Okay, so I need to figure out the geometrical position of the given density matrix in the qubit space. Let me start by recalling what a density matrix is and how it relates to quantum states. 

The density matrix ρ is given as (1/2)(|0⟩⟨0| + |1⟩⟨1|). Wait, that looks familiar. Oh right, that's a maximally mixed state. Because when you have equal probabilities for both basis states, the density matrix is diagonal with entries 1/2 each. 

But the question is about its position in the qubit space. Hmm, I think in the Bloch sphere representation, pure states are represented as points on the surface of the sphere. But a mixed state like this one would be inside the sphere. The density matrix here is diagonal, which usually corresponds to a state along the z-axis. But since it's a mixed state, maybe it's at the origin? Wait, let me think again.

Wait, the density matrix for a pure state |ψ⟩⟨ψ| would have a single 1 on the diagonal and the rest zeros, right? For example, if |ψ⟩ is |0⟩, then the density matrix is [[1,0],[0,0]], which corresponds to the point (0,0,1) on the Bloch sphere. Similarly, |1⟩ would be (0,0,-1). But this matrix is a mixture of |0⟩ and |1⟩ with equal probabilities. 

In the Bloch sphere, a mixed state would be represented by a point inside the sphere. The position is determined by the expectation values of the Pauli matrices. For a general density matrix ρ, the expectation values are given by Tr(σ_i ρ), where σ_i are the Pauli matrices. 

Let me compute those. The Pauli matrices are:

σ_x = [[0,1],[1,0]]
σ_y = [[0,-i],[i,0]]
σ_z = [[1,0],[0,-1]]

So, let's calculate Tr(σ_x ρ), Tr(σ_y ρ), and Tr(σ_z ρ).

First, ρ is (1/2)(|0⟩⟨0| + |1⟩⟨1|) which in matrix form is:

[[1/2, 0],
 [0, 1/2]]

So, Tr(σ_x ρ) = Tr([[0,1],[1,0]] * [[1/2,0],[0,1/2]]) / 2? Wait, no, wait. Wait, Tr(σ_x ρ) is the trace of σ_x multiplied by ρ. Let me compute the product first.

Multiplying σ_x and ρ:

σ_x * ρ = [[0*1/2 + 1*0, 0*0 + 1*1/2],
          [1*1/2 + 0*0, 1*0 + 0*1/2]]

Wait, no. Wait, matrix multiplication is row times column. Let me do it step by step.

σ_x is [[0,1],[1,0]]. ρ is [[1/2,0],[0,1/2]]. 

So the product σ_x * ρ is:

First row: [0*1/2 + 1*0, 0*0 + 1*1/2] → [0, 1/2]
Second row: [1*1/2 + 0*0, 1*0 + 0*1/2] → [1/2, 0]

So σ_x * ρ is [[0, 1/2], [1/2, 0]]. The trace of this matrix is 0 + 0 = 0. So Tr(σ_x ρ) = 0.

Similarly, Tr(σ_y ρ) would be the trace of σ_y * ρ. Let's compute σ_y:

σ_y is [[0, -i], [i, 0]]. Multiplying by ρ:

σ_y * ρ = [[0*1/2 + (-i)*0, 0*0 + (-i)*1/2],
          [i*1/2 + 0*0, i*0 + 0*1/2]]

Which is [[0, -i/2], [i/2, 0]]. The trace is 0 + 0 = 0. So Tr(σ_y ρ) = 0.

Now σ_z is [[1,0],[0,-1]]. Multiplying by ρ:

σ_z * ρ = [[1*1/2 + 0*0, 1*0 + 0*1/2],
          [0*1/2 + (-1)*0, 0*0 + (-1)*1/2]]

Which is [[1/2, 0], [0, -1/2]]. The trace is 1/2 + (-1/2) = 0. Wait, that can't be right. Wait, no. Wait, the trace of σ_z ρ is (1/2) + (-1/2) = 0. So all the expectation values are zero. 

In the Bloch sphere representation, the position vector is (x, y, z) where x = Tr(σ_x ρ), y = Tr(σ_y ρ), z = Tr(σ_z ρ). So here, x=0, y=0, z=0. That would correspond to the origin (0,0,0). Wait, but the options don't have that. Wait, looking back at the choices: D is (0,0,0). But the question says "geometrical position in the qubits space." Wait, but density matrices can't be directly represented as points in qubit space because they are mixed states. Wait, maybe the question is referring to the Bloch sphere representation of the state. 

