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Reduced density operator/matrix and partial trace

by Yee Wei Law - Wednesday, 7 June 2023, 9:37 AM

The reduced density operator is an application of the density operator.

The reduced density operator is so useful it is indispensable in the analysis of composite quantum systems [NC10, Sec. 2.4.3].

Consider a composite system, with density matrix $\rho_{AB}$ , consisting of Alice’s system and Bob’s system, there is often a need to express the state of Alice’s or Bob’s system in terms of $\rho_{AB}$ .

The operation that takes us from $\rho_{AB}$ to $\rho_A$ (denoting Alice’s density matrix) or $\rho_B$ (denoting Bob’s density matrix) is called the reduced density operator.

The reduced density operator for Alice’s system is defined by

$\rho_A \triangleq \text{tr}_B(\rho_{AB}),$

where $\text{tr}_B$ is a map of operators known as the partial trace over Bob’s system.

Suppose $|a_1\rangle$ and $|a_2\rangle$ are any two vectors in Alice’s state space, and furthermore, $|b_1\rangle$ and $|b_2\rangle$ are any two vectors in Bob’s state space, then the partial trace $\text{tr}_B$ is defined by the operation [NC10, (2.178)]:

$\text{tr}_B(|a_1\rangle\langle a_2|\otimes|b_1\rangle\langle b_2|) \triangleq |a_1\rangle\langle a_2|\text{tr}(|b_1\rangle\langle b_2|) = |a_1\rangle\langle a_2|\text{tr}(\langle b_2|b_1\rangle) = |a_1\rangle\langle a_2|\langle b_2|b_1\rangle.$

(1)

Above, the notation $|a_1\rangle\langle a_2|\otimes|b_1\rangle\langle b_2|$ is equivalent to

To see why the definition above makes sense, suppose a quantum system is in the product state $\rho_{AB} = \rho_A\otimes\rho_B$ , where $\rho_A$ and $\rho_B$ are the density matrices for subsystems $A$ and $B$ respectively, then

$\text{tr}_B(\rho_{AB}) = \text{tr}_B(\rho_A\otimes\rho_B) = \rho_A\text{tr}(\rho_B) = \rho_A,$

since the trace of any density matrix is 1. Similarly,

$\text{tr}_A(\rho_{AB}) = \rho_B.$

In general, if

$\rho_{AB} = \sum_{ijk\ell}c_{ij}^{k\ell}|i\rangle\langle j|_A\otimes|k\rangle\langle \ell|_B,$

where $\{|i\rangle_A,|j\rangle_A\}$ is an orthonormal basis of $A$ , and $\{|k\rangle_B,|\ell\rangle_B\}$ is an orthonormal basis of $B$ , then the partial trace over $B$ is [WN17, Definition 1.6.1]:

$\text{tr}_B(\rho_{AB}) = \sum_{ijk\ell}c_{ij}^{k\ell}|i\rangle\langle j|_A\text{tr}(|k\rangle\langle\ell|_B) = \sum_{ijkk}c_{ij}^{kk}|i\rangle\langle j|_A.$

For a quick summary of discussion up to this point, watch edX Lecture 1.7 “The partial trace” of Quantum Cryptography by CaltechDelftX QuCryptox.

Example 1 [KLM07, Sec. 3.5.2; NC10, p. 106; Qis21, Sec. 4]

Consider the pure (entangled) Bell state: $|\psi_{AB}\rangle = \dfrac{1}{\sqrt{2}}(|0_A0_B\rangle + |1_A1_B\rangle)$ .

The system comprises 1️⃣ single-qubit subsystem $A$ with basis vectors $|0_A\rangle$ and $|1_A\rangle$ , and 2️⃣ single-qubit subsystem $B$ with basis vectors $|0_B\rangle$ and $|1_B\rangle$ .

This system is entangled (i.e., not separable) because $|\psi_{AB}\rangle\neq|\psi_A\rangle\otimes|\psi_B\rangle$ , but using the reduced density operator, we can find a full description for subsystem $A$ and for subsystem $B$ .

