REVIEW
Distinguishability and Accessible Information in Quantum Theory
Quantum Physics · Submitted 23 Jan 1996 · 166 references
Pith's verdict: A solid dissertation: the fidelity-as-optimal-POVM-overlap derivation and the new lower bound M(t) are real, and the no-broadcasting theorem is the result people now cite it for — with one loose sentence in the overview and one overclaim in the abstract.
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Referee report
The paper
The paper claims
The dissertation proves that an unknown quantum state can be broadcast to two separate systems only when the possible states commute, and derives the fidelity measure that drives the proof.
The reading
What carries the argument
The load-bearing object is the fidelity $F( ho_0, ho_1)=operatorname{tr}sqrt{ ho_1^{1/2} ho_0 ho_1^{1/2}}$, derived by applying the Schwarz inequality with a unitary freedom in the trace. The equality conditions pick out the measurement basis diagonalizing $M= ho_1^{-1/2}sqrt{ ho_1^{1/2} ho_0 ho_1^{1/2}} ho_1^{-1/2}$ when $ ho_1$ is invertible. For the information bounds, the paper differentiates the mutual information twice and minimizes the second derivative using a lowering superoperator $L_ ho$ defined by $ ho L_ ho(A)+L_ ho(A) ho=2A$; the eigenbasis of $L_ ho(Delta)$ gives a concrete measurement and a lower bound. The no-broadcasting step uses the purification characterization of fidelity: the maximum overlap of purifications equals the fidelity expression, so fidelity cannot rise under any allowed quantum operation.
Extended reading notes
Core claim
The paper's central claim is that two quantum states can be broadcast—that is, a machine can produce a joint state whose two marginals are the two possible input states—if and only if the states commute. The proof runs through the fidelity $F( ho_0, ho_1)=operatorname{tr}sqrt{ ho_1^{1/2} ho_0 ho_1^{1/2}}$, which the dissertation derives as the minimum over all measurements of the classical statistical overlap. Because no quantum operation can increase this fidelity, a broadcasting device would have to increase the distinguishability of the two states unless they commute; hence noncommuting states cannot be broadcast. The same fidelity calculation identifies the optimal measurement and supplies the engine for bounds on accessible information and on the disturbance caused by information-gathering measurements.
Load-bearing premise
The load-bearing premise is that fidelity cannot increase under any allowed quantum operation; the no-broadcasting proof needs this monotonicity, which the dissertation inherits from the purification theorem it cites rather than deriving independently.
Editorial extensions
If this is right
- Mixed-state no-cloning follows: any device that tries to duplicate an unknown noncommuting state into two independent marginals must fail, so cryptographic protocols can rely on more than pure-state no-cloning.
- Commuting states are exactly the broadcastable ones, giving an operational meaning to commutativity as the boundary between classical and quantum copyability.
- The accessible-information upper bound is attained exactly when the signal states commute, so noncommutativity is the obstruction to extracting the full channel information.
- The fidelity expression gives a computable figure of merit for how well a measurement or channel preserves quantum distinguishability.
- The inference-disturbance formalism turns tradeoff questions into algebraic problems, making disturbance bounds amenable to the same operator techniques.
Reading between the lines
- The commuting/noncommuting dichotomy looks like a general template: any task that would require increasing fidelity is forbidden for noncommuting states, which may explain other no-go results beyond copying.
- Approximate broadcasting should degrade fidelity in a measurable way; a two-level experiment could map out how much fidelity gain is possible as a function of the commutator, turning the theorem into a quantitative boundary.
- Because the bounds on accessible information are stated as operator expressions, they suggest numerical tests on small-dimensional channels where the optimal measurement can be found by diagonalizing the lowering superoperator.
- The fidelity-based proof strategy may extend to multipartite broadcasting: if pairwise marginals are demanded, the same monotonicity may rule out noncommuting local states.
