basic requirements for building quantum computers. Hence, efficient experimental methods for detection, verification, and estimation of quantum entanglement are of great practical importance. For this task we need a complete theoretical framework which provides us with the tools for characterizing entanglement. It is well known that it is not difficult to say when a quantum pure state is entangled. On the other hand, it is usually very hard to identify whether a given mixed state is entangled. Indeed, it is more relevant for experimental applications to investigate the structure of mixed state entanglement, because in real settings we often have either incomplete information about the system or interactions with the environment, known as decoherence, both of which give rise to mixed states. The ultimate aim of this work is the characterization and detection of entanglement in mixed states. We will focus on the matrixproductdensityoperator (MPDO) representation for mixed states and deal with the characterization of entanglement by means of the partialtranspose (PT): one of the standard criterion for detecting entanglement. The reason of using these specifications lies on the fact that this criterion can be implemented efficiently to mixedquantumstates written in the MPDO representation, i.e. the PT can be easily handled with only a linear growth of computational effort on the size of the system. The key to this lies in the fact that having ρ written as an MPDO, it is only determined by a set of matrices {M}. Indeed, there is not only one set but a infinite number of them (Fig. 1.1) leading to the same state ρ. In view of this, in this thesis we are interested in answering these two questions: Does there exist a standard set {M} that completely represents the class of sets that define ρ? In other words, we are looking for a set {M} in onetoone 1
8 2 Chapter 1. Introduction and Motivation Figure 1.1: All these sets define the same ρ. correspondence with the state ρ. In order to do this, we will review the normal form in the matrixproduct formalism for pure states and extend this to a normal form for mixed states. Is it possible to study entanglement properties of ρ by means of studying properties of the set {M}? In particular, we would like to see if the positivity of a density operator ρ is a necessary and sufficient condition for the positivity of this set of standard matrices describing the state in the MPDO formalism. If this is the case, it would be sufficient to check the positivity of these matrices after PT for detecting entanglement; simpler than working directly with ρ. The structure of this thesis is as follows: In chapter 2 we will consider the PT as a mathematical tool for entanglement detection, together with an overview of the theory of entanglement. Although its roots date back from the end of the 80s and considerable advances have been achieved during the last couple of years, there exists no comprehensive publication summarizing the basic features of the matrixproduct formalism. The following two chapters give a complete, selfcontained description. In chapter 3 we start introducing the matrix product states (MPS) from different perspectives: The class of MPS first saw light as the ground state of the AKLT model, an exactly solvable model in condensed matter physics. In this section we briefly explain the model and how it leads to the matrixproduct representation. MPS appear as a natural and efficient way to represent states of slightly entangled systems. We also explain briefly the roll that MPS have played in the Density Matrix Renormalization Group (DMRG) method. The chapter follows with the explicit derivation of some formal aspects, namely the calculation of expectation values using transfer matrices and the normal form for MPS. For
9 completeness, the normal form is derived from a set of conditions imposed on a transfer matrix. Later, to give a flavor of what this states represent, some physical properties are derived. To close the chapter we apply MPS to the ground state calculation within the framework of the DMRG method. We summarize the most relevant features of this technique and give a detailed account of the algorithm using MPS. Finally, we give some results for the specific problem of the Ising model. In chapter 4 we follow more or less the same structure. Here matrix product density operators are introduced as an extension of matrix product states to the mixed state scenario. We derive the calculation of expectation values using