A universal programmable detector is a device that can be tuned to perform any desired measurement on a given quantum system, by changing the state of an ancilla. With a finite dimension d for the ancilla only approximate universal programmability is possible, with size d=f(epsilon(-1)) increasing the function of the "accuracy" epsilon(-1). In this Letter we show that, much better than the exponential size known in the literature, one can achieve polynomial size. An explicit example with linear size is also presented. Finally, we show that for covariant measurements exact programmability is feasible.
We prove that for a programmable measurement device that approximates every POVM with an error $\ensuremath{\leqslant}\ensuremath{\delta}$, the dimension of the program space has to grow at least polynomially with $\frac{1}{\ensuremath{\delta}}$. In the case of qubits we can improve the general result by showing a linear growth. This proves the optimality of the programmable measurement devices recently designed in G. M. D'Ariano and P. Perinotti, Phys. Rev. Lett. 94, 090401 (2005).
PACS. 03.65.Ta – Foundations of quantum mechanics; measurement theory. PACS. 03.65.Wj – State reconstruction, quantum tomography. Abstract. – We address the problem of achieving a universal measuring apparatus that allows to estimate the expectation value of any arbitrary operator by changing only the data processing of the outcomes. The “universal detector ” performs a joint measurement on the system and on a suitably prepared ancilla. We characterize such universal detectors, and show how they can be obtained either via Bell measurements or via local measurements and classical communication between system and ancilla. Quantum technology is nowadays turning from the stage of experimental setup design to that of quantum system engineering. The aim is to produce tools for communication, infor-mation processing, and storage, which rely on the principles of quantum mechanics, with the chance of achieving much higher speeds and capacities than those of classical devices. In this scenario, a new kind of quantum lab can be devised, in which universality and programmability are crucial features, with different tasks achieved by a basic set of devices. A “universal detector ” would allow the estimation of ensemble averages of arbitrary op-
Similar to a classical processor, which is an algorithm for reading a program and executing its instructions on input data, a universal programmable quantum processor is a fixed quantum channel that reads a quantum program
∣ψU⟩ that causes the processor to approximately apply an arbitrary unitary
U to a quantum data register. The present work focuses on a class of simple programmable quantum processors for implementing reflection operators, i.e.
U=eiπ∣ψ⟩⟨ψ∣ for an arbitrary pure state
∣ψ⟩ of finite dimension
d. Unlike quantum programs that assume query access to
U, our program takes the form of independent copies of the state to be reflected about
∣ψU⟩=∣ψ⟩⊗n. We then identify the worst-case optimal algorithm among all processors of the form
trProgram[V(∣ϕ⟩⟨ϕ∣⊗(∣ψ⟩⟨ψ∣)⊗n)V†] where the algorithm
V is a unitary linear combination of permutations. By generalizing these algorithms to processors for arbitrary-angle rotations,
eiα∣ψ⟩⟨ψ∣ for
α∈R, we give a construction for a universal programmable processor with better scaling in
d. For programming reflections, we obtain a tight analytical lower bound on the program dimension by bounding the Holevo information of an ensemble of reflections applied to an entangled probe state. The lower bound makes use of a block decomposition of the uniform ensemble of reflected states with respect to irreps of the partially transposed permutation matrix algebra, and two representation-theoretic conjectures based on extensive numerical evidence.