Quantum Information Can Be Negative

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Y the , 2 who proved that the amount of information that of state ρAB and vanishing but non-zero errors Bob needs is given by a quantity called the con- in the protocol. We find here that the amount of ditional entropy. It measures the partial infor- partial quantum information that Alice needs to mation that Alice must send to Bob so that he send Bob is given by the quantum conditional gains full knowledge of X given his previous entropy, which is exactly the same quantity as in knowledge from Y , and it is just the difference the classical case but with the Shannon entropy between the entropy of (X,Y ) taken together changed to the von Neumann entropy: (the total information) and the entropy of Y (the prior information). Of course, this partial infor- S(A|B) ≡ S(AB) − S(B) , (1) mation is always a positive quantity. Classically, where S B is the entropy of Bob’s state ρ there would be no meaning to negative informa- ( ) B and S AB is the entropy of the joint state ρ . tion. ( ) AB For quantum states, the conditional entropy can be negative4–6, and thus it is rather surprising In the quantum world, the first of our that this quantity has a physical interpretation two questions, how to quantify quantum infor- in terms of how much quantum communication 3 mation, was answered by Schumacher , who is needed to gain complete quantum information showed that the minimum number of quantum i.e., possession of a system in the total state ρAB. bits required to compress quantum information is given by the quantum (von Neumann) en- However, in the above scenario, the nega- tropy. To answer the second question, let us tive conditional entropy can be clearly inter- now consider the quantum version of the two- preted. We find that when S A B is nega- party scenario above: Alice and Bob each pos- ( | ) tive, Bob can obtain the full state using only sess a system in some unknown quantum state classical communication, and additionally, Al- with the total density operator being ρ and AB ice and Bob will have the potential to transfer each party having states with density operators additional quantum information in the future at ρ and ρ respectively. The interesting case is A B no additional cost to them. Namely, they end where Bob is correlated with Alice, so that he up sharing S A B Einstein-Podolsky-Rosen has some prior information about her state. We − ( | ) (EPR) pairs7, i.e. pure maximally entangled now ask how much additional quantum informa- states 1 (|00i + |11i), which can be used to tion Alice needs to send him, so that he has the √2 teleport8 quantum states between the two par- full state (with density operator ρAB). Since we want to quantify the quantum partial informa- ties using only classical communication. Nega- tion, we are interested in the minimum amount tive partial information thus also gives Bob the of quantum communication to do this, allowing potential to receive future quantum information unlimited classical communication – the latter for free. The conditional entropy plays the same type of information being far easier to transmit role in quantum information theory as it does in than the former as it can be sent over a telephone the classical theory, except that here, the quan- while the former is extremely delicate and must tum conditional entropy can be negative in an be sent using a special quantum channel. operationally meaningful way. One could say that the ignorance of Bob, the conditional entropy, if negative, precisely cancels the amount Since we are interested in informational by which he knows too much9; the latter being quantitities, we go to the limit of many copies just the potential future communication gained.

2 This solves the well known puzzle of how to interpret the quantum conditional entropy which has persisted despite interesting attempts to un- derstand it6. Since there are no conditional prob- R R abilities for quantum states, S(A|B) is not an entropy as in the classical case. But by going AB AB back to the deﬁnition of information in terms of storage space needed to hold a message or state, y y one can make operational sense of this quantity.