transfer matrices and work on the normal form for MPDO. To close the chapter, we study the implementation of PT on MPDO. Finally in chapter 5 we briefly summarize the results and give an outlook on future work. There are also three appendices at the end of this work. In appendix A we explain the basic mathematical notation that we use. We also include some selected topics in linear algebra to make this work more selfcontained, namely the definition of rank, unitary matrices, a brief comment on the determination of eigenvectors and eigenvalues and the singular value decomposition. In appendix B we treat the derivation of the normal form for MPS from the Schmidt decomposition used in chapter 3. Finally, in appendix C, we show that the normal form for MPS, in the case of OBC, implies normalized states. This fact simplifies in a very convenient way the formulation of the algorithm for the groundstate calculation, described at the end of chapter 3. 3
10 4 Chapter 1. Introduction and Motivation
11 Chapter 2 Entanglement If two systems interacted in the past it is, in general, not possible to assign a single state vector to either of the two subsystems [14]. This is also known as the principle of nonseparability and expresses much of what entanglement is about. First recognized by Einstein, Podolsky and Rosen [12] and Schrödinger [31], it is one of the most astonishing features of the quantum formalism. The main problem in Entanglement Theory is that we do not fully understand what entanglement is. More precisely, we only know is its mathematical definition and its manifestations [5, 7, 4]. Entanglement appears as the consequence of the combination of two of the quantum postulates: the state of a quantum system is described by a vector in a complex Hilbert space + the Hilbert space of a composite system is the tensor product of the two local spaces = superposition of pure states that cannot be written as the tensor product of pure states in each local space Antipodean to entangled states are the separable states, i.e., a state is entangled if and only if it is not separable. Whether a given state is entangled or just classically correlated is easy to determine for pure states. However, for arbitrary mixed states it is a hard problem [16]. We will see this later. 2.1 Separability Deciding whether several systems are entangled or whether they are just classically correlated is known as the separability problem. In this section we present the separability condition for pure and mixed states, i.e., the definition of entangled states. We will be referring to bipartite systems in a Hilbert space H = H A H B. 5
12 6 Chapter 2. Entanglement Pure States A pure state ψ is entangled if and only if it is not separable, i.e., it cannot be written as a product vector ψ = ψ A ψ B. In this case the criterion, for deciding if the state is entangled or not, is very simple. First we introduce an useful tool [24, 13]. Theorem 1 (Schmidt decomposition) Suppose ψ is a pure state of a composite system, AB. Then there are orthonormal states { i A } for system A, and orthonormal states { i B } of system B such that ψ = i λ i i A i B, where λ i are nonnegative real numbers satisfying i λ2 i = 1 known as Schmidt coefficients. If there is no degeneracy, this decomposition is unique up to arbitrary opposite phases in i A and i B. The Schmidt rank is defined as the number of nonvanishing Schmidt coefficients. Then, the criterion for pure states is ψ is pure ψ has Schmidt rank one. Mixed States A mixed state ρ is entangled if and only if it is not separable, i.e., it cannot be written as [39] N ] ρ = p i [ ψa ψ i A i ψb ψ i B i i=1 where N N + is arbitrary; ψ i A H A, ψ i B H B are arbitrary but normalized and p i 0 with N i=1 p i = 1. That is, a separable state can be prepared by two distant observers who receive instructions from a common classical source and prepare the different pure states ψ i A and ψi B with probability p i (Fig. 2.1). So, entangled states are those that cannot be created using local operations and classical communication. The criteria for entanglement of mixed state are many and diverse. Here we start introducing two of them [26, 20]. The symbol i indicates the transposition of subsystem i, i.e., partial transposition of the entire system with respect to i (see section 2.3) Theorem 2 (Peres) If ρ is separable then ρ A 0 and ρ B = (ρ A ) 0. Theorem 3 (Horodecki) A state ρ of a C 2 C 2 or C 2 C 3 system is separable if and only if its partial transposition is a positive operator.
