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International Journal of Quantum Information c World Scientific Publishing Company

INFORMATION EXTRACTION VERSUS IRREVERSIBILITY IN QUANTUM MEASUREMENT PROCESSES

FRANCESCO BUSCEMI ERATO-SORST QCI Project, Japan Science and Technology Agency Daini Hongo White Building 201, 5-28-3 Hongo, Bunkyo-ku, 113-0033 Tokyo, Japan [email protected] MASAHITO HAYASHI Graduate School of Information Sciences, Tohoku University Aoba-ku, Sendai, 980-8579, Japan [email protected] MICHAL HORODECKI Institute of Theoretical Physics and Astronomy, University of Gda´ nsk and National Quantum Information Center of Gda´ nsk, Sopot, Poland [email protected] Received Day Month Year Revised Day Month Year Accepted Day Month Year Communicated by (xxxxxxxxxx) A quantum measurement process, when non trivial, is not a closed evolution: the appearance of classical outcomes is usually interpreted as the evidence of some decoherencelike mechanism causing quantum superpositions to degrade into classical mixtures. Such mechanism is due to a net flow of information from the input system (measurement object), through the physical apparatus interacting with the object (measurement probe), into some environment, the latter representing all those degrees of freedom which are not directly accessible by the experimenter. For this reason, the phenomenon of state reduction induced by the measurement process generally entails an irreversible state change. The aim of our contribution is to answer the following questionsa : how much information a measurement is able to extract? “How much” irreversible is the state reduction due to a particular measurement process? In which way information gain and irreversibility are related? Keywords: completely positive instruments; information gain; quantum error correction a The

main results are from [F Buscemi, M Hayashi, and M Horodecki, arXiv:quant-ph/0702166v4, submitted to Physical Review Letters]. In this proceeding version, we report in particular about the interesting relations between our definition of information gain and previous analogous ones that we became aware of later, thanks to discussions held during the conference nic@qs07, Erice, November 2007. 1

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1. Completely positive quantum instruments A general measurement process M Q on the input system Q, described by the density matrix ρQ on the (finite-dimensional) Hilbert space H Q , can be described as a collection of classical outcomes X := {m}, together with a set of completely Q positive1 (CP) maps {Em }m∈X , such that, when the outcome m is observed with P Q Q probability p(m) := Tr[Em (ρ )], m p(m) = 1, the corresponding a posteriori 0 Q Q state ρQ := E (ρ )/p(m) is output by the apparatusb . This is the formalism of m m CP quantum instruments, introduced and developed by Ozawa2 . With a little abuse of notation, it is useful to express the average action of the measurement M Q on ρQ asc X 0 X M Q (ρQ ) := p(m)ρQ m ⊗m m (1) Q0 X := Θ , where mX is a shorthand for |mihm|X , being {|mX i}m a set of orthonormal (hence perfectly distinguishable) vectors on the classical register space X of outcomes. If 0 the outcomes are discarded before being read out, that is, if ΘQ X in Eq. (1) is P Q traced over X , then the resulting average map E Q := m Em is a channel, i. e. a CP trace-preserving (TP) map: quantum instruments contain quantum channels as a special case. If, on the other hand, we are not interested in the a posteriori states but only in the outcomes probability distribution p~(m) (that is equivalent to 0 trace ΘQ X over Q0 ), then the resulting average map ρQ 7→ p~(m) uniquely defines a positive-operator–valued measure (POVM) PQ , namely, a set of positive operators P Q Q Q = 11Q , such that p(m) = Tr[ρQ Pm ]: quantum PQ := {Pm }m∈X satisfying m Pm instruments contain POVMs as a special case. We now exploit a very useful representation theorem for CP quantum instruments2 : it states that whatever quantum measurement can be modeled as an indirect measurement, in which the input system first interacts with an apparatus (or probe) A, initialized in a fixed pure state φA , through a suitable 0 unitary interaction U QA ; subsequently, a particular measurement M A is performed on the apparatusd . In addition, by introducing a third reference system R purifying the input state as ΨRQ , TrR [ΨRQ ] = ρQ , we are in the following situation: right after the unitary interaction U QA , the global tripartite state is b We