Let us now turn to the protocol which allows Alice to transfer her state to Bob’s site in the Figure 1: Diagrammatic representation of the above scenario (we henceforth adopt the com- process of state merging. Initially the state |ψi is mon usage of refering directly to manipulations shared between the three systems R(eference), on “states” – meaning a manipulation on a phys- A(lice) and B(ob). After the communication Al- ical system in some quantum state). We call ice’s system is in a pure state, while Bob holds this quantum state merging, since Alice is ef- not only his but also her initial share. Note fectively merging her state with that of Bob’s. that the reference’s state ρR has not changed, Let us recall that in quantum information the- as indicated by the curve separating R from ory, faithful state transmission means that while AB. The protocol for state merging is as fol- the state merging protocol may depend on the lows: Let Alice and Bob have a large number n of the state ρ . To begin, we note that we density operator of the source, it must succeed AB only need to describe the protocol for negative with high probability for any pure state sent. S(A|B), as otherwise Alice and Bob can share An equivalent and elegant way of expressing nS(A|B) EPR pairs (by sending this number this criterion is to imagine that ρAB is part of of quantum bits) and create a state |ψiAA′BB′R a pure state |ψiABR, which includes a reference with S(AA′|BB′) < 0. This is because adding system R. Alice’s goal is to transfer the state an EPR pair reduces the conditional entropy by ρA to Bob, and we demand that after the pro- one unit. However, S(A|B) < 0 is equivalently tocol, the total state still has high ﬁdelity with expressed as S(A) > S(AB) = S(R), and it 10–12 |ψiABR (meaning they are nearly identical); see is known that measurement in a uniformly Figure 1 which includes a high-level description random basis on Alice’s n systems projects Bob of the protocol. The essentially element of state and R into a state |ϕiBR whose reduction to R merging is that ρR must be unchanged, and Al- is very close to ρR. But this means that Bob ice must decouple her state from R. This also can, by a local operation, transform |ϕiBR to means (seemingly paradoxically) that as far as |ψiABR. Finally, by coarse-graining the random measurement, Alice essentially projects onto a any outside party is concerned, neither the clas- 10–12 sical nor quantum communication is coupled good quantum code of rate −S(A|B); this still results in Bob obtaining the full state ρ , with the merged state. AB but now, just under −nS(A|B) EPR pairs are also created. These codes can also be obtained Let us now consider three instructive and simple by an alternative construction13. examples:

3 1. Alice has a completely unknown state the number of quantum codes in Alice’s pro- which we can represent as the maximally jection: the quantum mutual information I(A : 1 mixed density matrix ρA = 2 (|0ih0|A + R) = S(A) + S(R) − S(AR) between Alice |1ih1|A), and Bob has no state (or a known and the reference R. Secondly, the measure- state |0iB). In this case, S(A|B)=1 and ment of Alice makes her state completely prod- Alice must send one qubit down the quan- uct with R, thus reinforcing the interpretation tum channel to transfer her state to Bob. of quantum mutual information as the minimum She could also send half of an EPR state entropy production of any local decorrelating to Bob, and use quantum teleportation8 to process14, 15. This same quantity is also equal to transfer her state. the amount of irreversibility of a cyclic process: Bob initially has a state, then gives Alice her 2. The classically correlated state ρAB = share (communicating S(A)), which is ﬁnally 1 2 (|00ih00|AB + |11ih11|AB). We imag- merged back to him (communicating S(A|B)). ine this state as being part of a pure state The total quantum communication of this cycle with the reference system R, |ψiABR = is I(A : R) quantum bits. 1 √2 (|0iA|0iB|0iR + |1iA|1iB|1iR). In this case, S(A|B)=0, and thus no quantum Because state merging is such a basic prim- information needs to be sent. Indeed, Alice itive, it allows us to solve a number of other can measure her state in the basis |0i±|1i, problems in quantum information theory fairly and inform Bob of the result. Depending on easily. We now sketch four particularly striking the outcome of the measurement, Bob and applications. R will share one of two states |φ±iBR = 1 (|0i |0i ± |1i |1i ), and by a local √2 B R B R Distributed quantum compression: for a sin- operation, Bob can always transform the 1 gle party, a source emitting states with density ′ ′ state to |ψiA BR = √2 (|0iA |0iB|0iR + matrix ρA can be compressed at a rate given |1i ′ |1i |1i ) with A being an ancilla at A B R ′ by the entropy S(A) of the source by perform- Bob’s site. Alice has thus managed to send ing quantum data compression3. Let us now her state to Bob, while fully preserving consider the distributed scenario – we imagine their entanglement with R. that the source emits states with density matrix

+ 1 ρA1A2...Am , and distributes it over m parties. The 3. For the state |φ iAB = (|0iA|0iB + √2 parties wish to compress their shares as much |1i |1i ), S(A|B) = −1, and Alice and A B as possible so that the full state can be recon- Bob can keep this shared EPR pair to al- structed by a single decoder. Until now the low future transmission of quantum infor- general solution of this problem has appeared mation, while Bob creates the EPR pair 16 + intractable , but it becomes very simple once |φ i ′ locally. I.e. transferring a pure A B we allow classical side information for free, and state is trivial since the pure state is known use state merging. and can be created locally.