13 2.2. Entanglement Measures 7 Figure 2.1: Separablestates factory. A classical source gives with probability p i the output i, indicating far away partners which state to prepare. 2.2 Entanglement Measures Quantifying quantum entanglement is one of the central topics in quantum information theory. How can entanglement be measured or quantified, how can entanglement be classified, i.e., what physically different types of entanglement exist, and finally how does entanglement behave as a physical resource for quantum communication, quantum computation, etc.? First of all, we need to know what an entanglement measure is [28]. We answer this important question by stating the conditions that every measure of entanglement E has to satisfy: Entanglement is nonnegative. It is zero if and only if the state is separable E(ψ) 0 ψ, E(ψ) = 0 ψ is separable Entanglement of independent systems is additive E(ψ n ) = ne(ψ) Entanglement is conserved under local unitary operations ψ Uψ, U = U A U B : E(ψ) = E(Uψ) a local change of basis has no effect on E Its expectation value cannot be increased by local nonunitary operations ψ local nonunitary {p j, ψ j } : p j E(ψ j ) E(ψ) monotonicity under local operations and classical communication (LOCC) For more on this see the pioneering paper on entanglement measures [8]. A pure state s entanglement is measured by its entropy of entanglement E(ψ) j ψ = i p i ψ i A ψ i B : E(ψ) = S(ρ A ) = S(ρ B ) (2.1)
14 8 Chapter 2. Entanglement i.e., the apparent entropy of any of the systems considered alone, where S(ρ) = T r(ρ logρ) (2.2) is the von Neumann entropy, ρ A = T r B ψ ψ is the reduced density matrix of A, obtained after tracing over B s degrees of freedom, and the logarithm is to base two (the information is stored in qubits). The entropy measures how much uncertainty there is in the state of the physical system. For example, if ρ A and ρ B describe pure states (there is no uncertainty in the individual systems), then E(ψ) = 0 (there are no quantum correlations between them). We define an ebit as the amount of entanglement in a maximally entangled state of two qubits, for which E = 1. Another possibility is to use the rank of the Schmidt decomposition (SD) as a measure. If A is a subset of n qubits and B the rest of them, the SD of ψ with respect to the partition A : B reads ψ = χ A α=1 λ α ψ[a] α ψ[b] α The rank χ A of ρ A (the reduced density matrix for block A) is a natural measure [37] of the entanglement between the qubits in A and those in B. Therefore, a good measure to quantify the entanglement of state ψ would be the maximal value of χ A over all possible bipartite splits A : B of the n qubits, namely or the related entanglement measure E χ χ := max χ A A E χ := log 2 (χ) In the bipartite setting, E χ upper bounds the more standard measure entropy of entanglement. For mixed states we have a whole zoo of measures, there is not a unique measure of entanglement. The choice of one measure or another depends on what you need. We will see some examples in what follows. In principle, there are two approaches to quantify entanglement [8, 19]: Abstract approach A state function can be used to quantify entanglement if it satisfies the natural properties stated before as definition of a measure. 1. Von Neumann entropy S: already introduced in (2.2). 2. Relative Entropy of Entanglement E R : it is based on the idea of distance; the closer the state is to the set of separable states, the less entangled it is.
15 2.3. Entanglement Detection 9 3. Other measures: Squashed Entanglement E sq, Rényi Entropy E α, Logarithm of the Negativity E N, Concurrence C, etc. Operational approach The system is more entangled if it allows for better performance of some task impossible without entanglement. 1. Entanglement of Formation EoF : having a large number n of Bell states, we want to produce as many (highfidelity) copies ψ using LOCC, getting finally m copies, therefore ψ s E of formation is the limiting ratio n/m. 2. Distillable entanglement E D : performing the reverse process, it is the limiting ratio m/n, when having a large number m of copies of ψ and we want to distill as many Bell states using LOCC, getting finally n EPR pairs. 