put a prime on the output system to include situations where the input physical system (Q) gets transformed into something else (Q0 ): this is the case, for example, of so-called demolishing measurements. In fact, an outcome happens to occur if and only if there exists the corresponding a posteriori state—maybe carried by a different quantum system. c In Ref.3 , such a form is implicitly written inside a Bochner integral formula. d Also in this case we distinguish between the input apparatus A and the output apparatus A0 . In fact, after the interaction between the object and the probe, one can devise a second stage, called amplification, occurring before the readout. We consider the output apparatus A0 to be the amplified apparatus, by incorporating the amplification transformation, which is a local map, into U QA . For our analysis, however, we do not need to explicitly account for all the details of such an accurate description of the measurement process, called indirect measurement model 4 .

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0

|ΥRQ A i := (11R ⊗ U QA )(|ΨRQ i ⊗ |φA i). Then, without loss of generality, the action of the measurement on the apparatus can be written as X 0 0 0 0 RQ0 A00 (idRQ ⊗M A )(ΥRQ A ) := p(m)Υm ⊗ mX m (2) RQ0 A00 X := Θ , 0

00

0

00

RQ A RQ A Q where Υm are pure states such that TrA00 [Υm ] = (idR ⊗Em )(ΨRQ )/p(m) =: RQ0 RQ0 Q0 X ρm and TrR [ρm ] = ρm , and m are the classical register states, as before. The above equation is nothing but a particular extension of the analogous Eq. (1): in 0 00 0 fact, by tracing ΘRQ A X over R and A00 , one obtains the state ΘRQ in Eq. (1). (For this reason, in the following, where no confusion arises, we will adopt the convention that to omit indices in the exponent of a multipartite state means to trace over the same indices.)

1.1. Quantum information gain We define the quantum information gain ι(ρQ , M Q ) of the measurement M Q on the input state ρQ as ι(ρQ , M Q ) := I R:X (ΘRX ),

0

ΘRX := TrQ0 A00 [ΘRQ A

00

X

],

(3)

where I A:B (σ AB ) := S(σ A ) + S(σ B ) − S(σ AB ) is the usual quantum mutual information 5 . By standard algebra, it is possible to rewrite such a quantity as P R RQ0 ι(ρQ , M Q ) = S(ρR ) − m p(m)S(ρR m ), for ρm := TrQ0 [ρm ]. In other words, ι(ρQ , M Q ) is the χ-quantity6 of the ensemble induced on the system R by the measurement M Q . Notice that the information gain (3) only depends on the input state ρQ and on the measurement M Q performed onto it, regardless of the particular extension constructed in Eq. (2). In fact, ι(ρQ , M Q ) actually depends Q only on the input state ρQ and on the POVM {Pm }m∈X uniquely induced by M Q , Q regardless of the particular state reduction maps {Em }m∈X . In other words, the information gain does not depend on the a posteriori states, but only on the way in which classical information is extracted (by the POVM PQ )e . Incidentally, quantum information gain (3) always has the same numerical value of Hall’s “dual upper bound” on accessible information7 and of Winter’s “intrinsic information” of a POVM8 , thus acquiring two clear meaningsf . The first meaning is that, thanks to Hall’s bound, whatever encoding of a classical source {p(x), x}x∈X is Q Q performed on the input such that the input ensemble is {p(x), ρQ x }x∈X , ι(ρ , M ) e We

should then more rigorously write ι(ρQ , PQ ): for the sake of clarity, however, we will follow the notation introduced in Eq. (3). f Quantum information gain, Hall’s upper bound and Winter’s intrinsic information always return the same value, but in fact their definitions are slightly different. More explicitly, with our notation, p Hall andpWinter axiomatically define their as the χ-quantity of the ensemble p quantities p Q Q ρQ /p(m)}m∈X . Even though ρQ Pm ρQ /p(m) 6= ρR {p(m), ρQ Pm m , it is possible to prove that their von Neumann entropies are always equal.

