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Graduate studies in mathematics

Physically realizable transformations of density matrices

A. Kitaev, Alexander Khanievich Shen', M. Vyalyi

Published May 31, 2002• 0 citations
Publisher
DOI
10.1090/gsm/047/14

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Good mathematicians see analogies. Great mathematicians see analogies between analogies. Stefan Banach We have already discussed the static structure of our ‘Quantum Town’ – the set of density matrices – on the one hand, and the set of all physically realizable processes which may occur in it on the other hand. Now we are going to reveal a quite remarkable property: the set of all possible ways to travel in the ‘Quantum Town’ is equivalent to a ‘Quantum Country’ – an appropriately magnified copy of the initial ‘Quantum Town’! More precisely, the set of all transformations which map the set of density matrices of size N into itself (dynamics) is identical to a subset of the set of density matrices of size N 2 (kinematics). From a mathematical point of view this relation is based on the Jamiołkowski isomorphism , analysed later in this chapter. Before discussing this intriguing duality, let us leave the friendly set of quantum operations and pay a short visit to a neighbouring land of maps, as yet unexplored, which are positive but not completely positive. Positive and decomposable maps Quantum transformations which describe physical processes are represented by completely positive (CP) maps. Why should we care about maps which are not CP? On the one hand it is instructive to realize that seemingly innocent transformations are not CP, and thus do not correspond to any physical process. On the other hand, as discussed in Chapter 15, positive but not completely positive maps provide a crucial tool in the investigation of quantum entanglement.
Explicit separability of general two qubits density matrices is related to Lorentz transformations. We use the 4-dimensional form [Formula: see text] of the Hilbert–Schmidt (HS) decomposition of the density matrix. For the generic case, in which Lorentz transformations diagonalize [Formula: see text] into [Formula: see text], we give relations between the [Formula: see text] and the [Formula: see text]. In particular, we consider two cases: (a) Two qubits density matrices with one pair of linear terms in the HS decomposition. (b) Two qubits density matrices with two or three symmetric pairs of linear terms. Some of the theoretical results are demonstrated by numerical calculations. The four non-generic cases (which may be reduced to case (a) are analyzed and the non-generic property is related explicitly to Lorentz velocity [Formula: see text] which is not reachable physically.
In quantum mechanics, states are described by density matrices. Though their probabilistic interpretation is rooted in ensemble theory, density matrices embody a known shortcoming. They do not completely express the physical realization of an ensemble. Conveniently, the outcome statistics of projective and positive operator-valued measurements do not depend on the ensemble realization, only on the density matrix. Here, we show how the geometric approach to quantum mechanics tracks ensemble realizations. We do so in two concrete cases of a finite-dimensional quantum system interacting with another one with (i) finite-dimensional Hilbert space, relevant for quantum thermodynamics, and (ii) infinite-dimensional Hilbert space, relevant for state-manipulation protocols.