Original abstract
This document focuses on translating various information-theoretic measures of distinguishability for probability distributions into measures of distin- guishability for quantum states. These measures should have important appli- cations in quantum cryptography and quantum computation theory. The results reported include the following. An exact expression for the quantum fidelity between two mixed states is derived. The optimal measurement that gives rise to it is studied in detail. Several upper and lower bounds on the quantum mutual information are derived via similar techniques and compared to each other. Of note is a simple derivation of the important upper bound first proved by Holevo and an explicit expression for another (tighter) upper bound that appears implicitly in the same derivation. Several upper and lower bounds to the quan- tum Kullback relative information are derived. The measures developed are also applied to ferreting out the extent to which quantum systems must be disturbed by information gathering measurements. This is tackled in two ways. The first is in setting up a general formalism for describing the tradeoff between inference and disturbance. The main point of this is that it gives a way of expressing the problem so that it appears as algebraic as that of the problem of finding quantum distinguishability measures. The second result on this theme is a theorem that prohibits "broadcasting" an unknown (mixed) quantum state. That is to say, there is no way to replicate an unknown quantum state onto two separate quantum systems when each system is considered without regard to the other. This includes the possibility of correlation or quantum entanglement between the systems. This result is a significant extension and generalization of the standard "no-cloning" theorem for pure states.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
This document focuses on translating various information-theoretic measures of distinguishability for probability distributions into measures of distin- guishability for quantum states. These measures should have important appli- cations in quantum cryptography and quantum computation theory. The results reported include the following. An exact expression for the quantum fidelity between two mixed states is derived. The optimal measurement that gives rise to it is studied in detail. Several upper and lower bounds on the quantum mutual information are derived via similar techniques and compared to each other. Of note is a simple derivation of the important upper bound first proved by Holevo and an explicit expression for another (tighter) upper bound that appears implicitly in the same derivation. Several upper and lower bounds to the quan- tum Kullback relative information are derived. The measures developed are also applied to ferreting out the extent to which quantum systems must be disturbed by information gathering measurements. This is tackled in two ways. The first is in setting up a general formalism for describing the tradeoff between inference and disturbance. The main point of this is that it gives a way of expressing the problem so that it appears as algebraic as that of the problem of finding quantum distinguishability measures. The second result on this theme is a theorem that prohibits "broadcasting" an unknown (mixed) quantum state. That is to say, there is no way to replicate an unknown quantum state onto two separate quantum systems when each system is considered without regard to the other. This includes the possibility of correlation or quantum entanglement between the systems. This result is a significant extension and generalization of the standard "no-cloning" theorem for pure states.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | This document focuses on translating various information-theoretic measures of distinguishability for probability distributions into measures of distin- guishability for quantum states. | contribution 0.76 | 0.51 | 0.07 | asserted 0.98 |
| s1 | These measures should have important appli- cations in quantum cryptography and quantum computation theory. | interpretation 0.59 | 0.11 | 0.04 | hedged possibility 0.57 |
| s2 | The results reported include the following. | contribution 0.76 | 0.47 | 0.02 | unclear 0.60 |
| s3 | An exact expression for the quantum fidelity between two mixed states is derived. | result 0.99 | 0.95 | 0.16 | asserted 1.00 |
| s4 | The optimal measurement that gives rise to it is studied in detail. | contribution 0.88 | 0.91 | 0.05 | asserted 0.99 |
| s5 | Several upper and lower bounds on the quantum mutual information are derived via similar techniques and compared to each other. | result 0.94 | 0.95 | 0.08 | asserted 1.00 |
| s6 | Of note is a simple derivation of the important upper bound first proved by Holevo and an explicit expression for another (tighter) upper bound that appears implicitly in the same derivation. | result 0.90 | 0.86 | 0.08 | asserted 1.00 |
| s7 | Several upper and lower bounds to the quan- tum Kullback relative information are derived. | result 0.94 | 0.94 | 0.06 | asserted 1.00 |
| s8 | The measures developed are also applied to ferreting out the extent to which quantum systems must be disturbed by information gathering measurements. | contribution 0.86 | 0.88 | 0.07 | asserted 0.97 |
| s9 | This is tackled in two ways. | method 0.80 | 0.66 | 0.02 | asserted 0.62 |
| s10 | The first is in setting up a general formalism for describing the tradeoff between inference and disturbance. | method 0.68 | 0.89 | 0.05 | asserted 0.98 |
| s11 | The main point of this is that it gives a way of expressing the problem so that it appears as algebraic as that of the problem of finding quantum distinguishability measures. | interpretation 0.83 | 0.79 | 0.07 | asserted 0.98 |
| s12 | The second result on this theme is a theorem that prohibits "broadcasting" an unknown (mixed) quantum state. | result 0.97 | 0.88 | 0.19 | asserted 1.00 |
| s13 | That is to say, there is no way to replicate an unknown quantum state onto two separate quantum systems when each system is considered without regard to the other. | result 0.58 | 0.75 | 0.15 | asserted 1.00 |
| s14 | This includes the possibility of correlation or quantum entanglement between the systems. | interpretation 0.79 | 0.52 | 0.05 | asserted 0.79 |
| s15 | This result is a significant extension and generalization of the standard "no-cloning" theorem for pure states. | interpretation 0.79 | 0.78 | 0.16 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
Figures
Showing 3 of the paper's 4 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:quant-ph/9601020 by the authors.
