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Quantum, Classical, and Total Amount of Correlations in a Quantum State

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Quantum, Classical, and Total Amount of Correlations in a Quantum State PHYSICAL REVIEW A 72, 032317 ͑2005͒ Quantum, classical, and total amount of correlations in a quantum state Berry Groisman,1,* Sandu Popescu,1,2,† and Andreas Winter3,‡ 1H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, United Kingdom 2Hewlett-Packard Laboratories, Stoke Gifford, Bristol BS12 6QZ, United Kingdom 3Department of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom ͑Received 1 February 2005; published 13 September 2005͒ We give an operational definition of the quantum, classical, and total amounts of correlations in a bipartite quantum state. We argue that these quantities can be defined via the amount of work ͑noise͒ that is required to erase ͑destroy͒ the correlations: for the total correlation, we have to erase completely, for the quantum corre- lation we have to erase until a separable state is obtained, and the classical correlation is the maximal corre- lation left after erasing the quantum correlations. In particular, we show that the total amount of correlations is equal to the quantum mutual information, thus providing it with a direct operational interpretation. As a by-product, we obtain a direct, operational, and elementary proof of strong subadditivity of quantum entropy. DOI: 10.1103/PhysRevA.72.032317 PACS number͑s͒: 03.67.Mn, 03.65.Ud, 03.65.Yz I. INTRODUCTION 1 1 ␳ = ͉⌽+͗͘⌽+͉ + ͉⌽−͗͘⌽−͉, 2 2 Landauer ͓1͔, in analyzing the physical nature of ͑classi- cal͒ information, showed that the amount of information where stored, say, in a computer’s memory, is proportional to the 1 work required to erase the memory ͑reset to zero all the bits͒. ͉⌽−͘ ͉͑ ͘ ͉ ͘ ͉ ͘ ͉ ͘ ͒ = ͱ 0 A 0 B − 1 A 1 B . These ideas were further developed by other researchers 2 ͑most prominently Bennett͒ into a deep connection of clas- This mixed state is disentangled because it is identical sical information and thermodynamics ͑see ͓2͔ for a recent with a mixture of two direct product states survey͒. Here we follow Landauer’s idea in analyzing quan- tum information: we want to measure correlation by the 1 1 ͑ ͒ ͑ ͒ ␳ = ͉0͗͘0͉ ͉0͗͘0͉ + ͉1͗͘1͉ ͉1͗͘1͉ . thermodynamical effort required to erase destroy it. 2 A B 2 A B The main idea of our paper can be understood on a simple example. Consider a maximally entangled state of two qubits But although the entanglement is now gone, Alice’s and ͑equivalent to a singlet͒ Bob’s qubits are still correlated. Indeed, ␳ contains now 1 bit of purely classical correlations; furthermore, these correla- tions are secret since, given the procedure by which they 1 were obtained, they are not correlated with any third party, ͉⌽+͘ ͉͑ ͘ ͉ ͘ ͉ ͘ ͉ ͘ ͒ ͑ ͒ = 0 A 0 B + 1 A 1 B . 1 ͱ2 such as an eavesdropper. To also erase these classical correlations Alice has to Usually this state is seen as containing 1 ebit, i.e., one bit “work” more. She can do this by randomly applying a “bit ␳ ␴ of entanglement, based on the asymptotic theory of pure- flip” to , that is, applying either 1 or x at random, with state entanglement ͓3͔. The temptation is to think that it con- equal probability . This brings the state to tains 1 bit of correlation, and that this correlation is in pure 1 1 quantum form ͑which can be used either in a quantum way— ␳ Ј = 1A 1B, e.g., for teleportation—or to obtain one perfectly correlated 2 2 classical bit͒. where qubit A is completely independent of qubit B. We will argue however that this state contains in fact 2 To summarize, two bits of erasure ͑or, depending on the bits of correlation—1 bit of entanglement and 1 bit of ͑se- point of view, “bits of noise,” or “error”͒, are required to cret͒ classical correlations, as follows. completely erase the correlations in the singlet. The first bit Suppose that Alice wants to erase the entanglement be- erases the entanglement and the second erases the classical tween her bit and Bob’s. She can do this by applying 1 bit of secret correlations that are left after the entanglement is randomness: she applies to her qubit one of two unitary gone. We then say that the singlet contains 1 bit of pure transformations or ␴ with equal probability. By this the 1 z entanglement, and 1 bit of secret classical correlation. The pure state in Eq. ͑1͒ becomes a mixture total amount of correlation is 2 bits. We emphasize, however, that this total amount only makes sense in the above opera- tional description; obviously, Alice and Bob cannot make use *Electronic address: [email protected] of