Source-linked AI summary
Quantifying quantum resources with conic programming
Roope Uola, Tristan Kraft, Jiangwei Shang, Xiao-Dong Yu, Otfried Gühne
TL;DR
The paper asks when a quantum state or measurement can outperform all resourceless alternatives in a task. Using conic programming, it proves that generalized robustness quantifies such outperformance in discrimination tasks for convex free sets. The result applies to measurement and state-assemblage examples, subject to a Slater-condition assumption for the cone programs.
Problem
The paper asks which task, if any, lets a given quantum state or measurement outperform all resourceless alternatives, amid several candidate quantum resources.
Source: p. 1Method
The paper uses conic programming to relate generalized robustness with respect to convex free sets to discrimination tasks, including one-way LOCC-assisted subchannel discrimination for state assemblages.
Source: pp. 3–4Results
For a resource outside a convex compact free set, the paper establishes a discrimination task in which it strictly outperforms that set, with the outperformance exactly quantified by generalized robustness.
Source: pp. 3–4Takeaways & Limitations
The approach yields task characterizations for incompatible measurements, POVMs not simulable by PVMs, and state assemblages prepared from higher-Schmidt-number states.
Source: p. 1The results implicitly assume that the cone programs associated with the free set F satisfy Slater’s conditions.
Source: p. 3
Abstract
from arXiv · showhide
Resource theories can be used to formalize the quantification and manipulation of resources in quantum information processing such as entanglement, asymmetry and coherence of quantum states, and incompatibility of quantum measurements. Given a certain state or measurement, one can ask whether there is a task in which it performs better than any resourceless state or measurement. Using conic programming, we prove that any general robustness measure (with respect to a convex set of free states or measurements) can be seen as a quantifier of such outperformance in some discrimination task. We apply the technique to various examples, e.g. joint measurability, POVMs simulable by projective measurements, and state assemblages preparable with a given Schmidt number.
Appendix A: Incompatibility robustness SDP and its dual
The appendix casts incompatibility robustness as an SDP and gives its dual. It specifies the objective and equality mapping used in the conic-program form.
Appendix A: Incompatibility robustness SDP and its dual
- The incompatibility robustness 1 + IR is cast as an SDP with equality and inequality constraints.The number of constraints is |a|·|x| for each of the two constraint types.Source: Appendix A · p. 5
- The passage identifies d as the dimension of the space of the Gλ and states that their sum equals 1_d.The equation fragment also specifies the dimension of the block-diagonal matrix.Source: Appendix A · p. 5
- The SDP objective is the trace of the block-diagonal variable, summed over its blocks.The blocks are specified using ˜Gλ/d.Source: Appendix A · p. 5
- The dual of the robustness SDP is obtained from the general conic-program dual and includes a Hermitian variable Z.The passage gives a dual constraint involving Z and the identity operator.Source: Appendix A · p. 6
Appendix B: Upper bound on the success probability for sets on compatible POVMs
This appendix derives an upper bound on success probability for discrimination using jointly measurable POVMs, beginning from a constraint in the dual program.
Appendix B: Upper bound on the success probability for sets on compatible POVMs
- The appendix derives an upper bound on success probability when the optimal state assemblage is discriminated using jointly measurable POVMs.The supplied passage introduces this bound but does not include its expression.Source: Appendix B · p. 6
- The derivation uses the first constraint of the dual program, applied to each block labeled by λ.The resulting relation is stated for ˜Y λ, but the inequality is incomplete in the passage.Source: Appendix B · p. 6
- The resulting bound applies to any set of jointly measurable POVMs.Source: Appendix B · p. 6
Appendix C: Construction of the state discrimination task with prior information from the optimal dual variable
The appendix constructs a state-discrimination task with prior information from an optimal dual solution. It uses that construction to lower-bound a ratio of success probabilities.
Appendix C: Construction of the state discrimination task with prior information from the optimal dual variable
- The state-discrimination task with prior information is constructed from an optimal solution of the dual program.The construction begins by identifying an intermediate relation before inserting it into the dual objective.Source: Appendix C · p. 6
- The constructed task gives a lower bound on the ratio of success probabilities.The passage obtains this bound by using Eq. (29).Source: Appendix C · p. 6
- The appendix states that the inequality used in the bound follows from Eq. (29).The passage then says that Observation 1 follows.Source: Appendix C · p. 6
Appendix D: Robustness of sets of measurements and conic programming
This appendix formulates generalized robustness for sets of POVMs as a conic program and derives its dual. The dual constraint yields a witness condition, with a related form for state assemblages.
Appendix D: Robustness of sets of measurements and conic programming
- The generalized robustness for a set of POVMs is defined relative to a free set F and the number of POVMs |x|.Source: Appendix D · p. 6
- The robustness formulation is cast as a conic program, then reparameterized by defining ˜Oa|x = (1 + t)Oa|x.Solving for Na|x precedes the conic-program formulation.Source: Appendix D · p. 7
- The conic program uses CF, a cone with basis F, and specifies A, X, B, and Λ for its dual.The dual's first constraint is then translated into a condition on Y.Source: Appendix D · p. 7
- The dual condition implies tr[Y T] ≤ 1 for all T ∈ F, which the passage identifies as a typical witness property.Source: Appendix D · p. 7
- For state assemblages, the analogous expression drops the factor 1/d because the operators P_a ϱ_a|x have unit trace.Source: Appendix D · p. 7
Appendix E: Robustness of state assemblages and conic programming
The appendix relates assemblage robustness to performance in subchannel discrimination and establishes convexity and compactness for assemblages preparable with bounded Schmidt number.
Appendix E: Robustness of state assemblages and conic programming
- An assemblage’s subchannel-discrimination success probability, divided by the best free-assemblage success probability, is at most 1 + R_F.Here F is the free set of assemblages.Source: Appendix E · p. 7
- A robustness witness can be realized as a subchannel discrimination task with one-way LOCC measurements, completing the subchannels with an extra one when needed.The construction has one more subchannel than outputs.Source: Appendix E · p. 7
- Convex combinations of preparable assemblages can be realized by increasing Alice’s system dimension.The construction uses an auxiliary qubit to combine the preparations and their measurements.Source: Appendix E · p. 7
- For finite-dimensional Bob, assemblages preparable with Schmidt number n or smaller form a compact set.The proof uses finite convex combinations of extremal assemblages and convergent subsequences.Source: Appendix E · pp. 7–8