The concept of a(A,B)-quantum measurement is introduced and discussed.An equivalent relation between(A,B)-quantum measurement and g-frame with lower and upper bounds A and B is established.Some properties of direct sums and multiplications of(A,B)-quantum measurements are obtained.The measure frame operator and the dual quantum measurement of a(A,B)-quantum measurement are introduced and its canonical dual is constructed.Finally,a reconstruction formula of a quantum states is obtained in light of a(A,B)-quantum measurement.
The concept of generalized quantum measurement is introduced as a transformation that sets a one-to-one correspondence between the initial states of the measured object system and final states of the object-meter system with the help of a classical informational index, unambiguously linked to a classically compatible set of quantum states. It is shown that the generalized quantum measurement concept covers all key types of quantum measurement---standard projective, entangling, fuzzy, and generalized measurements with a partial or complete destruction of initial information associated with the object. A special class of soft quantum measurements as a basic model for the fuzzy measurements widespread in physics is introduced and its information properties are studied in detail. Also, a special class of partially destructive measurements mapping all states of the Hilbert space of a finite-dimensional quantum system onto the basis states of an infinite-dimensional quantum system is considered.
We establish a lower bound for estimating the von Neumann entropy from independent outcomes of any fixed rank-one POVM. A rotation-averaged van Trees argument gives a global minimax risk of at least
(d/n)log2{n/(4d)} when
d≥C and
n≥Cd, without a projective-design assumption. We also characterize risk on an operator-norm ball of radius
r around the maximally mixed state. We allow an approximate second moment: on the trace-zero Hermitian subspace, the measurement frame may differ by
ε<1 from the tight projective frame. A clipped estimator based on canonical dual shadows and the complete U-statistic for purity has risk at most
d3r2/n+d4/n2+d6r6. Lower bounds under the same frame control yield the local minimax rate
d3r2/n+d4/n2 when
n≥Cd2 and
r lies in an explicit matching range. For every fixed upper bound on
ε, approximation changes only the constants, not the powers of
d,n,r. At the critical radius
r=n−1/2, the local risk is asymptotically negligible relative to the global risk when
nlog2(n/d)≫d3. This separation holds for projective 2-designs, including global Clifford measurements in qubit dimensions, and for their uniformly well-conditioned frame approximations. Finite-sample experiments in dimension four illustrate the critical-radius benchmark and the effect of a nonexact frame.
Abstract Continuous frames and tensor products are important topics in theoretical physics. This paper combines those concepts. We derive fundamental properties of continuous frames for tensor product of Hilbert spaces. This includes, for example, the consistency property, i.e. preservation of the frame property under the tensor product, and the description of the canonical dual tensors by those on the Hilbert space level. We show the full characterization of all dual systems for a given continuous frame, a result interesting by itself, and apply this to dual tensor frames. Furthermore, we discuss the existence on non-simple tensor product (dual) frames. Continuous frame multipliers and their Schatten class properties are considered in the context of tensor products. In particular, we give sufficient conditions for obtaining partial trace multipliers of the same form, which is illustrated with examples related to short-time Fourier transform and wavelet localization operators. As an application, we offer an interpretation of a class of tensor product continuous frame multipliers as density operators for bipartite quantum states, and show how their structure can be restricted to the corresponding partial traces.
The concept of an entangling measurement as the most general form of ideal quantum measurement is introduced. The mathematical definition of an entangling measurement is given in the form of a superoperator transformation, which projects the initial states of an object-apparatus system on the corresponding entangled states appearing after a measurement. The mathematical properties of the superoperator of an entangling measurement are considered. The spectrum of superoperator eigenstates is studied analytically on the example of a two-level system. The algebraic relation ℳ 2 =ℳ AM between the superoperators of entangling and standard measurements is ascertained. The coherent information obtained in the course of measurement is analyzed and the relevant potentialities of entangling measurements are discussed.
We introduce two new complexity measures for Boolean functions, or more generally for functions of the form f:S->T. We call these measures sumPI and maxPI. The quantity sumPI has been emerging through a line of research on quantum query complexity lower bounds via the so-called quantum adversary method [Amb02, Amb03, BSS03, Zha04, LM04], culminating in [SS04] with the realization that these many different formulations are in fact equivalent. Given that sumPI turns out to be such a robust invariant of a function, we begin to investigate this quantity in its own right and see that it also has applications to classical complexity theory. As a surprising application we show that sumPI^2(f) is a lower bound on the formula size, and even, up to a constant multiplicative factor, the probabilistic formula size of f. We show that several formula size lower bounds in the literature, specifically Khrapchenko and its extensions [Khr71, Kou93], including a key lemma of [Has98], are in fact special cases of our method. The second quantity we introduce, maxPI(f), is always at least as large as sumPI(f), and is derived from sumPI in such a way that maxPI^2(f) remains a lower bound on formula size. While sumPI(f) is always a lower bound on the quantum query complexity of f, this is not the case in general for maxPI(f). A strong advantage of sumPI(f) is that it has both primal and dual characterizations, and thus it is relatively easy to give both upper and lower bounds on the sumPI complexity of functions. To demonstrate this, we look at a few concrete examples, for three functions: recursive majority of three, a function defined by Ambainis, and the collision problem.