The density matrix for the system is:

$\begin{split} \rho_{AB} = |\psi_{AB}\rangle\langle\psi_{AB}| &= \frac{1}{2}(|0_A0_B\rangle + |1_A1_B\rangle)(\langle0_A0_B| + \langle1_A1_B|) \\ &= \frac{1}{2}(|0_A0_B\rangle\langle0_A0_B| + |0_A0_B\rangle\langle1_A1_B| + |1_A1_B\rangle\langle0_A0_B| + |1_A1_B\rangle\langle1_A1_B|). \end{split}$

Applying Eq. (1), the reduced density operator for subsystem $A$ is:

$\begin{split} \rho_A &= \text{tr}_B(\rho_{AB}) \\ &= \frac{1}{2}\left\{\text{tr}_B(|0_A0_B\rangle\langle0_A0_B|) + \text{tr}_B(|0_A0_B\rangle\langle1_A1_B|) + \text{tr}_B(|1_A1_B\rangle\langle0_A0_B|) + \text{tr}_B(|1_A1_B\rangle\langle1_A1_B|)\right\} \\ &= \frac{1}{2}\left\{\text{tr}_B(|0_A\rangle\langle0_A|\otimes|0_B\rangle\langle0_B|) + \cdots \right\} \\ &= \frac{1}{2}\left\{|0_A\rangle\langle0_A|\langle0_B|0_B\rangle + |0_A\rangle\langle1_A|\langle1_B|0_B\rangle + |1_A\rangle\langle0_A|\langle0_B|1_B\rangle + |1_A\rangle\langle1_A|\langle1_B|1_B\rangle\right\} \\ &= \frac{1}{2}\left\{|0_A\rangle\langle0_A| + |1_A\rangle\langle1_A|\right\} = \frac{1}{2}\mathbf{I}. \end{split}$

Since $\text{tr}(\rho_A^2) = \text{tr}(\rho_B^2) = 0.5 \lt 1$ , both $\rho_A$ and $\rho_B$ are mixed states.

🤔 How do we reconcile the preceding observation ☝ with the fact that $\rho_{AB}$ is a pure state?

The result of calculating the reduced density operator for $A$ is equivalent to the representation we obtain for $\psi_A$ when measurements were taken over the qubit of $B$ .
When measuring $B$ ’s qubit in the standard basis, the outcome is $|0\rangle$ or $|1\rangle$ at equal probabilities.
Due to entanglement, $A$ ’s qubit is $|0\rangle$ or $|1\rangle$ at equal probabilities.
Similarly for the reduced density operator for $B$ . Hence the mixed states.
We can say the reduced density operator is a way of describing the statistical outcomes of a subsystem when the measurement outcome of the other subsystem (in a bipartite system) is averaged out — this is in fact what “tracing out” the other subsystem means.

The strange property, that the joint state of a system can be pure (completely known) yet a subsystem be in mixed states, is a hallmark of quantum entanglement.

⚠ Caution [KLM07, Exercise 3.5.5]

The partial trace $\text{tr}_B(\rho)$ contains all the relevant information about subsystem $A$ if subsystem $B$ is discarded.

Similarly, $\text{tr}_A(\rho)$ contains all the relevant information about subsystem $B$ if subsystem $A$ is discarded.

These local descriptions do not in general contain enough information to reconstruct the state of the whole system.

Expressing a bipartite vector in the Schmidt basis makes it much easier to compute the partial trace of either subsystem. For this reason, let us discuss Schmidt decomposition.

References

[Gra21]	F. Grasselli, Quantum Cryptography: From Key Distribution to Conference Key Agreement, Quantum Science and Technology, Springer Cham, 2021. https://doi.org/10.1007/978-3-030-64360-7.
[KLM07]	P. Kaye, R. Laflamme, and M. Mosca, An Introduction to Quantum Computing, Oxford University Press, 2007. Available at https://ebookcentral.proquest.com/lib/unisa/reader.action?docID=415080.
[NC10]	M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information, 10th anniversary ed., Cambridge University Press, 2010. Available at http://mmrc.amss.cas.cn/tlb/201702/W020170224608149940643.pdf.
[Qis21]	Qiskit, The density matrix and mixed states, Qiskit textbook, June 2021. Available at https://learn.qiskit.org/course/quantum-hardware/density-matrix.
[WN17]	S. Wehner and N. Ng, Lecture Notes: edX Quantum Cryptography, CaltechDelftX: QuCryptox, 2017. Available at https://courses.edx.org/courses/course-v1:CaltechDelftX+QuCryptox+3T2018/pdfbook/0/.

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Reduced density operator/matrix and partial trace

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