Matrix Product Operators and Central Elements: Classical Description of a Quantum State

I. Arad, “A note about a partial no-go theorem for quantum PCP”, arXiv:1012.3319

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Cited in the paper.

To briefly summarize the definitions of mpo ; mpo2 and to fix notation: given L L sites on a line, labelling the sites by integers 0 , 1 , … , L − 1 0,1,...,L-1 , a matrix product operator representation of an operator O O is given by writing O = ∑ α ​ β ​ γ ​ … . τ O 0 α ​ O 1 α ​ β ​ O 2 β ​ γ ​ … ​ O L − 1 τ O=\sum_{\alpha\beta\gamma....\tau}O_{0}^{\alpha}O_{1}^{\alpha\beta}O_{2}^{\beta\gamma}...O_{L-1}^{\tau} where α , β , γ , … \alpha,\beta,\gamma,... are discrete indices, and O 0 α O_{0}^{\alpha} is an operator supported on site 0 0 (there is one such operator for each choice of α \alpha ), O 1 α ​ β O_{1}^{\alpha\beta} is an operator supported on site 1 1 (there is one such operator for each choice of the pair of indices α , β \alpha,\beta ), and so on. The number of values that index α \alpha can take is called the bond dimension on the bond connecting sites 0 0 and 1 1 , the number of values that β \beta can take is called the bond dimension on the bond connecting sites 1 1 and 2 2 . The bond dimension of the matrix product operator representation is the maximum over all bonds of the bond dimension on each bond. Every operator O O can be written as a matrix product operator but for an arbitrary O O the bond dimension needed might be exponentially large; we are interested in this paper in the case where the matrix product operator representation has a small bond dimension. Note that the same operator O O might have different matrix product operator representations with different bond dimensions, and strictly speaking we should talk about the bond dimension of a matrix product operator representation rather than a matrix product operator. However, we will often speak of the bond dimension of a matrix product operator meaning by this the bond dimension of a particular representation of that operator. Also, we should note that the above definition applies to a line, while if the sites are on a circle then a slightly different definition is more natural: O = ∑ α ​ β ​ γ ​ … . τ ​ μ O 0 μ ​ α ​ O 1 α ​ β ​ O 2 β ​ γ ​ … ​ O L − 1 τ ​ μ O=\sum_{\alpha\beta\gamma....\tau\mu}O_{0}^{\mu\alpha}O_{1}^{\alpha\beta}O_{2}^{\beta\gamma}...O_{L-1}^{\tau\mu} , and we use this in Eq. 38

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Using the Hilbert-Schmidt inner product, introduce an orthonormal basis of operators X e ​ x ​ t α X_{ext}^{\alpha} on ℋ e ​ x ​ t {\cal H}_{ext} and another orthonormal basis of operators X i ​ n ​ t β X_{int}^{\beta} on ℋ i ​ n ​ t {\cal H}_{int} . Then, O O can be written as a sum of products of these operators as O = ∑ α ​ β A α ​ β ​ X i ​ n ​ t α ​ X e ​ x ​ t β O=\sum_{\alpha\beta}A_{\alpha\beta}X_{int}^{\alpha}X_{ext}^{\beta} , where A α ​ β A_{\alpha\beta} is a complex scalar depending upon α , β \alpha,\beta . Then, Eq. ( 45

Cited in the paper.