But if all the expectation values are zero, then the state is at the origin. But wait, the origin is (0,0,0), which is option D. However, I thought that the maximally mixed state would be at the origin. Wait, but wait, the maximally mixed state is 1/2 (|0⟩⟨0| + |1⟩⟨1|), which is indeed a completely mixed state. But in the Bloch sphere, the pure states are on the surface, and mixed states are inside. The completely mixed state (maximally mixed) would correspond to the center of the sphere, which is (0,0,0). 

But let me verify. For a pure state |ψ⟩, the expectation values are (x, y, z) = (2⟨ψ|σ_x|ψ⟩, 2⟨ψ|σ_y|ψ⟩, 2⟨ψ|σ_z|ψ⟩), but the density matrix for a pure state is |ψ⟩⟨ψ|, so Tr(σ_i ρ) would be 2⟨ψ|σ_i|ψ⟩, which for a pure state gives x, y, z as coordinates on the Bloch sphere. 

But in our case, the density matrix is 1/2 (|0⟩⟨0| + |1⟩⟨1|). Let's compute Tr(σ_i ρ) for each σ_i. As we did earlier, all of them are zero. So the expectation values are zero, so the state is at the origin. Therefore, the answer should be D. But wait, looking at the options, D is (0,0,0). So that's the position. 

But wait, the question is asking for the geometrical position in the qubits space. But qubits space is often represented by the Bloch sphere, where pure states are on the surface and mixed states are inside. The maximally mixed state is at the center. So yes, the answer would be D. 

Wait, but the options given are A. (1,1,0), B. (1,1,1), C. (0,0,1), D. (0,0,0). So D is the origin. But the problem is that the density matrix is a mixture of |0⟩ and |1⟩, which are antipodal points on the Bloch sphere. The completely mixed state is indeed the origin. 

But wait, maybe there's a misunderstanding here. Let me double-check. The density matrix ρ = 1/2 (|0⟩⟨0| + |1⟩⟨1|) can be written as (|+⟩⟨+| + |−⟩⟨−|)/2, where |+⟩ = (|0⟩ + |1⟩)/√2 and |−⟩ = (|0⟩ − |1⟩)/√2. But that's not directly helpful. 

Alternatively, the density matrix can be written in terms of the Pauli matrices. Let's see: 

|0⟩⟨0| = (I + σ_z)/2, because σ_z |0⟩ = |0⟩, so ⟨0|σ_z|0⟩ = 1. Similarly, |1⟩⟨1| = (I - σ_z)/2. So adding them and multiplying by 1/2 gives:

ρ = 1/2 [ (I + σ_z)/2 + (I - σ_z)/2 ] = 1/2 [ (I + σ_z + I - σ_z)/2 ] = 1/2 [ (2I)/2 ] = 1/2 * I = I/2. 

So ρ is the maximally mixed state, which is I/2. The maximally mixed state corresponds to the completely mixed state, which in the Bloch sphere is the origin, since all the expectation values are zero. 

Therefore, the correct answer should be D. (0,0,0). But let me check the options again. The choices are A, B, C, D as given. So D is (0,0,0). 

Wait, but the question says "geometrical position in the qubits space". Qubits space is typically the Bloch sphere, so yes, the origin. 

But wait, maybe the question is referring to the state vector in the computational basis. But the density matrix is given, so it's a mixed state, not a pure state. So the position in qubit space would be inside the Bloch sphere. But the options given don't have (0,0,0) as a pure state. Wait, but the options are all points on the surface except D. 

Wait, perhaps the question is referring to the state vector. But no, the density matrix is given. Wait, maybe I made a mistake. Let me think again. 

Wait, the density matrix ρ is I/2. So in the Bloch sphere, the density matrix corresponds to a state with x=0, y=0, z=0. But the Bloch sphere's center is (0,0,0), which is the origin. So the answer is D. 

But let me check again. If the density matrix is I/2, then the expectation values are all zero. So yes, the state is at the origin. Therefore, the correct answer is D. 

But wait, looking back at the question, the choices are A. (1,1,0), B. (1,1,1), C. (0,0,1), D. (0,0,0). So D is the origin. Therefore, the answer is D. 

But wait, another way to think: the density matrix being diagonal with entries 1/2, 1/2. For pure states, the density matrix is |ψ⟩⟨ψ|, which is a projector. For mixed states, it's a statistical mixture. The trace is 1, which it is. 

In the Bloch sphere, the pure states are on the surface (r has length 1), and mixed states are inside (r has length less than 1). The origin is the most mixed state. So yes, D is correct. 

Therefore, after all that, the answer should be D.
Therefore, the correct answer is $\boxed{D}$.
</think>