Consider a multi-qubit system consisting of two qubits. There are four possible final states: $|00\rangle, |01\rangle, |10\rangle, |11\rangle$; and it makes sense to express the quantum state of this two-qubit system as the linear combination:

$|a\rangle = a_{00}|00\rangle + a_{01}|01\rangle + a_{10}|10\rangle + a_{11}|11\rangle,$

where the normalisation rule still applies:

$|a_{00}|^2 + |a_{01}|^2 + |a_{10}|^2 + |a_{11}|^2 = 1.$

An alternative representation of $|00\rangle,\ldots,|11\rangle$ is

$|0\rangle_2, \quad |1\rangle_2, \quad |2\rangle_2, \quad |3\rangle_2,$

where the subscript indicates the order of the system.

Some authors call the set $\{|00\rangle, |01\rangle, |10\rangle, |11\rangle\}$, and equivalently $\{|0\rangle_2, |1\rangle_2, |2\rangle_2, |3\rangle_2\}$, the tensor base [Des09, p. 316].

Watch Microsoft Research’s presentation on “Quantum Computing for Computer Scientists”:

In the vector representation, given two separated qubits:

$|a\rangle = \begin{bmatrix}a_0\\a_1\end{bmatrix}, \qquad |b\rangle = \begin{bmatrix}b_0\\b_1\end{bmatrix},$

their collective state can be expressed using the Kronecker product (also called the matrix direct product and tensor product):

$|b\rangle\otimes|a\rangle \equiv \begin{bmatrix} b_0\begin{bmatrix}a_0\\a_1\end{bmatrix} \\ b_1\begin{bmatrix}a_0\\a_1\end{bmatrix}\end{bmatrix} = \begin{bmatrix}b_0a_0 \\ b_0a_1 \\ b_1a_0 \\ b_1a_1\end{bmatrix}.$

Multiple shorthands exist for $|b\rangle\otimes|a\rangle$: 1️⃣ $|b\rangle|a\rangle$, 2️⃣ $|b,a\rangle$, 3️⃣ $|ba\rangle$ [Mer07, p. 6; NC10, Sec. 2.1.7]; we have used the third shorthand earlier.

In general, we can compose the Hilbert space of a multi-qubit system using the vector space direct product (also called tensor direct product) of lower-dimensional Hilbert spaces [NC10, Sec. 2.1.7]:

• The vector space direct product is a specialisation of the direct product, and should be differentiated from Kronecker product because the latter operates on vectors and matrices.
• Suppose $V$ and $W$ are Hilbert spaces of dimensions $m$ and $n$ respectively, then $V \otimes W$ (read “$V$ tensor $W$”) is an $mn$-dimensional Hilbert space.
• Suppose $|i\rangle$ and $|j\rangle$ are othonormal bases for $V$ and $W$ respectively, then $|i\rangle\otimes|j\rangle$ is a basis for $V \otimes W$.
• Thus the elements of $V \otimes W$ are linear combinations of the tensor products of the orthonormal bases for $V$ and $W$. For example, suppose $V$ is a two-dimensional Hilbert space with basis vectors $|0\rangle$ and $|1\rangle$, then every element of $V \otimes V$ is a linear combination of $|0\rangle\otimes|0\rangle$, $|0\rangle\otimes|1\rangle$, $|1\rangle\otimes|0\rangle$ and $|1\rangle\otimes|1\rangle$.

• Suppose $|v\rangle\in V$ and $\mathbf{A}$ is a linear operator on $V$. Similarly, suppose $|w\rangle\in W$ and $\mathbf{B}$ is a linear operator on $W$. Then we can define the composite linear operator $\mathbf{A}\otimes\mathbf{B}$ on $V \otimes W$ by:

  $(\mathbf{A}\otimes\mathbf{B})(|v\rangle\otimes|w\rangle) \equiv \mathbf{A}|v\rangle\otimes\mathbf{B}|w\rangle.$

  The above implies

  $(\mathbf{A}\otimes\mathbf{B})\left(\sum_i a_i|v\rangle_i\otimes|w\rangle_i\right) \equiv \sum_ia_i\mathbf{A}|v\rangle \otimes \mathbf{B}|w\rangle.$

  Some authors write $\mathbf{A}\tilde{\otimes}\mathbf{B}$ to differentiate $\tilde{\otimes}$ (representing a Kronecker product) from $\otimes$ (representing a direct product) [Zyg18, p. 16].

• Inner product in $V \otimes W$ is defined as

  $\left\langle \sum_ia_i|v\rangle_i\otimes|w\rangle_i \Bigg| \sum_jb_j|v'\rangle_j\otimes|w'\rangle_j \right\rangle \equiv \sum_{i,j}a_i^\ast b_j\langle v_i|v'_j \rangle\langle w_i|w'_j \rangle.$

Classical-quantum state

In quantum cryptography, we often encounter states that are partially classical and partially quantum.

A classical state (c-state for short) is a state defined by a density matrix $\rho_X$ that is diagonal in the standard basis of the $d$-dimensional state space of $X$, i.e., $\rho_X$ has the form:

$\rho_X = p_0|0\rangle\langle0| + p_1|1\rangle\langle1| + \cdots + p_{d-1}|d-1\rangle\langle d-1|,$

where $\sum_{i=0}^{d-1}p_i=1$.

Suppose we prepare the following states for Alice and Bob: with probability 1/2, we prepare $|0\rangle\langle0|_X\otimes\rho_0^Q$ and with probability 1/2, we prepare $|1\rangle\langle1|_X\otimes\rho_1^Q$. Then, the joint state is the so-called classical-quantum state (cq-state for short):

$\rho_{XQ} = \dfrac{1}{2}\sum_{x\in\{0,1\}}|x\rangle\langle x|_X\otimes\rho_x^Q.$

In quantum-cryptographic convention,

• $x$ typically denotes some (partially secret) classical string that Alice creates during a quantum protocol,
• $X$ denotes a classical register (in fact, the symbols $X,Y,Z$ are reserved for classical registers),
• $Q$ denotes a quantum register, and
• $\rho_x^Q$ denotes some quantum information that an adversary may have gathered during the protocol, and which may be correlated with the string $x$.

Formally,