Remarkably, the parties can compress the Let us now make a couple of observations state at the total rate S(A1A2 ...Am) – the 3 about state merging. First, the amount of clas- Schumacher limit for collective compression – sical communication that is required is given by even though they must operate seperately. This

4 is analogous to the classical result, the Slepian- Wolf theorem2. We describe the quantum solution for two parties and depict the rate region in Figure 2.

RA Noiseless coding with side information: re- solely RA quantum lated to distributed compression is the case regime where only Alice’s state needs to arrive at the S(A) S(A) RA +RB =S(AB) decoder, while Bob can send part of his state S(A|B) to the decoder in order to help Alice lower her S(B|A) S(B) 0>S(B|A) RB R rate. The classical case of this problem was in- B troduced by Wyner17. For the quantum case, we RA RA demand that the full state ρAB be preserved in the protocol, but do not place any restriction on what part of Bob’s state may be at the decoder and what part can remain with him. For one-

way protocols, we find using state merging that RB RB if ρA and ρB are encoded at rates Ra and Rb re- Figure 2: The rate region for distributed com- spectively, then the decoder can recover ρA if pression by two parties with individual rates RA and only if R S A U and R E AU a ≥ ( | ) b ≥ p( : and RB. The total rate RAB is bounded by R) − S(A|U) with R being the purifying ref- S(AB). The top left diagram shows the rate erence system, U being a system with its state region of a source with positive conditional en- produced by some quantum channel on ρB, and tropies; the top right and bottom left diagrams Ep(AU : R) ≡ minΛ S(AΛ(U)) being the en- show the purely quantum case of sources where tanglement of purification18. The mimimum is S(B|A) < 0 or S(A|B) < 0. It is even possible taken over all channels Λ acting on U. that both S(B|A) and S(A|B) are negative, as shown in the bottom right diagram, but observe that the rate-sum S(AB) has to be positive. If Quantum multiple access channel: in addi- one party compresses at a rate S(B), then the tion to the central questions of information the- other party can over-compress at a rate S(A|B), ory we asked earlier, how much is there to by merging her state with the state which will know, and how great is our ignorance, informa- end up with the decoder. Time-sharing gives the tion theory also concerns itself with communi- full rate region, since the bounds evidently can- cation rates. In the quantum world, the rate at not be improved. Analogously, for m parties Ai, which quantum information can be sent down a and all subsets T ⊆ {1, 2,...,m} holding a com- noisy channel is related to the coherent infor- bined state with entropy S(T ), the rate sums ¯ mation I(AiB) which was previously defined RT = Pi T RAi have to obey RT >S(T |T ) with as19 max{S(B) − S(AB), 0}. This quantity T¯ = {1, 2∈,...,m} \ T the complement of set T . is the quantum counterpart of Shannon’s mutual information; when maximized over input states, it gives the rate at which quantum information can be sent from Alice to Bob via a noisy quantum channel10–12. As with the classi-

