es

Generally speaking, a tensor is defined as a series of some entries (for instance, numbers, functions, etc.) labeled by N indexes, with N called the order of the tensor. In this context, a scalar, which is one number and labeled by zero index, is a zeroth-order tensor. Many physical quantities are scalars, including energy, free energy, magnetization, and so on. Graphically, we use a dot to represent a scalar.

A D-component vector consists of D entries/numbers labeled by one index, and thus is a first-order tensor. For example, one can write the state vector of a spin-1/2 in a chosen basis (say the eigenstates of the spin operator S) as

\[ \mid \psi \rangle = C_1 \,\mid 0 \rangle + C_2 \,\mid 1 \rangle = \sum_{s =0,1} C_s \,\mid s \rangle , \]
with the coefficients C a two-component vector. Here, we use |0〉 and |1〉 to represent spin up and down states.

A matrix is in fact a second-order tensor. Considering two spins as an example, the state vector can be written under an irreducible representation as a four- dimensional vector. Instead, under the local basis of each spin, we write it as

\begin{align*} \mid \psi \rangle &= C_{00} \,\mid 0 \rangle \, \mid 0 \rangle + C_{01} \,\mid 0 \rangle \,\mid 1 \rangle + C_{10} \,\mid 1 \rangle \, \mid 0 \rangle + C_{11} \,\mid 1 \rangle \,\mid 1 \rangle \\ 7= \sum_{s,t =0}^1 C_{st} \,\mid s \rangle \, \mid t \rangle , \end{align*}
with Cst a matrix with two indexes. Here, one can see that the difference between a (D ×D) matrix and a D²-component vector in our context is just the way of labeling the tensor elements.

It is then natural to define an N-th order tensor. Considering, e.g., N spins, the 2N coefficients can be written as an N-th order tensor C, satisfying

\[ \mid \psi \rangle = \sum_{s_1 , s_2 , \ldots , s_N = 0}^2 C_{s_1 \cdots s_N} \,\mid s_1 \rangle \, \mid s_2 \rangle \,\cdots \,\mid s_N \rangle . \]
Similarly, such a tensor can be reshaped into a 2N-component vector.

In above, we use states of spin-1/2 as examples, where each index can take two values. For a spin-S state, each index can take d = 2S + 1 values, with d called the bond dimension. Besides quantum states, operators can also be written as tensors. A spin-1/2 operator Sα (α = x, y, z) is a (2 × 2) matrix by fixing the basis, where we have Sαt ₁ t ₂ e ₁ s ₂ = ⟨ t ₁ t ₂ ∣ Sα ∣ s ₁ s ₂ ⟩. In the same way, an N-spin operator can be written as a 2N-th order tensor, with N bra and N ket indexes.

We would like to stress some conventions about the “indexes” of a tensor (including matrix) and those of an operator. A tensor is just a group of numbers, where their indexes are defined as the labels labeling the elements. Here, we always put all indexes as the lower symbols, and the upper “indexes” of a tensor (if exist) are just a part of the symbol to distinguish different tensors. For an operator which is defined in a Hilbert space, it is represented by a hatted letter, and there will be no “true” indexes, meaning that both upper and lower “indexes” are just parts of the symbol to distinguish different operators.



  1. Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International
  2. Cartan, E., "Sur certaines expressions différentielles et le problème de Pfaff", Annales Scientifiques de l'École Normale Supérieure, 16: 239–332, doi:10.24033/asens.467