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Physical Review A

Ancilla dimensions needed to carry out positive-operator-valued measurement

Ping-Xing Chen, János A. Bergou, Shi-Yao Zhu, Guang‐Can Guo

Published Dec 20, 2007• 39 citations
Publisher

Abstract

To implement a positive-operator-valued measurement (POVM), which is defined on the ${d}_{S}$-dimensional Hilbert space of a physical system, one has to extend the Hilbert space to include ${d}_{A}$ additional dimensions (called the ancilla). This is done via either the tensor product extension (TPE) or the direct sum extension (DSE). The implementation of a POVM utilizes the available resources more efficiently if it requires fewer additional dimensions. To determine how to implement a POVM with the least additional dimensions is, therefore, an important task in quantum information. We have determined the necessary and sufficient (hence minimal) number of the additional dimensions needed to implement the same POVM by the TPE and the DSE, respectively. If the POVM has $n$ elements and ${r}_{i}$ is the rank of the $i$th element, then the dimension of the minimal ancilla is ${d}_{A}={\ensuremath{\sum}}_{i=1}^{n}{r}_{i}\ensuremath{-}{d}_{S}$ for the DSE implementation, and this represents a lower bound for the added dimensions in the TPE implementation. In the proof, we explicitly construct the DSE implementation of a general POVM with elements of arbitrary rank. As an example, we determine ${d}_{A}$ for the unambiguous discrimination of $N$ linearly independent states and provide the full DSE implementation of a state-discriminating POVM for $N=2$.

