**4. Transformations of obscure qubits**

Let us consider the obscure qubits in the vector representation, such that

$$|\mathbf{0}\rangle = \begin{pmatrix} \mathbf{1} \\ \mathbf{0} \end{pmatrix}, \qquad |\mathbf{1}\rangle = \begin{pmatrix} \mathbf{0} \\ \mathbf{1} \end{pmatrix} \tag{32}$$

are basis vectors of Hð Þ<sup>2</sup> *<sup>q</sup>* . Then a standard quantum computational process in the quantum register with *L* obscure qubits (qudits (1)) is performed by sequences of unitary matrices *<sup>U</sup>*^ of size 2*<sup>L</sup>* � <sup>2</sup>*<sup>L</sup>* (*nL* � *nL*), *<sup>U</sup>*^ † *<sup>U</sup>*^ <sup>¼</sup> ^*I*, which are called quantum gates (^*<sup>I</sup>* is the unit matrix). Thus, for one obscure qubit the quantum gates are 2 � <sup>2</sup> unitary complex matrices.

In the vector representation, an obscure qubit differs from the standard qubit (8) by a 2 � 2 invertible diagonal (not necessarily unitary) matrix

$$\left|\boldsymbol{\mu}\_{ab}^{(2)}\right\rangle = \hat{\boldsymbol{M}}(a\_0, a\_1) \left|\boldsymbol{\mu}^{(2)}\right\rangle,\tag{33}$$

$$
\hat{M}(a\_0, a\_1) = \begin{pmatrix} a\_0 & 0 \\ 0 & a\_1 \end{pmatrix}. \tag{34}
$$

We call *<sup>M</sup>*^ ð Þ *<sup>α</sup>*0, *<sup>α</sup>*<sup>1</sup> a membership matrix which can optionally have the property

$$\text{tr}\hat{M}^2 = \mathbf{1},\tag{35}$$

if (15) holds.

Let us introduce the orthogonal commuting projection operators

$$
\hat{P}\_0 = \begin{pmatrix} \mathbf{1} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{pmatrix}, \qquad \hat{P}\_1 = \begin{pmatrix} \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{1} \end{pmatrix}, \tag{36}
$$

$$
\hat{\boldsymbol{P}}\_0^2 = \hat{\boldsymbol{P}}\_0, \quad \hat{\boldsymbol{P}}\_1^2 = \hat{\boldsymbol{P}}\_1, \quad \hat{\boldsymbol{P}}\_0 \hat{\boldsymbol{P}}\_1 = \hat{\boldsymbol{P}}\_1 \hat{\boldsymbol{P}}\_0 = \hat{\mathbf{0}},\tag{37}
$$

where 0 is the 2 ^ � 2 zero matrix. Well-known properties of the projections are that

$$
\hat{P}\_0|\boldsymbol{\psi}^{(2)}\rangle = \mathfrak{a}\_0|\mathbf{0}\rangle, \qquad \hat{P}\_1|\boldsymbol{\psi}^{(2)}\rangle = \mathfrak{a}\_1|\mathbf{0}\rangle,\tag{38}
$$

$$
\left\langle \boldsymbol{\nu}^{(2)} \middle| \hat{P}\_0 \middle| \boldsymbol{\nu}^{(2)} \right\rangle = \left| \boldsymbol{a}\_0 \right|^2, \qquad \left\langle \boldsymbol{\nu}^{(2)} \middle| \hat{P}\_1 \middle| \boldsymbol{\nu}^{(2)} \right\rangle = \left| \boldsymbol{a}\_1 \right|^2. \tag{39}
$$

Therefore, the membership matrix (34) can be defined as a linear combination of the projection operators with the membership amplitudes as coefficients

$$
\hat{M}(a\_0, a\_1) = a\_0 \hat{P}\_0 + a\_1 \hat{P}\_1. \tag{40}
$$

We compute

$$
\hat{\mathcal{M}}(a\_0, a\_1) \Big| \boldsymbol{\nu}\_{ab}^{(2)} \Big) = a\_0^2 a\_0 |\mathbf{0}\rangle + a\_1^2 a\_1 |\mathbf{1}\rangle. \tag{41}
$$

We can therefore treat the application of the membership matrix (33) as providing the origin of a reversible but non-unitary "obscure measurement" on the standard qubit to obtain an obscure qubit (cf. the "mirror measurement" [40, 41] and also the origin of ordinary qubit states on the fuzzy sphere [42]).

An obscure analog of the density operator (for a pure state) is the following form for the density matrix in the vector representation

$$\rho\_{ab}^{(2)} = \begin{vmatrix} \boldsymbol{\nu}\_{ab}^{(2)} \end{vmatrix} \begin{pmatrix} \boldsymbol{\nu}\_{ab}^{(2)} \end{pmatrix} = \begin{pmatrix} a\_0^2 |a\_0|^2 & a\_0 a\_0^\* a\_1 a\_1 \\ a\_0 a\_0 a\_1 a\_1^\* & a\_1^2 |a\_1|^2 \end{pmatrix} \tag{42}$$

with the obvious standard singularity property det *ρ* ð Þ2 *ob* ¼ 0. But tr*ρ* ð Þ2 *ob* ¼ *α*2 <sup>0</sup>j j *a*<sup>0</sup> <sup>2</sup> <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>1</sup>j j *a*<sup>1</sup> <sup>2</sup> 6¼ 1, and here there is no idempotence *<sup>ρ</sup>* ð Þ2 *ob* � �<sup>2</sup> 6¼ *ρ* ð Þ2 *ob* , which distincts *ρ* ð Þ2 *ob* from the standard density operator.