3. Other measures: Entanglement Cost E C, Entanglement of Assistance EoA, etc. All these measures are equivalent in certain limits, e.g. [17]. We have so many definitions not only due to the diverse interpretations, but because calculating some of them are of the Big Open Problems of QIT. 2.3 Entanglement Detection Entangled states of many qubits are needed for quantum information tasks such as measurement based quantum computation [29], error correction [15] or quantum cryptography [10], to mention only few. Thus, it is important to study, both theoretically and experimentally, multipartite entanglement and to provide efficient methods to verify if in a given experiment entanglement is really present. Although, to detect entanglement is not an easy job. Here we introduce some ideas of two formalisms that deal with entanglement detection: positive maps and entanglement witnesses Positive Maps Any admissible physical transformation of a density matrix can be specified through some operators {K i } such that, ρ ρ i = K iρk i p i : p i = T r(k i ρk i ) (2.3) {K i } are known as Kraus operators [21]. These transformations define what is called a completely positive map (CPM), κ(ρ) = i K i ρk i which fulfils the following properties:
16 10 Chapter 2. Entanglement 1. Sends positive operators into positive operators, i.e., a positive map. 2. It s also positive for composite systems, ρ 0, κ(ρ) 0 ρ AB 0, (I A κ)(ρ AB ) 0 because any physical transformation should still remain meaningful when it is just performed on a subset of the parties. Maps that are positive, but not completely positive, define unphysical operations. This property makes them useful for the detection of entanglement: Any positive map acting on a product state gives a positive operator. Therefore, the same is valid for separable states. But if acting on some ρ AB this map produces to a nonpositive operator, then one can conclude that the state is entangled. We can see this in more detail, consider we have a separable state ρ AB = N ( ) p i ρ i A ρ i B and we apply the positive, but noncompletely positive, map κ on subsystem B ρ AB ρ AB = i=1 N ( p i ρ i A κ(ρ i B) ) i=1 given that κ is a positive map, κ(ρ i B ) will also be legitimate density matrices. So, it follows that none of the eigenvalues of ρ AB is negative. This is a necessary condition for ρ AB being separable. For every entangled state there is a positive map detecting it. This is an straightforward consequence of the following theorem [20]. Theorem 4 (Horodecki) A state ρ H A H B is separable if and only if for all positive maps ε : H B H C we have (I A ε)ρ 0 This translates the problem of detecting entanglement to the characterization of all the positive maps. We now move to the study of a concrete positive map.
17 2.3. Entanglement Detection 11 Partial Transposition The most known positive map which is not completely positive is the matrix transposition. The transposition θ is the map θ(c) C where C is the matrix obtained by exchanging C s rows and columns and it satisfies the identity (C ) 1 = (C 1 ) writing the matrix elements From this we define the map (c) i,j = c j,i. (θ I) (C) C A which is called partial transposition. If we work with the matrix elements we would write (c) A i A i B,j A j B = c ja i B,i A j B. As we saw before, right after the definition of a completely positive map, if we apply (θ I) (ρ) = ρ and ρ is a nonpositive operator, then ρ is entangled. Our criterion reduces to transpose part of ρ and diagonalize the resultant matrix. This is an easily computable criterion for entanglement in mixed states [26], seen in theorem 2. A drawback of PT is that it is not a sufficient condition; it has only been proved to be a sufficient condition for pure states and for composite systems having dimensions 2 2 and 2 3 [20]. Regardless of the fact that PT is a nonphysical operation (consequently, it cannot be use to detect entanglement experimentally), it can be understood as antiunitary time inversion operation in one subsystem; it means that e.g. Alice inverses time while Bob does not. We can understand more this and the effectiveness of PT in the following. According to Wigner s theorem [41], every symmetry transformation should always be implemented by a unitary (U) or antiunitary (A) matrix. If we are working with a binary composite system, i.e., H = H a H b, the direct product of unitary matrices U a U b (or antiunitary matrices A a A b ) is a unitary (or antiunitary) matrix in H. Nevertheless, the combination of a unitary and an antiunitary transformation U a A b (or A a U b ) results in a transformation which is neither unitary nor antiunitary in H, whose action on a general ket of the composite system ψ H, furthermore, cannot be properly defined [30]. However, its action on a product state is, but for a phase ambiguity, well defined. As a separable state ρ s H can always be rewritten as a statistical mixture of product states, ρ s = p i ( a i a i b i b i ), 1 p i 0, p i = 1 i i the action of such operations on ρ s leads to a ρ s ρ s ρ s = i p i ( a i a i b i b i )
