We introduce the concept of a physical process that purifies a mixed quantum state, taken from a set of states, and investigate the conditions under which such a purification map exists. Here, a purification of a mixed quantum state is a pure state in a higher-dimensional Hilbert space, the reduced density matrix of which is identical to the original state. We characterize all sets of mixed quantum states, for which perfect purification is possible. Surprisingly, some sets of two noncommuting states are among them. Furthermore, we investigate the possibility of performing an imperfect purification.
Purification of mixed states in Quantum Mechanics, by which we mean the transformation into pure states, has been viewed as an {\it Operation} in the sense of Kraus et al and explicit {\it Kraus Operators} \cite{kra1,kra2,kra3} have been constructed for two seperate purification protocols. The first one, initially due to Schrodinger \cite{sch} and subsequently elaborated by Sudarshan et al \cite{sudar}, is based on the {\it preservation of probabilities}. We have constructed a second protocol here based on {\it optimization of fidelities}. Both purification protocols have been implemented on a single qubit in an attempt to improve the fidelity of the purified post measurement state of the qubit with the initial pure state. We have considered both {\it complete} and {\it partial} measurements and have established bounds and inequalities for various fidelities. We show that our purification protocol leads to better state reconstruction, most explicitly so, when partial measurements are made.
The classification of the multipartite entanglement is an important problem in quantum information theory. We propose a class of two qubit mixed states
σAB=p∣χ1⟩⟨χ1∣⊗ρ1+(1−p)∣χ2⟩⟨χ2∣⊗ρ2, where
∣χ1⟩=α∣0⟩+β∣1⟩,
∣χ2⟩=β∣0⟩+(−1)nα∣1⟩. We have shown that the state
σAB represent a classical state when
n is odd while it represent a non-classical state when
n is even. The purification of the state
σAB is studied and found that the purification is possible if the spectral decomposition of the density matrices
ρ1 and
ρ2 represent pure states. We have established a relationship between three tangle, which measures the amount of entanglement in three qubit system and the quantity
⟨χ1∣χ2⟩, which identifies whether the two qubit mixed state is classical or non-classical. The three qubit purified state is then classified as a separable or biseparable or W-type or GHZ-type state using the quantum correlation, measured by geometric discord, of its reduced two qubit density matrix.
We find the necessary and sufficient condition under which two two-qubit mixed states can be purified into a pure maximally entangled state by local operations and classical communication. The optimal protocol for such transformation is obtained. This result leads to a necessary and sufficient condition for the exact purification of
n copies of a two-qubit state.
A longstanding belief in quantum tomography is that estimating a mixed state is far harder than estimating a pure state. This is borne out in the mathematics, where mixed state algorithms have always required more sophisticated techniques to design and analyze than pure state algorithms. We present a new approach to tomography demonstrating that, contrary to this belief, state-of-the-art mixed state tomography follows easily and naturally from pure state algorithms.
We analyze the following strategy: given
n copies of an unknown state
ρ, convert them into copies of a purification
∣ρ⟩; run a pure state tomography algorithm to produce an estimate of
∣ρ⟩; and output the resulting estimate of
ρ. The purification subroutine was recently discovered via the "acorn trick" of Tang, Wright, and Zhandry. With this strategy, we obtain the first tomography algorithm which is sample-optimal in all parameters. For a rank-
r d-dimensional state, it uses
n=O((rd+log(1/δ))/ε) samples to output an estimate which is
ε-close in fidelity with probability at least
1−δ. This algorithm also uses poly
(n) gates, making it the first gate-efficient tomography algorithm which is sample-optimal even in terms of the dimension
d alone. Moreover, with this method we recover essentially all results on mixed state tomography, including its applications to tomography with limited entanglement, classical shadows, and quantum metrology. Our proofs are simple, closing the gap in conceptual difficulty between mixed and pure tomography. Our results also clarify the role of entangled measurement in mixed state tomography: the only step of the algorithm which requires entanglement across copies is the purification step, suggesting that, for tomography, the reason entanglement is useful is for consistent purification.