5 cal conditional entropy, Shannon’s classical mu- help her partner achieve the highest rate. The tual information1 is always positive, and indeed protocol is for one of the parties to merge their it makes no sense to have classical channels with state with the state held by the decoder. The ex- negative capacity. However, the relationship be- pressions in (2) are in formal analogy with the tween the coherent information and the quantum classical multiple access channel. channel capacity contained a puzzle. As the latter was thought to be meaningful as a positive Entanglement of assistance (localizable en- quantity, the former was defined as the maxi- tanglement): consider Alice, Bob and m − 2 mum of 0 and S(B) − S(AB) since it could be other parties sharing (many copies of) a pure negative. quantum state. The entanglement of assistance 21 EA is defined as the maximum entanglement We will see that negative values do make that the other parties can create between Al- sense, and thus propose that I(AiB) should not ice and Bob by local measurements and clas- be defined as above, but rather as I(AiB) = sical communication. For many parties, this is −S(A|B). It turns out that negative capacity, often referred to as localizable entanglement22, impossible in classical information theory has although here we work in the regime of many its interpretation in a situation with two senders. copies of the shared state. This problem was re- cently solved for up to four parties23, and can be We imagine that Alice and Bob wish to send generalized to an arbitrary number m of parties independent quantum states to a single decoder using state merging (using universal codes de- Charlie via a noisy channel which acts on both pending only on the density matrix of the helper, inputs. This problem is considered by Yard et as described in Figure 1). We find that the max- al.20. Our approach using state merging pro- imal amount of entanglement that can be dis- vides a solution also when either of the channel tilled between Alice and Bob, with the help of capacities are negative, and gives the following the other parties, is given by the minimum en- better achievable rates: tanglement across any bipartite cut of the system which separates Alice from Bob: RA ≤ I(AiCB) ,

RB ≤ I(BiCA) , EA = min{S(AT ),S(BT )} (3) T RA + RB ≤ I(ABiC) , (2) where the minimum is taken over all possible where RA and RB are the rates of Alice and Bob partitions of the other parties into groups T and for sending quantum states. Here, we use our re- its complement T = {1,...,m − 2}\ T . To deﬁnition of the coherent information, in that we achieve this, each party in turn merges their state allow it to be negative. In achieving these rates, with the remaining parties, preserving the mini- one party can send (or invest) I(AiC) quantum mum cut entanglement. bits to merge her state with thedecoder. The second party then already has Alice’s state at the decoder, and can send at the higher rate I(BiAC). We have described a fundamental quan- This provides an interpretation of negative chan- tum information primitive, state merging, and nel capacities: if the channel of one party has demonstrated some of its many applications. negative coherent information, this means that There are also conceptual implications. For she has to invest this amount of entanglement to example, the celebrated strong subadditivity

6 of quantum entropy24, S(A|BC) ≤ S(A|B), Trans. Inf. Theory 19, 461–480 (1971). receives a clear interpretation and transparent proof: having more prior information makes 3. Schumacher, B. Quantum coding. Phys. state merging cheaper. Our results also shed new Rev. A 51, 2738–2747 (1995). light on the foundations of quantum mechanics: 4. Wehrl, A. General properties of entropy. it has long been known that there are no condi- Rev. Mod. Phys. 50, 221–260 (1978). tional probabilities, so defining conditional entropy is problematic. Just replacing classical 5. Horodecki, R. & Horodecki, P. Quantum entropy with quantum entropy gives a quan- redundancies and local realism. Phys. Lett. tity which can be positive or negative. Quite A 194, 147–152 (1994). paradoxically, only the negative part was under- stood operationally, as quantum channel capac- 6. Cerf, N. J. & Adami, C. Negative en- ity, which, if anything, made the problem even tropy and information in quantum mechan- more obscure. State merging ”annihilates” these ics. Phys. Rev. Lett 79, 5194–5197 (1997). problems with each other. It turns out that the 7. Einstein, A., Podolsky, B. & Rosen, N. Can puzzling form of quantum capacity as a condi- quantum-mechanical description of physi- tional entropy is just the flip-side of our interpre- cal reality be considered complete? Phys. tation of quantum conditional entropy as partial Rev. 47, 777–780 (1935). quantum information, which makes equal sense in the positive and negative regime. The key 8. Bennett, C. H. et al. Teleporting an un- point is to realize that in the negative regime, known quantum state via dual classical and one can gain entanglement and transfer Alice’s Einstein-Podolsky-Rosen channels. Phys. partial state, while in the positive regime, only Rev. Lett 70, 1895–1899 (1983). the partial state is transfered. 9. Schrödinger, E. Die gegenwärtige Sit- On a last note, we wish to point out that re- uation der Quantenmechanik. Naturwis- markably and despite the formal analogy, the senschaften 23, 807–821; 823–828; 844– classical scenario does not occur as a classi- 849 (1935). cal limit of the quantum scenario – we consider 10.Shor, P. The quantum chan- both classical and quantum communication, and nel capacity and coherent infor- there is no meaning to preserving entanglement mation. Lecture notes online at in the classical case. We have only just begun to www.msri.org/publications/ln/ grasp the full implications of state merging and msri/2002/quantumcrypto/shor/1/ negative partial information; a longer technical (2002). Talk at MSRI Workshop on Quan- 25 account , with rigorous proofs and further ap- tum Computation, Berkeley 2002. plications is in preparation. 11. Lloyd, S. The capacity of the noisy quan- 1. Shannon, C. E. A mathematical theory of tum channel. Phys. Rev. A 55, 1613–1622 communication. Bell Syst. Tech. J. 27, 379– (1997). 423; 623–656 (1948). 12. Devetak, I. The private classical capacity 2. Slepian, D. & Wolf, J. K. Noiseless cod- and quantum capacity of a quantum channel ing of correlated information sources. IEEE (2003). Eprint, quant-ph/0304127.