18 12 Chapter 2. Entanglement where a i := U a a i H a, b i := A b b i H b, which is also physical (a positive defined hermitian matrix with normalized trace). Separable states are characterized by this: any local 1 symmetry transformation, which obviously transforms local physical states into local physical states, also transforms the global physical state into another physical state. There is only one independent antiunitary symmetry which physical meaning is time reversal. Any other antiunitary transformation can be expressed as the product of a unitary matrix times time reversal. Therefore, quantum separability of composite systems implies the lack of correlation between the time arrows of their subsystems, as if separable systems do not have memory of a unique time direction in the sense entangled states have and they are thus compatible with a time evolution which factorizes into the product of two opposed time evolutions still leading to a physical state Entanglement Witnesses Entanglement detection in an experiment is a hard problem, since reconstructing the whole density matrix is usually not possible and the quantum state is only partially known. One can typically measure a few observables and still one would like to detect some of the entangled states. In this direction appears another approach for detecting entanglement, the socalled entanglement witnesses (EW). An EW is a hermitian operator (an observable) W such that if T r[w ρ] < 0, then ρ is entangled. For every entangled state there is an EW W detecting it. This is a consequence of a special formulation of the HahnBanach theorem: Theorem 5 (HahnBanach) Let S be a convex set in a finite dimensional Banach space. Let ρ be a point in the space with ρ / S. Then there exists a hyperplane 2 that separates ρ from S. Therefore, for every entangled ρ / S 3 there exists a hyperplane, described by a Hermitian operator W 4, which separates ρ from S, such that T r[ρw ] < 0, whereas σ S : T r[σw ] 0. See Fig. (2.2) As theorem 4 for positive maps, this theorem is quite powerful from a theoretical point of view. However, we have the same handicap: it is not useful for constructing witnesses that detect entanglement in a given state ρ. An entanglement witness only gives one condition at detecting entanglement, while for a map has to be positively definite (i.e., there are many that have to be fulfilled). Thus 1 Local means that it refers to the subsystem 2 A linear subspace with dimension one less than the dimension of the space itself. 3 S, set of separable ρ s, is a convex and closed set. 4 In operator space, Hermitian operators define planes: {ρ : T r[ρw ] = const.}.
19 2.3. Entanglement Detection 13 Figure 2.2: Geometric HahnBanach theorem. W is a valid entanglement witness for all entangled ρ s in the redlined zone. a map is much stronger. However, EW are able to provide a more detailed classification of entangled states [32].
20 14 Chapter 2. Entanglement
21 Chapter 3 Matrix Product States A positiondependent unnormalized matrix product state for a onedimensional system of size N is defined as, ψ mps = d s 1,...s N =1 T r (A[1] s 1 A[2] s 2...A[N] s N ) s 1,...s N (3.1) A[i] s i : Matrix associated to site i and its state s i, whose dimension is bounded by some fixed number D i D i+1. They parametrize the state. d : Dimension of the Hilbert space corresponding to the physical system. They are a class of states that yields local descriptions of multipartite quantum states, giving a very good approximation [33] with only a polynomial number of parameters in some 1D problems. In the special case of open boundary conditions (OBC) we have D 1 = D N+1 = 1. In this chapter we start introducing matrix product states from three different perspectives. First, from their roots as the ground state of the AKLT model, where they were originally introduced as ValenceBond Solid (VBS) states. Second, in a more mathematical scenario through the Schmidt decomposition. Third, we give a short description of their roll in the Density Matrix Renormalization Group (DMRG). We end this first introductory section with some examples. Later, we treat formal aspects of the formalism concerning calculations of expectation values and the definition of the matrices A for a given state. Then, we study some physical properties of these states. To close this chapter, we deal with an application of all the machinery we have built: the calculation of ground states and correlation functions. 3.1 The Basics We acquaint the reader three different pictures of matrix product states. We start with a physical approach, follow later in a more mathematical direction and end with 15