References
How later papers used it
68 later papers cite it
2 later papers by other authors are listed below: 2 used its method or tool.
Uses its method or tool 2 of 5 papers with this label
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The results in Fig. [ref] show the HWP's response is in good agreement with the theoretical estimation during long-term automatic rotation, with the average fidelity estimated by the squared statistical overlap [ref] [equation] as $99.88%$ ($p_i,q_i$ are the theoretical and experimental $|H⟩$ probability in the outcome photon, respectively, during the 10000 step rotation.).
Active Learning on a Programmable Photonic Quantum Processor (2022) · Data Encoding · 0.92
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The performance of the compass state preparation is quantified by calculating the statistical overlapping [ref] fidelity $F=∑_n√(P_n^measP_n^ideal)$ between the measured and ideal photon number populations of the compass states.
Quantum-Enhanced Dark Matter Search Using Cat States (2025) · Introduction · 0.88
How the literature uses it
68 arXiv papers cite this in 93 sentences
Papers citing it · 54 sentences without a confident label
Contests it 1 paper
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While the specific results of Refs. [ref] do not depend on this observation, the observation per se is clearly incorrect (see again Ref. [ref] ).
On Shor's conjecture on the accessible information of quantum dichotomies · Introduction · contest 0.82
Builds on it 1 paper
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Further, the fidelity of the mixed state case [ref] can be generalized via the Bures metric [ref] .
Stereographical visualization of a polarization state using weak measurements with an optical-vortex beam · SUMMARY AND DISCUSSION · 0.81
Uses its method or tool 4 papers
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If an adversary, Eve, wants to decide what the plaintext $b$ is, given the ciphertext $|Φ_k,b⟩$ and all $(T-1)$ possible copies of the public key $|Ψ_k⟩$, then she is ultimately faced with the problem of deciding which of the following two states she has: [equation] The optimal procedure (“POVM”) for solving this “binary quantum decision problem” is given in Refs. [ref] and depends on $ρ_0$, $ρ_1$, and the prior probability distribution $(p,1-p)$ of the plaintext $b$ (i.e. $P[b=0]=p$).
Deterministic quantum-public-key encryption: forward search attack and randomization · Forward search attack based on a symmetry test · 0.90
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In the next subsection, we shall opt for two of them, namely the Bhattacharyya coefficient and the Kolmogorov distance [ref] .
Assessments of macroscopicity for quantum optical states · The notion of distinguishability · 0.92
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The performance of the compass state preparation is quantified by calculating the statistical overlapping [ref] fidelity $F=∑_n√(P_n^measP_n^ideal)$ between the measured and ideal photon number populations of the compass states.
Quantum-Enhanced Dark Matter Search Using Cat States · Introduction · 0.88
Compares with it 1 paper
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However, the actual optimal measurement used in this case will generally be different than that which is optimal for the statistical distance [ref] .
Local and global statistical distances are equivalent on pure states · Conclusion · 0.94
Cites it as background 27 papers
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Worst pairwise states and phase transition.—The worst pairwise states, denoted by $ρ_w,0$ and $ρ_w,1$, are a pair of states that attain the minimum quantum Chernoff divergence over all pairs belonging to two sets of quantum states, $ρ∈𝒞_0$ and $σ∈𝒞_1$, where the quantum Chernoff divergence is defined as $C_Q(ρσ) = -lnmin_0≤s≤1 Tr( ρ^sσ^1-s )$ [ref] .
Universal Scaling of the Minimum Error Probability in Qualification of Quantum States · Introduction · 0.89
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An usual and standard definition of distinguishability relies on the following expressions [ref] .
Quantum distinguishability measures: projectors vs. states maximization · Introduction · 0.92
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It was shown that approaching the regions of criticality the fidelity between two neighboring ground states exhibits a dramatic drop [Fidelity is closely related to various distance measures on the set of quantum states: the closer two states are, the more similar (less distinguishable) they are (for an overview of different distance measures induced by the fidelity, see [ref] ).].
Macroscopic Distinguishability Between Quantum States Defining Different Phases of Matter: Fidelity and the Uhlmann Geometric Phase · Introduction · 0.82
Discussion
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