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represents a Holevo-like9 upper bound to the classical mutual information I(X : X ) between the input alphabet and the measurement outcomes. The second meaning is that, thanks to Winter’s intrinsic information, the quantum information gain ι(ρQ , M Q ) exactly represents the rate at which “useful” information, defined as the net information expurgated of randomness that is statistically independent of the input states, can be extracted from raw data, in the limit of an asymptotically large number of repetition of the same experiment. Having clarified the relation existing between quantum information gain, Hall’s bound, and Winter’s intrinsic information, we are left to understand the relation with another definition of information gain, namely, the one dating back to Groenewold10 and Ozawa11 . The Groenewold-Ozawa information gain, in our notaP 0 11 tion, is defined as ιGO (ρQ , M Q ) := S(ρQ ) − m p(m)S(ρQ gave a comm ). Ozawa plete characterization of those instruments with ιGO (ρQ , M Q ) ≥ 0, explicitly pointing out that the general formalism allows situations, notably including most of experimentally realizable non-demolishing measurements, in which ιGO (ρQ , M Q ) < 0 strictly. In Groenewold-Ozawa definition, the feature making negative values possible is that, contrarily to ours, it depends on the particular a posteriori states 0 {ρQ m }m∈X . Nonetheless, there are situations (to be shown in the following) where ιGO (ρQ , M Q ) = ι(ρQ , M Q ). 1.2. Quantum disturbance As we anticipated in the abstract, our notion of disturbance is closely related to the notion of coherent information. A first step in this direction is due to Maccone12 , who however used a different definition, not suitable for general quantum measurements. For a quantum channel E Q , from Q to Q0 , acting on the input state ρQ , 0 it is known that the quantity IcR→Q ((idR ⊗E Q )(ΨRQ )), where IcA→B (σ AB ) := S(σ B ) − S(σ AB ) is the so-called coherent information 13 , plays a central role in quantifying how well the channel preserves quantum coherence. The degree of irreversibility of the channel is then effectively quantified by the coherent information 0 loss13,14,15 S(ρQ ) − IcR→Q ((idR ⊗E Q )(ΨRQ )). We treat quantum instruments exactly as channels of the form (1), paying attention to the fact that the output of an instrument is not only Q0 , but also X . We then define the disturbance δ(ρQ , M Q ) caused by the measurement M Q on the input state ρQ as 0

0

δ(ρQ , M Q ) := S(ρQ ) − IcR→Q X (ΘRQ X ),

(4)

The quantity δ(ρ , M ) accurately describes the degree of irreversibility due to the state reduction, in the sense that the following theorem holds Q

Q

Theorem 1 (Approximate instrument correction). There exists a family of 0 recovering operations {RQ m }m∈X such that ! q X e Q Q0 Q F ρ , Rm ◦ Em ≥ 1 − 2δ(ρQ , M Q ), m

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where F e (ρQ , E Q ) := hΨRQ |(idR ⊗E Q )(ΨRQ )|ΨRQ i is the entanglement fidelity 16 of a channel E Q : Q → Q with respect to the input state ρQ .  In other words, the optimal strategy to deterministically invert the state reduc0 tion is to perform a correction, described by the channel RQ m , conditionally upon observing the outcome m. The value of the quantum disturbance (5) then tells how good this correction is in average, that is, “how much irreversible” is the state reduction caused by M Q . It is worth stressing that also the converse statement is true, namely, an approximately reversible instrument is almost undisturbing. In 0 fact, a sort of quantum Fano inequality holds14 for every set of channels {RQ m }m∈X X 0 Q δ(ρQ , M Q ) ≤ f[1 − F e (ρQ , RQ m ◦ Em )], m

where f(x) is an appropriate positive, continuous, monotonic increasing function such that f(0) = 0. 2. Closed information balance We easily find that δ(ρQ , M Q ) = I R:A

00

X

00

(ΘRA

X

).