22 16 Chapter 3. Matrix Product States their appearance within a numerical method that during the last years has become very successful simulating condensedmatter systems. The purpose of this section is merely to give an idea of where do the MPS come from MPS in the AKLT model The AKLT [2, 3] is an exactly solvable model of an antiferromagnetic spin1 chain exhibiting strong quantum fluctuations. Proposed by Affleck, Kennedy, Lieb and Tasaki, they were able to establish the existence of a Haldane gap (and related phenomena) rigorously. This was the first rigorous demonstration that exotic behavior in integer spin chains predicted by Haldane is indeed possible. We will briefly summarize it here. The goal was to create an instance for a Hamiltonian, for a onedimensional isotropic spin chain, with a continuous symmetry, exponentially decaying correlation functions, a gap and a unique (in the thermodynamic limit) ground state. Figure 3.1: A Valence Bond. The key to the model is the idea of a valence bond. Given two spin1/2 s, a valence bond is formed by putting them in the singlet state, as represented in Fig As depicted in Fig. 3.2, consider a spin1 chain. Each spin can be regarded as the symmetric part of the product of two virtual spin1/2 s = 0 1 Antisym. Symmetric We construct a state with a valence bond between each pair of adjacent sites i and i + 1 by forming a singlet out of one of the spin1/2 s at site i and one at site i + 1 (this also preserves the translational invariance in a simple way) [1]. After doing this we must symmetrize the two spin1/2 s at each site to restore a spin1 at each site. is If we label the virtual spins at each site α and β, the state of the spin at a given site
23 3.1. The Basics 17 Figure 3.2: Valence Bond Solid state (VBS). s αβ := 1 2 ( α β + β α ) for α, β =,, while s αβ = s βα denote the same state. This gives us an orthogonal basis in the symmetric part of the tensor product (LHS refers to the real space, RHS refers to the virtual space): +1 = s 2 0 = s = s 1 = s 2 or equivalently, we can apply the following projector over each site to map the state to the real space P = +1 s + 0 s + 0 s + 1 s Generalizing to different projectors P [i] for different sites i: P [i] = s i,α,β A[i] s i α,β s i αβ (3.2) where the elements of the matrix A denote the coefficients in the projector P. For a system of two particles we can make the following construction
24 18 Chapter 3. Matrix Product States AKLT [N=2] α1 β 2 = β 1,α 2 ε β 1α 2 s α1 β 1 s α2 β 2 (3.3) where ε is the LevyCivita antisymmetric tensor, it preserves the rotational invariance and contracts two adjacent virtual spins from different sites to form a singlet. This kind of state has only spin 0 or 1. We construct the Hamiltonian H as the sum of link Hamiltonians H i for each site. If we want to have (3.3) as ground state, we choose H as the sum of projection operators onto spin2 for each neighboring pair H = H i = [ 1 S 2 i S i ( S i S i+1 ) ] (3.4) 3 i i where S i are spin1 operators. Obviously H 0, since H i are projectors. Therefore, if we find an state such that H i s = 0 for all i, it will be the ground state. Thus, (3.3) is effectively the exact ground state for (3.4) with groundstate energy zero. Considering a chain of arbitrary size N, this ground state can be written as the sum of tensor products of states s 1 s n with the coefficients c s1...s N expressed as products of matrices A[1] s 1... A[N] s N, i.e., a matrixproduct state. This can be easily seen when we map explicitly (3.3) to the real space using (3.2) s at each site for generic coefficients A MPS from slightly entangled states Another approach to MPS is [37] by G. Vidal, developed independently to previous works. There, a particular decomposition for the coefficients of an arbitrary state is introduced ψ H2 n in the computational basis { 0, 1 } c i1 i 2 i n = ψ = 1 i 1 =1 1 c i1 i n i 1 i n i n=0 α 1,α n 1 Γ[1] i1 α1 λ[1] α1 Γ[2] i 2 α1 α 2 λ[2] α2 Γ[3] i 3 α2 α 3 Γ[n] in α n 1 (3.5) It is obvious that by contracting Λ λ s or λ Λ s together we obtain (3.1). The motivation of [37] is to show that any quantum computation with pure states can be efficiently simulated with a classical computer provided the amount of entanglement involved is sufficiently restricted. We will only describe the state decomposition and finish with a brief justification of the work. Decomposition (3.5) employs n tensors {Γ[1],..., Γ[n]} and n 1 vectors {λ[1],..., λ[n 1]}, whose indices i l and α l take values in {0, 1} and {1,..., χ}, respectively.
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