7 13. Devetak, I. & Winter, A. Relating quantum 22. Verstraete, F., Popp, M. & Cirac, J. I. Entan- privacy and quantum coherence: an opera- glement versus correlations in spin systems. tional approach. Phys. Rev. Lett 93, 080501 Phys. Rev. Lett 92, 027901 (2004). (2004). 23. Smolin, J. A., Verstraete, F. & Win- 14. Groisman, B., Popescu, S. & Winter, A. ter, A. Entanglement of assistance On the quantum, classical and total amount and multi-partite state distillation (2005). of correlations in a quantum state (2004). quant-ph/0505038. Eprint, quant-ph/0410091. 24. Lieb, E. H. & Ruskai, M. B. Proof 15. Horodecki, M. et al. Local versus non-local of the strong subadditivity of quantum- information in quantum information theory: mechanical entropy. with an appendix by formalism and phenomena (2004). Eprint, B. Simon. J. Math. Phys. 14, 1938–1941 quant-ph/0410090. (1973).

16. Ahn, C., Doherty, A., Hayden, P. & Win- 25. Horodecki, M., Oppenheim, J. & Winter, A. ter, A. On the distributed compression Quantum state merging (2005). In prepara- of quantum information (2004). Eprint, tion. quant-ph/0403042. 17. Wyner, A. On source coding with side information at the decoder. IEEE Trans. Inf. Acknowledgements This work was done at the Isaac Newton Institute (Cambridge) August- Theory 21, 294–300 (1975). December 2004 and we are grateful for the institute’s 18. Terhal, B. M., Horodecki, M., DiVin- hospitality. We thank Bill Unruh for his helpful cenzo, D. P. & Leung, D. The entan- comments on an earlier draft of this paper, and glement of purification. Journal of Math- Roberta Rodriquez for drawing Figure 1. We ematical Physics 43, 4286–4298 (2002). acknowledge the support of EC grants RESQ, No. IST-2001-37559, QUPRODIS, No. IST-2001- quant-ph/0202044. 38877, PROSECCO (IST-2001-39227), MH was 19. Schumacher, B. & Nielsen, M. A. Quantum additionally supported by the Polish Ministry of data processing and error correction. Phys. Scientific Research and Information Technology Rev. A 54, 2629–2635 (1996). under grant No. PBZ-MIN-008/P03/2003, JO was additionally supported by the Cambridge-MIT Insti- 20. Yard, J., Devetak, I. & Hayden, P. Capac- tute and the Newton Trust, and AW was additionally ity theorems for quantum multiple access supported by the U.K. Engineering and Physical channels — Part I: classical-quantum and Sciences Research Council. quantum-quantum capacity regions (2005). Eprint, quant-ph/0501045. Competing Interests The authors declare that they have no competing financial interests. 21. DiVincenzo, D. P. et al. Entanglement of Assistance. In Williams, C. P. (ed.) Proc. 1st Correspondence Correspondence and requests NASA International Conference on Quan- for materials should be addressed to Jonathan tum Computing and Quantum Communi- Oppenheim. (Email: j.oppenheim (at) cation, LNCS 1509, 247–257 (Springer, damtp.cam.ac.uk) 1999).