(5)

Then, by using the chain rule for quantum mutual information17 , valid for all (generally mixed) tripartite states σ ABC , that is I A:C (σ AC ) + I A:B|C (σ ABC ) = I A:BC (σ ABC ), where I A:B|C (σ ABC ) := S(σ AC ) + S(σ BC ) − S(σ ABC ) − S(σ C ) is the quantum conditional mutual information, we can put together Eqs. (3) and (5), thus obtaining the global balance of information in a quantum measurement as ι(ρQ , M Q ) + ∆(ρQ , M Q ) = δ(ρQ , M Q ),

(6)

where the positive quantity 00

00

∆(ρQ , M Q ) := I R:A |X (ΘRA X ) X 00 00 = p(m)I R:A (ρRA m ),

(7)

m 00 ρRA m

0 00 A TrQ0 [ΥRQ ], m

for := measures the “missing information” in terms of the hidden correlations between the reference system and some inaccessible degrees of freedom—internal degrees of freedom of the apparatus or environmental degrees of freedom which interacted with the apparatus during the measurement process— which cannot be controlled by the experimenter. The existence of the tradeoff ι(ρQ , M Q ) ≤ δ(ρQ , M Q ) is then a direct evidence of the appearance of such correlated hidden degrees of freedom, causing an irreversible loss of information. The quantity ∆(ρQ , M Q ) is null if and only if, for every outcome m, the ref00 A00 erence and the apparatus are in a factorized state, that is, ρRA = ρR m m ⊗ ρm , ∀m. This is the case, for example, of the so-called “single-Kraus” or “multiplicity Q free” instruments, for which every map Em is represented by a single contraction

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Q Q † † as Em (ρ ) = Em ρQ Em , with Em Em ≤ 11Q . Hence, this kind of measurements maximize the information gain for a fixed disturbance, or, equivalently, minimize the disturbance for a fixed information gain: they are optimal measurements— in a sense, noiseless—closely related to the notion of “clean” measurements18 . Single-Kraus measurements satisfy ι(ρQ , M Q ) = δ(ρQ , M Q ), while, in general cases, the tradeoff ι(ρQ , M Q ) ≤ δ(ρQ , M Q ) holds. Moreover, for single-Kraus measurements, Groenewold-Ozawa information gain coincide with ours, namely, ιG (ρQ , M Q ) = ι(ρQ , M Q ), as anticipated before.

Acknowledgments The authors are grateful to A Barchielli and A Winter for discussions during the conference nic@qs07, in which they pointed out the relation of our approach with Hall’s bound and the “intrinsic information”. F B in particular acknowledges M Ozawa for having explained the structure of completely positive quantum instruments and having clarified many subtle points in the discussion. M Horodecki is supported by EC grant IP SCALA. References 1. M-D Choi, Lin. Alg. Appl. 10, 285 (1975). 2. M Ozawa, J. Math. Phys. 25, 79 (1984). 3. M Ozawa, in Quantum Communications and Measurement, ed. V P Belavkin et al. (Plenum Press, New York, 1995). 4. M Ozawa, Phys. Rev. A 62, 062101 (2000). 5. R L Stratonovich, Prob. Inf. Transm. 2, 35 (1965); C Adami and N J Cerf, Phys. Rev. A 56, 3470 (1997). 6. J P Gordon, in Quantum Electronics and Coherent Light, Proc. Int. Schoool Phys. “Enrico Fermi”, ed. by P A Miles (Academic, New York, 1964); D S Lebedev and L B Levitin, Inf. and Control 9, 1 (1966). 7. M J W Hall, Phys. Rev. A 55, 100 (1997); A Barchielli and G Lupieri, Q. Inf. Comp. 6, 16 (2006). 8. A Winter, Comm. Math. Phys. 244, 157 (2004). 9. A S Holevo, Probl. Inf. Transm. 9, 110 (1973). 10. H J Groenewold, Int. J. Theor. Phys. 4, 327 (1971). 11. M Ozawa, J. Math. Phys. 27, 759 (1986). 12. L Maccone, Europhys. Lett. 77, 40002 (2007). 13. B Schumacher and M A Nielsen, Phys. Rev. A 54, 2629 (1996); S Lloyd, ibid. 55, 1613 (1997). 14. H Barnum, M A Nielsen, and B Schumacher, Phys. Rev. A 57, 4153 (1998). 15. B Schumacher and M D Westmoreland, Quant. Inf. Processing 1, 5 (2002). 16. B Schumacher, Phys. Rev. A 54, 2614 (1996). 17. M Hayashi, Quantum Information: an Introduction (Springer-Verlag, Berlin Heidelberg, 2006). See Eq. (5.75). 18. F Buscemi, G M D’Ariano, M Keyl, P Perinotti, and R F Werner, J. Math. Phys. 46, 082109 (2005).