both bits of correlation simultaneously. †Electronic address: [email protected] A couple of remarks concerning the connection to Land- ‡Electronic address: [email protected] auer’s theory of information erasure. Just as Landauer did for 1050-2947/2005/72͑3͒/032317͑11͒/$23.00032317-1 ©2005 The American Physical Society GROISMAN, POPESCU, AND WINTER PHYSICAL REVIEW A 72, 032317 ͑2005͒ information ͑entropy͒, our approach quantifies correlations II. TOTAL BIPARTITE CORRELATIONS via their robustness against destruction. However, there As explained in the Introduction, we want to add ran- seems to be a contradiction: whereas Landauer considers re- domness to a state ␳=␳ of a bipartite system AB ͑with setting the memory to a standard state ͑and we take for AB local Hilbert space dimensions d ,d Ͻϱ͒ in such a way as granted that one can generalize his argument to quantum A B to make it into a product state. In fact, we shall consider memory͒, effectively exporting—“dissipating”—the entropy n→ϱ many copies of ␳, and be content with achieving of the system, we inject entropy into it. This is actually only decorrelation ͑product state͒ approximately ͑but arbitrarily an apparent contradiction, as can be seen easily once we well in the asymptotic limit͒. realize that in the above example we tacitly assumed that In detail, the randomization will be engineered by an en- Alice forgets which Pauli operator she has applied. Indeed, semble of local unitaries ͕p ,U V ͖N , to which is associ- we can present what she does in more detail as follows. She i i i i=1 has a reservoir of random bits, which she uses to apply one ated the randomizing map of the Pauli operators as above in a reversible way ͑by a N ͒ ͑ †͒ ␶͑͒ ͑ ۋ ␶ ͒ quantum-controlled unitary . This step does not affect the R: ͚ pi Ui Vi Ui Vi . 2 correlations between A and B. Only when she decides to i=1 erase ͑forget͒ the random bits, then the correlations are af- We call the class of such completely positive and trace pre- fected, as we have shown above. Now it is evident that the serving ͑CPTP͒ maps on AB “coordinated local unitary ran- entropy pumped into the state is equal to the Landauer era- domizing” ͑COLUR͒ maps. Considering that our object is to sure cost of the random bits. study the correlation between A and B, it may seem a bit In this paper we develop these ideas, as follows. suspicious to allow coordinated application of U and V at For an arbitrary bipartite quantum state ␳ the quantum i i AB the two sites. Hence we define A-LUR to be those maps mutual information is defined as where all Vi =1, and B-LUR those where all Ui =1—because they can be implemented by application of noise strictly lo- A B ͑ ͒ ͑␳ ͒ ͑␳ ͒ ͑␳ ͒ cally at or alone, respectively. The combination of an I A:B = S A + S B − S AB . A-LUR with a B-LUR map ͑i.e., independent local noise at either side͒ we call simply “local unitary randomizing” ͑LUR͒. ͑ ͓ ͔ The name is taken from Cerf and Adami 4 , but Stratonov- We say that R ⑀-decorrelates a state ␳ if there is a product ich ͓5͔ considered this quantity already in the mid-1960s.͒ ␻ ␻ state A B such that While this definition is formally very simple, an opera- ͓ ͔͑ ʈ ͑␳͒ ␻ ␻ ʈ ഛ ⑀ ͑ ͒ tional interpretation for it was hitherto missing 6 at least R − A B 1 , 3 not for the quantity itself with given state; however, it plays ʈ ʈ a crucial role in the formula for the entanglement-assisted where · 1 is the trace norm of an operator, i.e., the sum of capacity of a quantum channel ͓7͔͒. We show here that the the absolute values of the eigenvalues. For technical reasons, when we study the asymptotics of such transformations ͑i.e., total amount of correlation, as measured by the minimal rate ␳͒ of randomness that is required to completely erase all the acting on n copies of the state , we will demand that the ␳ ͑ ͒ output of the map R ͑and similar maps studied below͒ is correlations in AB in a many-copy scenario , is equal to the n quantum mutual information. This is the main result of Sec. supported on a space of dimension d , for all n, with some II. As an important consequence of this result we shall dem- finite d. onstrate that it leads to the strong subadditivity of von Neu- How shall we account for the amount of noise intro- mann entropy. duced? From the point of view of the ensemble of unitaries, the most conservative option will be to take log N ͑all loga- In our above example this amount of total correlation di- ͒ vides neatly into the amount required to obliterate the quan- rithms in this paper are taken to base 2 , the space required to ͑ ͒ identify the element i uniquely.
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