DOI
10.1103/physreva.76.060303

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To implement a positive-operator-valued measurement (POVM), which is defined on the dS{d}_{S}dS​-dimensional Hilbert space of a physical system, one has to extend the Hilbert space to include dA{d}_{A}dA​ additional dimensions (called the ancilla). This is done via either the tensor product extension (TPE) or the direct sum extension (DSE). The implementation of a POVM utilizes the available resources more efficiently if it requires fewer additional dimensions. To determine how to implement a POVM with the least additional dimensions is, therefore, an important task in quantum information. We have determined the necessary and sufficient (hence minimal) number of the additional dimensions needed to implement the same POVM by the TPE and the DSE, respectively. If the POVM has nnn elements and ri{r}_{i}ri​ is the rank of the iiith element, then the dimension of the minimal ancilla is dA=\ensuremath∑i=1nri\ensuremath−dS{d}_{A}={\ensuremath{\sum}}_{i=1}^{n}{r}_{i}\ensuremath{-}{d}_{S}dA​=\ensuremath∑i=1n​ri​\ensuremath−dS​ for the DSE implementation, and this represents a lower bound for the added dimensions in the TPE implementation. In the proof, we explicitly construct the DSE implementation of a general POVM with elements of arbitrary rank. As an example, we determine dA{d}_{A}dA​ for the unambiguous discrimination of NNN linearly independent states and provide the full DSE implementation of a state-discriminating POVM for N=2N=2N=2.
Optimization of the mean efficiency for unambiguous (or error-free) discrimination among NNN given linearly independent nonorthogonal states should be realized in a way to keep the probabilistic quantum-mechanical interpretation. This imposes a condition on a certain matrix to be positive semidefinite. We reformulated this condition in such a way that the conditioned optimization problem for the mean efficiency was reduced to finding an unconditioned maximum of a function defined on a unit NNN sphere for equiprobable states and on an NNN ellipsoid if the states are given with different probabilities. We established that for equiprobable states a point on the sphere with equal values of Cartesian coordinates, which we call symmetric point, plays a special role. Sufficient conditions for a vector set are formulated for which the mean efficiency for equiprobable states takes its maximal value at the symmetric point. This set, in particular, includes previously studied symmetric states. A subset of symmetric states, for which the optimal measurement corresponds to a positive-operator-valued measure (POVM) requiring a one-dimensional ancilla space is constructed. We presented our constructions of a POVM suitable for the ancilla space dimension varying from 1 to NNN and the Neumark extension differing from the existing schemes by the property that it is straightforwardly applicable to the case when it is desirable to present the whole space system+ancilla\text{system}+\text{ancilla}system+ancilla as the tensor product of a two-dimensional ancilla space and the NNN-dimensional system space.
Unambiguous-nonorthogonal-state discrimination has a fundamental importance in quantum information. Moreover, it can be used for entanglement distillation and secure communication. The discrimination is carried out by a positive operator-valued measure (POVM) generalized measurement, which is typically implemented by coupling the system to an ancilla. We find a trade-off between the needed energy resources and the evolution time needed to implement the POVM and express it in terms of an actionlike cost inequality. We find the realization that minimizes this actionlike cost and show that, in this case, the cost is determined by the maximal population transfer from the system to the ancilla. We demonstrate our findings in an example of a three-level system coupled to a laser.
Quantum operations (QO's) describe any state change allowed in quantum mechanics, such as the evolution of an open system or the state change due to a measurement. We address the problem of which unitary transformations and which observables can be used to achieve a QO with generally different input and output Hilbert spaces. We classify all unitary extensions of a QO and give explicit realizations in terms of free-evolution direct-sum dilations and interacting tensor-product dilations. In terms of Hilbert space dimensionality the free-evolution dilations minimize the physical resources needed to realize the QO, and for this case we provide bounds for the dimension of the ancilla space versus the rank of the QO. The interacting dilations on the other hand, correspond to the customary ancilla-system interaction realization, and for these we derive a majorization relation which selects the allowed unitary interactions between system and ancilla.
Informationally complete (IC) positive operator-valued measures (POVMs) are generalized quantum measurements that offer advantages over the standard computational basis readout of qubits. For instance, IC-POVMs enable efficient extraction of operator expectation values, a crucial step in many quantum algorithms. POVM measurements are typically implemented by coupling one additional ancilla qubit to each logical qubit, thus imposing high demands on the device size and connectivity. Here, we show how to implement a general class of IC-POVMs without ancilla qubits. We exploit the higher-dimensional Hilbert space of a qudit in which qubits are often encoded. POVMs can then be realized by coupling each qubit to two of the available qudit states, followed by a projective measurement. We develop the required control pulse sequences and numerically establish their feasibility for superconducting transmon qubits through pulse-level simulations. Finally, we present an experimental demonstration of a qudit-space POVM measurement on IBM Quantum hardware. This paves the way to making POVM measurements broadly available to quantum computing applications.
Abstract We propose a scheme to implement general quantum measurements, also known as Positive Operator Valued Measures (POVMs) in dimension d using only classical resources and a single ancillary qubit. Our method is based on probabilistic implementation of d-outcome measurements which is followed by postselection of some of the received outcomes. We conjecture that success probability of our scheme is larger than a constant independent of d for all POVMs in dimension d. Crucially, this conjecture implies the possibility of realizing arbitrary nonadaptive quantum measurement protocol on d-dimensional system using a single auxiliary qubit with only a constant overhead in sampling complexity. We show that the conjecture holds for typical rank-one Haar-random POVMs in arbitrary dimensions. Furthermore, we carry out extensive numerical computations showing success probability above a constant for a variety of extremal POVMs, including SIC-POVMs in dimension up to 1299. Finally, we argue that our scheme can be favorable for experimental realization of POVMs, as noise compounding in circuits required by our scheme is typically substantially lower than in the standard scheme that directly uses Naimark’s dilation theorem.
I construct a positive-operator-valued measure (POVM) which has 2d2d2d rank-1 elements and which is informationally complete for generic pure states in ddd dimensions, thus confirming a conjecture made by Flammia, Silberfarb, and Caves (e-print quant-ph∕0404137). I show that if a rank-1 POVM is required to be informationally complete for all pure states in ddd dimensions, it must have at least $3d\ensuremath{-}2$ elements. I also show that, in a POVM which is informationally complete for all pure states in ddd dimensions, for any vector there must be at least $2d\ensuremath{-}1$ POVM elements which do not annihilate that vector.